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Local and global Markoff fields
Vol. 19 (1984)
REPORTS
ON MATHEMATICAL
No. 2
PHYSICS
LOCAL AND GLOBAL MARKOFF FIELDS S. ALBEVERIOand R. H@EGH-KROHN Fakultlt fur Mathematik, Universitdt Bielefeld, F.R.G. Mathematisches Institut, Ruhr-Universitat Bochum, F.R.G. CNRS - CPT, Marseille, Universite d’Aix-Marseille II, UER de Luminy, France Matematisk Institutt, Universitetet i Oslo, Norway (Received February 12, 1980)
We discuss the relation between local and global Markoff property for processes and (generalized) random fields, both in the case of discrete and continuous “time parameters”. In particular, we give a short account of our proof of the global Markoff property for the Euclidean Sine-Gordon fields in two dimensions. We also give an account of our proof of the uniqueness of the solutions of the DLR equations in this case (extremality of the Gibbs state, implying in particular uniqueness of the phase). We also discuss more generally the relation between the uniqueness and global Markoff properties, and discuss some consequences of the global Markoff property. Finally, we indicate shortly how the global Markoff property is proven (by ourselves together with G. Olsen) in the case of lattice systems.*
1. Markoff processes and Markoff fields In this paper we shall discuss problems connected with the Markoff property of Euclidean (and lattice) random fields. Random fields are an extension to the case of “higher dimensional time” of stochastic processes, so let us start with the latter. Among stochastic processes the class of Markoff processes is a very important one (it contains e.g. the processes describing Brownian motion and more generally it contains the diffusion processes) [l]. Of particular interest are processes and fields which are homogeneous (in the sense that the corresponding measures are invariant with respect to the Euclidean group in Rd). In the case of processes this means that they are given by a Markoff semigroup P, (such that 0
*Modified version of Lectures given at the Winter School in Theoretical 1978 (Dir. W. Karwowski).
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Physics, Karpacz, February
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S. ALBEVERIO
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but can also be e.g. a Polish space or, more generally, a Suslin space of distributions like S’(R”) or D’(R”)). The requirement of invariance of the process under the time reversal implies that P, is a symmetric semigroup on L’(dp), i.e. Pf = P,, * meaning adjoint. We also assume the process is conservative i.e. P, 1 = 1. Then we have a stationary process with invariant measure ,u (one has J(P,~~)(x)&fx) = s(P, l)(x)~,(x)@x) = &I)). Observe that since P, maps 1 into 1 it also maps L” into L” and by interpolation we have that P, is a Markoff semigroup on all L’(dp),l < p < co. Let t--t 5, be the corresponding process (which can be set up according to Kolmogorov’s construction, e.g. by identifying the algebra L”(S) with the continuous functions on its spectrum). Then the Markoff property in the sense of the theory of stochastic processes can be expressed e.g. by E(f+f-
I&) = Edf+ IB,)ECf-
I&),
(1.1)
where E(.(B,) means conditional expectation with respect to the o-algebra generated by the time zero variable to(.) and f+, f_ are bounded functions measurable with respect to B, resp. B_, where B, are the a-algebras generated by the variables &, I E( co,O) resp. t E (0, co). In short, “the future is independent of the past given the present” (if instead of E(~IB,) we had (1.1) for E(e) we would have the complete independence of future from past. Thus the Markoff property is an expression of conditional independence). Equivalently (see e.g. [l] a)) EU-+ I&d =
EU-+IB- VW,
(1.2)
where B- VB, is the a-algebra generated by B_ and B,. The Markoff property is actually equivalent with having a Markoff semigroup P,, i.e. a semigroup s.t. 0
LOCAL AND MARKOFF
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227
quantum fields, can also be useful constructive tools (see [8]). Processes with the Osterwalder-Schrader positivity condition ([9]) have in general only weaker “markoflian properties” (this is analyzed completely in the case d = 1 [lo], [47]; note also that generalized free fields are in the class but are not MarkofT). Let us now mention a property which is weaker than the Markoff property as defined above but in another sense than “higher order Markoff’ or “OsterwalderSchrader positive (for processes)“. It is the property ECflB,_.,,,) = ECflB_,V B,) for anyf’bounded measurable with respect to the o-algebra B, _ , ,u,~ B,,, x,j. In analogy with the case of generalized random fields ([ 111, [ 121) this property has been called “local Markoff property” ([ 131 whereas (1.1) itself is then called the “global Markoff property”. Note that the corresponding property to (1.2) E cfl B,,,,,) = ECfl B,v Bb) for anyfB, _ ,u.bl-measurable is what is often understood as Markoff property of a process (not necessarily stationary) indexed by I c [O,~Z). The corresponding property for random fields (with [a.b] replaced by the interior of a smooth closed bounded curve and I replaced by some subset of Rd) is also what is often called in the theory of random fields (Gibbsian fields) (see e.g. [ 141) a “Markoff property”. The global Markoff property (1 .l) is actually the most suitable concept for homogewous processes and fields, but is also difficult to verify; see below. It is however only the global Markoff property that permits to associate to Euclidean random field Markoff semigroups yielding also a complete description of the associated quantum fields by a Schrddinger (canonical, fixed time) representation. Since we are making a sort of a “tour d’horizon” concerning Markoff-type properties, let us mention the “strong Markoff property” [ 151: in the case of homogeneous processes this amounts to (1 .l) with B, replaced by B,. B, being a random time. The above mentioned diffusion processes (associated with Dirichlet diffusion forms) have realizations which have the strong Markoff property (are in fact even Hunt processes [5] .b), [5] c)). For homogeneous random fields a complete extension of this concept has not yet been elaborated, however a partial one is discussed in [5] b), [5] c), [5] d). Let us now discuss in more detail some of the extensions of the concepts discussed above for processes to the case of random fields. These are defined analogously as processes but with the time parameter (which runs over some interval of R) replaced by a point running in some subset of Rd (or Zd, in the discrete case, corresponding to random walk on Z in the one dimensional case). If s(x) (.) for every x E Rd is a random variable on some probability _ space (0,~) with values in a state space S we call x -+ <(X)(C)) a random field with values in S [7], [ 161, [ 171. Then E(x)(u)) is, for fixed U)E Q, an S-valued function of .x. In the case where S = R one speaks of ordinary random field. If S is a distribution space, so that ;(X)(W) is a generalized function in x, for p-a.e.o, then we speak of x -+ c(x)(.) as a generalized random field [17]. In this case (t,(p) (.) = J<(x)(.)cp(x)dx, for cp in a test
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S. ALBEVERIO and R. HOEGH-KROHN
function space, e.g. CF(ZP’), and where the integral is understood in the sense of distributions, is an ordinary random variable. For applications in quantum field theory and statistical mechanics one is interested particularly in homogeneous (generalized) random fields, i.e. random fields which are invariant under the Euclidean group E(Rd) in Rd i.e. such that the corresponding measure ~1is invariant under E(Rd). Let us first recall such homogeneous generalized random fields in the Gaussian case. It is well known that ford = 1 the only Gaussian field which is Euclidean invariant and has in addition the Markoff property and continuous paths (stationary symmetric diffusion process) is the Ornstein-Uhlenbeck velocity process t --+&, i.e. the Gaussian (with measure p) process with mean zero and covariance
(-$+m’)-‘,m>O(see
e.g. [l] b). In this case Q can by taken to be C(R) and one has E,cf(<,)) = (P,f)(x) for any fEL2(R,dpo), with E, the conditional expectation given &, =x and with P, = e-‘r’Hmt~ R,H,,,, being the diffusion operator defined as the self-adjoint positive operator uniquely associated with the closed diffusion Dirichlet form g --f +JI Vg(‘dpo, V being the closed gradient operator (closure of the gradient as defined on C,?,(R)) and p,, is the Gaussian measure obtained by restricting p to the a-algebra generated by to (this is the Gauss measure with mean 0 and covariance 1/2m (see [l]b). For d > 2 the corresponding objects are the homogeneous generalized Gaussian random fields with mean zero and covariance (- d + m2)-l in L2(Rd,dx), with A the Laplacian in L2(Rd,dx), m > 0 for d > 3, m > 0 for d 6 2. The probability space of such fields can be taken to be (S’(Rd),dpo) where p” is the Gaussian measure with support within S’(Rd), and is the normal distribution with covariance the scalar product in the real Hilbert space H_ ,(Rd) (i.e. the closure of C;(Rd)) in the norm given by the scalar product given by jq(x)G(x - y)cp(y)dxdy, with G(x-y)=(-A +m2)-’ (x-y)). This measure and the corresponding generalized random field are called the free Markoff lield. Their relevance to quantum fields became clear particularly through the work of Symanzik, Nelson and Guerra connected with the “Markoff property”.’ Let us describe this property. We first remark that the free Markoff field cp --f (5,~)(.), cpES(R~),
‘This property is actually due to the fact that the covariance is the inverse of an elliptic differential operator of second order. Markofian properties of higher order elliptic operators and more generally of pseudodifferential operators are studied e.g. in [7], [8].
LOCAL AND MARKOFF
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229
respect to PO. Define, for any Bore1 subset A c Rd, B(A) as the o-subalgebra of the pmeasurable subsets generated by the linear functions (<,p)(.) with suppp c A. Let now C be a (d - 1)“dimensional piecewise C’ hypersurface which divides Rd - C into two disjoint components Q, and sZ_. Let f+ ,f- be bounded B(S2, ) resp. B&-)-measurable functions. Then we have ECf+f- IC) = Ecf+ IC)Ecf-
IC)>
(2.1)
where E(.JC) is the conditional expectation with respect to the measure p” and the oalgebra B(C). In the present case, where ,u’ is the measure of the free Markoff field, (2.1) holds for all C, even C which extend to infinity, like e.g. hyperplanes ((x~,...,x~)E Rd(x, = O}. We say, because of this, that the free Markoff field has the global Markoff property. Consider now more generally a generalized random field t(x) with corresponding probability space (S’(Rd),p) and suppose 5 has a regularity property in thesensethatE(J(Sr,cp)l2)~C(U~S~(x)G(x-y)cp(y)dxdyforanycpED(U),cp3Oany bounded open set U, with C(U) a constant independent of cp and where E means expectations with respect to p. In this case we can define for any Bore1 subset A of Rd, B(A) in a way corresponding to the one we did before, with ,u’ replaced by p. Then if (2.1) holds for all (d - 1)-dimensional piecewise C’ hypersurface dividing Rd-C into two disjoint components Q+ and Q_ we say that p (and 5) have the global Markof property. Remark. In the case d = 1 this property corresponds to the usual Markoff property for processes (l-dimensional time parameter), in an obvious sense. Moreover the global Markoff property implies the Markoff property with respect to half planes [ 181, [ 191, [20], which is known, see below, to yield a Markoff process in the orthogonal direction. In the case where (2.1) holds only for all (d - 1)-dimensional piecewise C’, bounded hypersurfaces dividing Rd-C into two disjoint components 92, and Q_ we say that p (and 5) have the local Markoff property. Remark. In the case d = 1 the interpretation of local Markoff property, formulated above, is not quite unique. If we take C reduced to a point then the local Markoff property coincides with the global one. However, looking at the statement of the local Markoffproperty for arbitrary d taking C to be spheres and letting Q,, Q_ be the outer resp. inner of these spheres, we find a less trivial l-dimensional analogue in C: the extremes a, h of an interval [a, b] and Sz,_ = ( - x ,a) u (b, x ), 92_ = (a, b). This is the property involving B,, B, discussed above. The local Markoff property understood this way, is then in general not the same as the global one, see [21].”
*For a relation between Markoff property and Mackey’s concept of imprimitivity discussions of concepts related to the Markoff property see also e.g. [45]b), c).
see [45]a). For other
230
S. ALBEVERIO
and R. H@EGH-KROHN
Let us illustrate the local versus global Markoff property schematically ford = 2 as the conditional independence of observations within Sz, and Sz- given all the observations on C in the two typical situations
(global Markoff)
(local Markoff)
Remark. The concepts of local and global Markoff properties of Euclidean tields were discussed originally by Nelson and Newman [ 181, [ 121. Markoffian properties of other types, like the n-Markoff property (corresponding to conditional independence when not only the values of j; on C are given but also the derivatives up to order n - 1: a concept which applies to ordinary (not generalized) random fields [7], [S] or the GMarkoff property (in which 5 in a whole neighborhood of C has to be prescribed to get conditional independence) have been discussed in the mathematical literature (e.g. [22]). These other Markoffian random fields do not, however, yield directly relativistic local quantum fields (see however the recent work [S] for a very useful applications of nMarkoflian fields). An intermediate notion of Markoff property between the local and the global one is that of Markoff property with respect to hyperplanes ((2.1) requested to hold for all hyperplanes C). This is actually the Markoff property which is sufficient for yielding relativistic lields from Euclidean ones, as first remarked by Nelson [ 1 l] [ 181. Later Osterwalder and Schrader [9] realized that a weaker positivity condition (which holds e.g. for generalized random fields which are weak limits of generalized random fields t,(x) which are symmetric Markoff processes x1 --, &(xl,xZ,...,xd)) is also in fact sufficient for yielding relativistic fields from Euclidean ones. For related works see also e.g. [23]. Let us now come back to the case of the free Markoff field measure ,n” and the corresponding generalized random field t(x). If we write x = (x1 ,...,x,), x1 E R and look at the process xi -+ {(x1,x 2r...,~d) from xi E R with values in S’(Rdpl), we have (see e.g. [19]) that 5(x1,x2 ,..., xd) -, 5(x1 + s,x 2,...,xd is represented in L2(dpo) by a unitary map U, and if we call E, the conditional expectation with respect to the hyperplane C = {(x1 =0,x, ,..., x,)>,we have
E, USE, = eCsH,
s 3 0,
where eesH is a Markoff semigroup in EoL2(dpo) ~L’(dpz), p” to B(C). Consider now [S] the Dirichlet form
p: being the restriction
of
LOCAL AND MARKOFF
FIELDS
231
first defined in a natural way on cylinder functions
with &C, (Rk,R). This positive quadratic form in L2(d&) is known to be closable ([5]a), b) and its closure, which is ij[ Vf12d& is uniquely associated with the operator H in the sense that (H1’2f;H”2f)
= +JIVjJ’d&
for all f’~D(2?“‘)
= D(V).
As proven in [5] a), [5] b), the physical Hilbert space (Fock space) associated with the free Markoff field is identifiable with EoL2(dpo) rL2(d&) and the physical Hamiltonian is identifiable with H. The time evolution of the relativistic field is given by the one parameter group generated by H. Moreover, the Lorentz-boost is represented by the Dirichlet form
This is the canonical picture of the physics of quantum fields in the interaction free case. Note that much more is known about the process 5, associated with eesH, e.g. it can be realized as a Hunt process [5] b), c) (see also e.g. [5], [24] for other results). Is it possible to have something like this for Euclidean measures different from the Gaussian one? In the cased = 2 the existence of Euclidean measures on S’(R2) is known in several cases, see e.g. [25], [26], [27]. S UCh measures are constructed as follows. One first constructs a family of measures pL, on S’(R’), attached to the family of open bounded regions /1 in R2. The p,, are all absolutely continuous with respect to the free Euclidean field measure 1~’and dp,,/dp’ is of the form dp,,/dpO = e- “,d/je- “.fdp’, with U,, is measurable with respect to the o-algebra B(n) U,, EL’(dp’), e- “4 EL’(d,u’), generated by the fields with support in /1, additive (i.e. U,, 1v,,2 = U,, 1 + U,,, whenever /1, n ,4, = p) and covariant (U,,(T[) = U,,(t) for all Euclidean transformations 7). One then shows that the weak limit p of p,, as /iT R2 (along the net of bounded open subsets of R2) exists (p is then invariant under the Euclidean group in R2, since the limit is taken along arbitrary open bounded sets). We can construct U, as follows. Let x,(x) be the function on Rd whose Fourier transform is the characteristic function Xl,(k) of the ball of radius K > 0 i.e. iK(k) = 1 for (kl < K, iK(k) = 0 for Ikl > K. Let t,(x) = (<*x,)(x), where * denotes convolution of the distribution YES’ with x,. Then t,(x) is a C”-function ofx, hence exp(a<,(x)) is a well defined continuous function of c( and x, for any K > 0 and YES’. Detine:exp(a<,(x)) :- exp(-$a’G,(O)) exp(a&(x)), with G,(O) the value at 0 of xK*G*xK, i.e.
232
S. ALBEVERIO
G,(O)=
and R. H@EGH-KROHN
(2n)Lpk)l’&5.
Then: exp(a&_(x)):EL2(d~o). Let P, be the orthogonal projection in the Hilbert space L*(dp’) onto the subspace generated by C&(X).Then P,,P, = P, for K’ > K i.e. P,, 3 P,. Since :exp(a~,(x)):~P,L*(d~~) and for the o-algebras B, generated by t,(x) we have B, 1 B,, it follows that {exp(a&(x)), K > 0} is a positive martingale with respect to {BJC 2 0). Also j :exp( g<,(x)): dx for any bounded open ,4 is a martingale in the same sense. As K + cc thi orthogonal martingale L*(dp’)
projections
convergence
to a function
P, converge strongly in L*(dp’) to the identity. The
theorem
in L*(dp’)
gives convergence
of U: = j :exp(cc<,(x)):dx
if and only if ~(~:exp(~&(x)):dx)*d~~
in
is uniformly
bounded in K, for fixed A This is satisfied if z2 < 47c since, using the definition of:exp(a<,(x)): and p”, we see that = s ~e’2G~~“-Y~dxdy+ J Jea2Gcx-y)dxdy, AA AA
J( j:exp(&(x)):dx)IdyU A and this integral = -&lnlx-yl
is finite since G(x + 0(1x-yl)
y) is continuous
as x-y-+0
for x - y # 0 and G(x - y)
(and c1*~47~). Let us call U;, with V(S)
E exp(as), the limit for K + co of J:exp(a{,(x)):d
x and write (in the usual suggestive
A
way, based on t,(x) -+ <(x ) as distributions,
as K+ cc) US; = J:exp(N<(x)):dx.
Vi EL*(d,u’) (a fortiori
Moreover
thus U> ELl(d,u’)).
Then
U 1 > G (as L*-limit of non-
negative functions), hence e -‘i E L”,(dp’). Clearly, U y1is additive (since the U > are additive by construction). CJZ is also B(A)-measurable [26], [18]-[20], (see below). Thus we have verified that { U .Y,,A bounded open) is an interaction in the above sense. It is called the “exponential interaction”, v being an exponential function (it was first studied by the authors in [26]; see also [35]f) for a recent investigation of the exponential interaction on Rd,d B 2). Other interactions can be constructed from this. be the projection of J:exp<(x):dx onto P’“‘L2(dpo) = Let j:[“:(x)dx E H(“~6H(“-‘~, where HCk) is the subspace of L2(d,po) generated
by the monomials
LOCAL AND MARKOFF
fi (r,cpi), i=l
qiES(R’), j = l,...,k, and constants.
over, it is B( /l)-measurable,
Clearly,
as limit of B( Il)-measurable
= L2(dpo) -
limS:exp(&,(x)):dx,
exp([,(x)):dx
and%“):exp<,(x):
Then
FIELDS
thus
233
j:<“:(x)dx A
is additive. More-
quantities (since j:expt(x):dx A
P’“) j:exp&(x): =
lim
S P?exp(<,(x)):dx A
is, for XE /1, &&measurable).
U);, with v(s) now defined as V(S)= 3, $ a,$,
a,,;l E R, defined by U f; =
k=O x Id x is also B(/i)-measurable, additive and in L2(dpo). From the type of A estimates originally due to Nelson [28], Glimm [29] and Guerra [30] one proves also =
i.kg?kJ:g’:(
that e-“~~~p(d~o) for all 1
+ j:exp(A
i(at(x)
+ 0):dx - iok
the
+ B)):dx} with
,~~(5)P”‘Bi!:i”j:(x)dx. ‘J
Such interactions were introduced in a regularized version by us in [31] and the regularization was removed by Frohlich in [32]. Again it follows easily that US; is additive, B(n)-measurable and in [32] it is shown that e - ‘: E L’ @PO), hence U: is an interaction, “the trigonometric interaction” (“Sine-Gordon interaction”). For a very interesting study of this interaction see also [33], [27]. Let now U, be any of the interactions U: described above. It is easy to show that the corresponding measure dp, = e-“~d,u”Ije -“Ad,u” has the global Markoff property. This was proved first (for the case of polynomial interactions) by Nelson [18]. The observation is, as formulated in [34], that due to the global Markoff property of p” and the additivity of U,, U, = Vi + U; + U:, where Uj are B(Q,)-measurable, UC, is B(C)-measurable, R, being the components of R2 into which R2 - C is divided by C, hence
E,,cf+f- ICI = QJ.f+ I’3EJ.L
IO
However p” is not Euclidean invariant, hence the Markoff generalized random field (4,~“) is not homogeneous. The interesting question is now whether one can use pu, to
S. ALBEVERIO
234
and R. H@EGH-KROHN
construct a homogeneous Markoff field. As mentioned above one can indeed show that there exist Euclidean invariant measures p on S’(R2) which are weak limits of pu, as Al R2. This is so in particular for u of one of the forms u(s) = Ilchasdv(a), u(s) = 2 ;
a,s j , a2n > 0, i/a,
suppv c (-2&,2&),
i~R[26],
small [25] a), L;(S)= Acos(as + Q), a2 < 2&,i
small
j=O
[27], (and suitable superpositions thereof). For further references concerning such (and related) Euclidean measures see e.g. [ 191. In particular, they are known to be non-Gaussian. Other Euclidean measures are obtained replacing p” 1B(A) in the definition of p,, by fi?no, where dpop,, is the Gaussian measure with covariance (- A?,, + m2)- ‘, where d;,, is the Laplacian with self-adjoint boundary conditions on 8 /1 (e.g. Dirichlet ones, see e.g. [36]). More generally, one can consider a probability measure p (Gibbs state for the interaction U,) on S’(R’) such that pIR(n,) coincides with ,u~ for any /b open in /i. The question of uniqueness of ,U (for given v, hence p,,) arises. A general uniqueness result of this kind is described by Dobrushin and Minlos in [37] for the case of polynomial interactions (with a sketch of proof). A complete proof for the case of trigonometric interactions is provided by us in [34]. Moreover, we show in this case that p also has the global Markoff property [34]. Let us now describe these results, starting from the one about uniqueness. 2. The proof of uniqueness of the Gibbs state and the global Markoff property for Euclidean fields with trigonometric interaction To prove uniqueness of the Gibbs state we first make the following elementary remarks. Let G(pn) be the set of Gibbs states (probability measures) on {pcl,,/1 bounded open} and call ALEG(p.,,) extreme if it has the property that VEG(pCln)and absolute continuity of v with respect to p implies p = v. Let B, = n B( A), A bounded AEB,
b-p@)
open,
2 = R2 - A_ Call
B,
trivial
for p if
:O,l.
1. We have: (B,
trivialfor
p)q(p
extreme).
Proof If p is not extreme then there exists a function h 2 0, hEL’(dp), h $1, such that hpEG(p,,). But by the definition of conditional expectations, E,(hl &fp I B(A) = h,u I B(A), but PLEG(~~), hence E,(hl A) = E,,(hl A), thus hp I B(n) = E,,(hl n)p I I{( A), thus h = E,,(hl
?i) ,u-a.e.,
i.e. h is B( &measurable,
hence h is B,( = n B( 2?))A
LOCAL AND MARKOFF
FIELDS
235
measurable. By assumption h + 1,~a.e., i.e. 3 E > 0, AEB, s.t. p(A) # O,l, h(r) 2 E, for
E,(W)
’ g conditional expectations, b em
with respect to B,.
Proof: Using EPA = E, on BAo, A0 c A (since ,UEG&J,
we see that E,,(.j ;i) is a monotone decreasing family of projections in L’(dp) (the family A being partially ordered by inclusion). Thus ENA(*I;i) converges strongly as operators in L2(dy) to a projection F. Then E (-1CD) d E_,(./ A), thus E,(.l “c) 6 F. On the other hand, for any gEFL2(dp) we have SEE,,” (.I A,,)L2(d,u) for all n sufficiently large thus g is B(a)measurable, thus B,-mea&able i.e. in the range of E,(.lco), which then implies F d E,,(.lco), so that with the former inequality in the other direction we get F = E,(*l a):q.e.d. 3. For arbitrary subalgebras E,(.lBo)
B, and measures
trivial for v.
= E,(.)==Bo
Proof: =c-: If v(A) # 0,l for some AEB,
v one has:
then E&)
# 0,l. But E,(x~(B,)
= xA
= OJ, thus x,., #E&J&, contrary to the assumption. =:v(A) = 0,l for all AEB, implies E,(xAIBo) = xA = E&J. By taking limits of sums of characteristic function this implies E,(.IB,) = E,(s). 4. E,A(.ln)JE,(*)oB,
trivialfor
p.
Proof: 3 : Immediate from 3. =: From 3 we have E,(.lx) = E,(.) 5. p extreme
= B,
and then, by 2,E,,,(.l;i)lE,(.).
trivial for p.
Proofi If B * is non-trivial for p then 3 A E B 1 , y(A) # 0,l. Thenx, is B , -measurable, xA $ 1 and xAp <
6. p extreme o E,(.l,$JE,(*). Proof: By 1, 4, 5. 7.
E,(~l~)lE,(9.
In order to prove this the following main observation is used. By the fact that p is a Gibbs measure we have Efl(fl A) = EPA (JI 2) for any B,,-measurable function, A, c A (“the local Markoff property” of p [ll], [ 121). Now observe that
~%,(fl~) = EpO(fe-“AI~)l~,de-“nl A). From the local Markoff property
of ~1’ we have
E,,(ff-““I-$
= Epo(fe-““IdA).
236
S. ALBEVERIO
and R. HGEGH-KROHN
We now use the following basic observation ([34]; related ideas are in [37] b), [S]) E,o(g18A)(p) = Ep;A(g + $dn), for p-lo-a.e. y and any gEL’(dp’), where
the integration being in the sense of generalized functions, where P,(x,dy) is the Poisson kernel for the Dirichlet problem (- A + m’)h(x) = 0 in Ah(x) = h,(x), h,~CF(ii A), for x~dA (so that h(x) = j Pc(x,dy)h,(y)). Using this we get PA
where for any g EL’ (dp’) we define g”“V (0 = g(
- UC;:,,)) we
= Ep,,,(8,“’ ).
Thus to show E,,,(*[ n)J E,(.) it is sufficient to prove that ~5:~ converges weakly as AtR2 for p-a.e. q andpVAconverges tofin the sense of distributions, for p - a.e. q, as A tR2. These results have been indeed proven by us [34] in the case where v is a trigonometric interaction with A2 < 27c and i sufficiently small. The proof of the weak convergence of &fV as At RZ is rather technical and involves an extension of the cluster expansion method of Frohlich and Seiler [27] (the reason being that py,, is the measure for the interaction UyV(<) EEU,(< + $v), which is not U, itself but its translate by $i;n) (Frohlich’s and Seiler’s method is based on the fundamental paper by Glimm, Jaffe and Spencer [25]a) on the cluster expansion for polynomial interactions). The proof thatp$A
for some subsequence A,,,t R2 such that exp( - m’d(A,,84,)) A,,? R2, a, c A,,, by using the inequality E,
s I$,““(x)12dx *0
< ue-‘n’df’@n)l~j >
--, 0 for some m’ > 0, as
LOCAL AND MARKOFF
FIELDS
237
for any & c /1, /lo n being open bounded regular a some constant, which is proven by using the regularity property of p together with potential theoretical methods, see [34] for details. Note that this basic ingredient, namely that tj”;’ -+ 0,is model independent, in the sense that all what is used is that one has a regular random field measure p. We have thus the following result [34]. THEOREM 2.1. Let tx2 < 271then there is a 1, > 0 depending on CI(and m, > 0) such thatfor 111 < 1, ifn~S’(R~) is such that $,8”(x) --+0 locally uniformly as AT R2 (and this
is the casefor p-a.e. n~s’(R~)) then d,u$ z e-“ncT)dpOan(~l~)/E(e-“~l8A)(~) converges weakly as AT R2, where ,tt is the Euclidean measure for the trigonometric interactions constructed in [27] (Gibbs measurefor the interaction given below). Here ,t.P”(<1n) is the free Euclidean field with condition 4 = n on 8A (such that jf(t)dpO”“([(q) = = jf(e + ti”,”)dpouoa”(5), dcl0~ being the free field measure with Dirichlet boundary condition on 8 A). Moreover,
there is one and only one regular measure p, regular in the
above sense, which is a Gibbs statefor
U, = Lj:cos(aS(x)
state (equivalently: the o-algebra at infinity*B, EJ.1 A) -+ E,(o) as nt R’).
+ 0):dx. ,u is an extreme Gibbs
- n Bx is trivial for p; equivalently:
Remark. This proof gives in particular the uniqueness of solutions of the DLR equations [38], [20]. Uniqueness results of this type in models of statistical mechanics have been proven particularly by Dobrushin, see e.g. [14] and [39]. Independence of particular boundary conditions, (obviously properly contained in our class) has been proven for Pi models in [36] b), c). We now turn to the proof of the global Markoff property for the Euclidean measure p. Again we restrict ourselves to the above case of trigonometric interactions. We want to show that
EJf+f-
IC) = E,cf,
IC)E,(f-
I C)
for arbitrary fk which are bounded and B,* -measurable and arbitrary (piecewise C ‘) curves C (not necessarily bounded), separating R2 into two regions A+ u C and /i_ u C with /1+ n A_ = Q. Using now that p is a Gibbs state (measure) we see that E,(f 1Av C) = Eqn(fl AvC) for any f which is BAO-measurable, no c /1/1 bounded open, hence using that ,u,, is global Markoff, so that E,,(f+ f- 1/iv C) = E,,(f+ ) AvC) E,“Cf_ I Av C) whenever fk are B, O-measurable, with AC = A, n A’, we see that it is enough to prove that EWn(fl AV C) converges as AT R2, for any f which is Bno-measurable for arbitrary open bounded A’, to E,(f 1C). Now one uses the fundamental formula E,,(f
I Av C)(v)= E,o(fe~uAIAv Ch)IEpo(_UAl Av C)(v) = Eflo(f ::,“A e_‘C,il””)/EWo(e-“%bd ) = E,,,9CU
aA(f,,c”an)
S. ALBEVERIO
238
and R. H@EGH-KROHN
(similar to the one derived before for Au C replaced by /1, and proven in the same way), with UcUaA >f f;UJ” , pC,“f” defined as the corresponding quantities defined before for the case% = $9,i.e. with t+QUaA replacing now I@. As for the proof the uniqueness it is thus enough to prove that ~5%~~ converges weakly as /It R2, for p - a.e. q (which we have proven [34] for the weak trigonometric interactions we are considering by the cluster expansion method for the interaction UzU,“” instead of LJaA n,rt ) and, on the other hand, that i@Jafx) --*$:(x) as /iT R2, locally uniformly in x for all q ES’(R’) (which is proven similarly as 1+9? 4 0 by potential theoretical methods, for any regular random field measure p). See [34] for details. Hence we arrive at the following result:
THEOREM
2.2. Let ~1~< 2n. Then there is a I, > 0 depending only on CI(and m, > 0)
such that, for IAl < 3L1, the unique Euclidean interactions
(of Theorem
Gibbs measure
2.1) has the global Markoff
u of the trigonometric
property
in the sense
that
E,(f+fIC) = E,(f+ IW,cf- I’2 f or any piecewise Cl-curve (which can extend to infinity) dividing R2 into two regions Q, UC and (;2_ UC with Sz, n& = $4 and uny integrable functions f+ which are B,,-measurable, BA, being the o-algebras generated by p -+ ([,p),p being any measure such that suppp c A, and Jdp(x)dp(y)G(x -y)
Let us now make some remarks concerning the proof of the global Markoff property. It might be tempting to think that once the uniqueness result (Theorem 2.1) has been proven the global Markoff property would follow immediately. This is however not so in general (what is true is that the proof of the global Markoff property is similar to that of the uniqueness). The difficulty consists in the following. The uniqueness theorem implies that the o-algebra at infinity B, is trivial for p. However this does not imply (unless the Bore1 o-algebra at infinity is countably generated, which is too strong an assumption (counterexamples to similar situations are known, e.g. [40])) that B, is trivial for the conditioned measure pc(t Iq) (which exists, incidentally, by the fact that S’(Rd) is a Suslin space [41]) for ,u - a.e.q. In fact for each A E B, one has p&4 1~) = 0 for all q 4 S,, p(S,) = 0, but the exceptional set S, depends on A, hence one can not deduce in general ,u&*(~) = 0,l for all ~ES,S some set, with p(S) = 0 (unless B, is countably generated). The global Markoff property is however equivalent to EJ.1 Au C) +E,(*IC), hence to the triviality of B, for pc. Note that in general one has, by monotonicity, convergence in the sense of L2(dp)-projections, of Ep(.l Au C) to some projections in L’(dp) as ArR*. The identification of the limit with E,(.jC) needs however further work (like that done above), unless B, is countably generated. If the Bore1 o-algebra at infinity were countably generated we would have the following much simpler situation:
239
LOCAL AND MARKOFF FIELDS
LEMMA. Let (X, B,p) be a measurable space. Let B, be a net of sub-a-algebras, directed so that $X < a’, B,, c B,, then there exists a”, X’ < ci’ such that B,.. c B,, c B,. Let B” be afixed sub-o-algebra
such that the regular disintegration pso of p with respect
to B” exists (i.e. ECfl B’)(q) = Jf(S)dpao(<[q),for p 1B” - a.e.q, with pL,o(
,LL-
generated.
IThen, for
i.s. and strongly in L’(d,u) to E,cfjB”),
any fE L’(dp),
E,,(flB,V
B”)
as c1-+ x.
Proof:
1. For any sub- o-algebra B’ and any measure v one has, as observed above, E,(.lB’) = E,(e) o v(A) = 0,l for all A E B’. as a -+ co, as projections in L*(dv) (by monotonicity). 2. E,(.lB,) 1 U.lB,) 3. E,(.(B,)JE,(.)
as projections
4. For any AEB&,&I) disintegration).
in L*(dv)ov(A)
= jp,o(Alq)dpBO(q)
5. p(A) = 0,l for all AEB,
= 0,l for all AEB,.
with d,uBo = d,u 1B” (by the regular
=pLBO(Alq) = 0,l for all YES, with $‘(S,)
= 0.
6. B, countably generated implies pBo(A 1~) = 0,l for all q E S and all q ES and all AEB,, with $‘(S) = 0, S independent of A. 7. By 1 for pBOinstead of v we get EpgO(.IB,) = E,,, (.I 8,) 8. Identify (using the definition) 9. Conclude
E,,,(.Iq)(.IB,)
with E(.(B,v
= EpSO(.). B’).
the proof of the Lemma using 6, 7, 3 and 8.
This lemma then shows, that if the Bore1 a-algebra at infinity were countably generated, E,,(.( /iv C) -+ E,(.(C), which by what we saw above would imply the global Markoff property. It can also be proven that, for generalized random fields on Rd, the local Markoff property and “the property of short range correlations in d independent directions” (in the sense that E(.l/1;)1 E(-) as 14 co, as projections in L*(dp), with J 3 {x~ Rdllxal 2 l} for d independent vectors c( in Rd) together with the countable generation of the Bore1 o-algebra at infinity would imply already the global Markoff property [42]. In turn any measure which is locally Markoff, Euclidean invariant, satisfies the Osterwalder-Schrader positivity condition [9] and the cluster property (in the sense of the intimum of the spectrum of the Hamiltonian being a simple eigenvalue) would already imply the property of short range correlations in d independent directions, hence, by the above, it would also imply the global Markoff property [42]. Remark. 3. If we take C = (x’,x*)E R*)x’ = 0}, i.e. C is the “time zero” axis in R*, in Theorem 2.2, then we see that “the global Markoff property for hyperplanes” [ 111, [18]-[20] is a particular case of the global Markoff property proven above, for trigonometric interactions [34]. The weaker property E + E _ = E, = E _ E +, with E f
240
S. ALBEVERIO
and R. H@EGH-KROHN
defined as E(.ISZ”,), Sz, = {(x’,x2)~R21x1 >has beendiscussed by Accardi [43] (see also [S] d)). Let us now mention shortly the main consequences of the global Markoff property we proved for the trigonometric interactions. 1. The “axioms” discussed by Nelson [ll], [18], Simon [19], Friihlich [23] and others (see e.g. [23]) hold for the trigonometric interactions of Theorem 2.2. (The global Markoff property was the missing step in such a proof). 2. Write y = (t,x), FERN, t, XER, r(y) = [(LX) and look at t + &,x) as a (generalized) stochastic process indexed by R, with state space S’(R) (a point of view used particularly in [S], [lo], [44]). By the global Markoff property with respect to C = ((t,x)~R’lt = 0} we have that t + 5,((p) 3 J<(t,x)cp(x)dx is a t-homogeneous stationary Markoff process ([5]b), c)) with invariant measure cl0 E p 1B(C) (which exists by the regularity of p [5] b). By the time-reflection invariance of p we see ([ 111, [ 191, [51) that G is a symmetric Markoff process with a symmetric Markoff transition semigroup eetH, t > 0 (0
S’(R)
R
where V is the closure of the gradient FC’
s
dxd/Jo(O = 3 Ff 12dPO>
t
in L2(dpo)! L’(R)
operator
defined as a map from
(i.e. (Bf)(c,)
df
=
md~). s
This Dirichlet form can be identified with a regular Dirichlet diffusion form in the sense of [5]a) on C(X), where X is the spectrum of the uniformly closed algebra F C2 in L”(dpu,). For these results see [5] b), [5]c) (part of these results is written for the case of to the case of trigonometric polynomial interactions; however the adaptation interaction is easy). We have dx= R
-$A-p.V,
LOCAL AND MARKOFF
(this is the mathematical
241
sense in which formally Hz_
s +--_
a2
Ghd2
R
s
dx + &,(x))dx R
over the “flat space” (S’(R),l?d<,,(x)) (See [5] for this point). x the infinitesimal generator of the Lorentz boost is given on FC2 by
is a Schrodinger Analogously,
FIELDS
operator
(4X50) = -
+x
& +B(C,(x))&)dx.
R
For more details see [5], [24]. 4. p,, is S(R)-quasi invariant, abelian group S’(R) in L2(dpo)
hence there exist unitary representations of the by multiplication cp+ ei(e’~@)and translation cp
+(e”(@“f)(&J = [do&, + GQGJ11’2f(~0 + cp) and (5~)~ I satisfy the Weyl commutation relations. rc((p) is essentially self-adjoint on {ei(cb@), $ES(R)) and coincides with the physical momentum operator. This gives the whole “Schrodinger interactions. The representation” of canonical field theory3 with trigonometric equation of motion is i[n(qo),H] = :0’:(q) + (.,( - d + m2)q) with:v’:(q) = - a;lJ: sin(ax + O):q(x)dx ([5], [24], [34]). 5. HP0 is a.n elliptic operator in infinite dimensions, one expects the ground state to have the usual properties of ground states of elliptic finite dimensional operators (Frobenius theorem, regularity properties). In fact one has that the inlimum of the spectrum of HP0 in L’(dp,) is zero, simple, isolated, with eigenvector Q(tO) = 1. Szis an analytic vector for 7t((p).~~(5~ + TV)is analytic in t and CL,,is strictly positive (in the sense that the conditional measures with respect to arbitrary finite dimensional subspaces have strictly positive density with respect to Lebesgue measure, on compacts) (for these results see [5] b)). 6. The process t + qt associated with the Dirichlet form iJ( yf12dpo can be realized as a strong Markoff, in fact a Hunt process with state space differing from X by a polar set. Both 5, and qt satisfy the stochastic equation d<, = /&,)dt + dw,, where w, is the Brownian motion on S(R) c L2(R) c S’(Rd), and have the same infinitesimal generator on FC2.([5]a)-c)).
3The trigonometric interaction models are the first class of models for which the ideas of canonical quantum field theory (like those expressed e.g. in [46]) are verified.
242
S. ALBEVERIO
and R. H@EGH-KROHN
Remark. We shall end with a remark concerning the global Markoff property for lattice systems. In [48] we have shown, in collaboration with G. Olsen, that for a class of lattice systems for which Dobrushin’s method [ 141 a)-c) yields uniqueness of Gibbs states one can also prove the global Markoff property, adapting the method used by us for Euclidean fields. Let us shortly describe our result. Consider a lattice Zd in d-dimensions. Let @ be a (finite range) interaction on Zd i.e. a real-valued function defined on the subsets of Zd, with support on finitely many finite subsets of Zd and such that for A c Zd, @(A) = @(- A). Define, for any subset A c Zd, 11@(A) (1= J.JcP((/i)l(lAI- l), where the sum is over all finite subsets of Zd. Let oi be the function oi(A) = 1 if i is a point in A and ~i( n) = 0 if i$ A and let B, be the a-algebra generated by the oi,ie A. Define the interaction measure (note the analogy with the measures pn for the
Euclidean dpA(a)
fields
in the
E exp(-
bounded
C OiC @(A’) II itA
j-
A’
itA*
open
where da(a)
GE ll {O,l}(i, by
iEz*
oj))d~(o)
C OiC@(A’)
itA
A c Rd)dpn(a),
region
A’
fl
j- ic,l’
= II dcci( cri), d cli being a probability
oj)da(a)
1 -l,
measure on {O,l},i,, the i-th copy of
i&
the 2-point space {OJ}. Note that oic @(A’) n
,
crj is the interaction
between the
lattice point i and the rest of the la&e (on&;i;‘,se points j give a non-vanishing contribution which are such that they can be obtained by adding to i the vector corresponding to a point in some of the finite sets on which @has support. E.g. if, for d = 2, @ has only support on the finite set {(O,O),(- LO), (OJ), (1,O)) then only the nearest neighbors ofj contribute. ,Uis called a Gibbs measure to (plno, A,, bounded} if E,Cfl Jo) = &I,(fl A) for any bounded shin’s results about
existence
lows that if 11@/I< ie-’ limE(fl$ d
= E(f)
Bore1 function f on II {O,l},,. From Dobru-
and uniqueness
of Gibbs measures
then p is unique, p I B,
(this is the corresponding
[14]a)-c)
it fol-
is trivial, where B,, z n Bn, and
statement
for lattice systezs
to our
%iqueness result Theorem 2.1). We shall now see that analogous considerations as in the case of Euclidean fields lead to the global Markoff property. To state the latter (and the corresponding local Markoff property) we consider decompositions Zd -= Q + u C v ii_, where Q +, C, Q _ are subsets of Zd which are pairwise disjoint
LOCAL AND MARKOFF
FIELDS
243
and D +, O_ are such that one cannot reach any finite subset of R, from Q_ by translation by vectors corresponding to a set in the support of @(i.e. there is no nonvanishing interaction between the random variables of the regions Sz, and a_, and C separates Q+ and Q_ in this sense. E.g., if @is a nearest neighbors interaction, C must be such that between points of Sz+ and L?_ there must be at least one of C, if @is a next nearest neighbors interaction, between points of Q_ and Sz, there must be a least 2 of C etc.). We say then that C insulates Q + and s2 _ . As for Euclidean fields it follows easily, see [48] for details, that the local Markoff property E,(f+f_ IC) = E,(f+ jC)E,(f_ IC) holds, where Q + is finite andf, are Q k -measurable, and ,Uis any Gibbs measure, and C insulates Q + and Q _. Moreover, the application of the ideas used for proving the global Markoff property for Euclidean fields, described above in the case of trigonometric interactions, yields first Ercc_,V,Cf+f-I JoI = &d_f+
I ‘JoKL,dL
Ia,
where ,~+(.j~) is the conditional probability measure defined by /J and the o-algebra B,. By the uniqueness result of Dobrushin, applied this time to the Gibbs measure &.lvl), under the assumption 11@I(< fe- ‘, we get that E,cc+,,(~l&) + EPc&.) as /i,, r Zd, and then we get fi,~+JJ+f- 1 = Epd.Iqd_f+)E,,tl,df), CL- a.s., i.e. E,cf,fI WI) = E,(f+ I C)(n)E,(f_ IC)(q) for p - a.e. q, which is the global Markoff property, s2 +, C, Sz- being arbitrary (not necessarily finite) such that C insulates Q, and Q_. We thus have the following THEOREM.
< $e-I.
Let @ be any interaction offinite range on the lattice Zd, such that //@II Then to the interaction measures
{P~,~}, dP~(~)=Z~‘exP(- C Oix @3(A) n k,l
,4'
oj)do(aL
j- isA’
where Z, is the number such that dpA is a probability measure, and do is any product of probability measures on {O,l), there is one and only one Gibbs measure ,tt,as proven by Dobrushin. Moreover,for s2, from
any disjoint partition Zd = Q+ v C v Q _ such that C insulates
52_ with respect
H iO-'I,i,eft BQ+-measurable,
to the interaction
and any bounded
we have the global Murkof
Bore1 function
on
property
isz"
EJf+fE,(.IC)
IC) = EJY+ IC)E,,cf-IC),
being the conditional expectation
with respect to p and B,.
The global Markoff property can be used to deduce consequences corresponding to the one we derived for the case of Euclidean fields. In particular, one has, for any splitting Zd = Z x Zd- ’ with i = (k,l), ke Z, 1E Zd- ’ a symmetric stationary Markoff chain k + & with values in {O,l} and invariant distribution p 1B,, C being the
244
S. ALBEVERIO
and R. H@EGH-KROHN
hyperplane k = 0. The corresponding symmetric transition matrix e-H, H > 0 on L2(d,u 1B,) is the “Onsagerian” (“transfer matrix”) postulated in many papers of classical statistical mechanics (see especially [49]). The relation of this transfer matrix and the one constructed using an adaptation [50] of Osterwalder-Schrader method is entirely similar to the one discussed above concerning the relation of the global Markoff property of Euclidean fields and Osterwalder-Schrader construction of the physical Hilbert space and the Hamilton operator. Remark. Recently J. Bellissard and P. Picco have extended our results on the global Markoff property to more general lattice systems, using their uniqueness results for such systems [39] c) (as for uniqueness results see also [39] a), [39] b). H. Follmer also remarked [51] that the global Markoff property for lattice systems follows whenever Dobrushin’s uniqueness conditions are satisfied using similar ideas as in our work [48]. Moreover, he shows that if the interaction is attractive and admits a “high density state” and a “low density state” then each of the states has the global Markoff property. Added in proof: The global Markoff property has been proven recently for exponential interactions by R. Gielerak (J.M.P. 24 (1983), 347-355). For further recent references see the authors’ contribution to Adu. Prob., ed. M. Pinsky, M. Dekker (1983).
Acknowledgement )JJe are very grateful to Professor Witold Karwowski and the Organizing Committee of the Karpacz 1978 Winter School of Theoretical Physics for a very kind invitation, where the first version of this work was presented in lectures, and for doing everything to arrange an extremely pleasant stay in Karpacz and Wroclaw. We are also grateful to the Centre de Physique Theorique, CNRS, Marseille and the UER, Universite d’Aix-Marseille II, Luminy, for their hospitality and financial support.
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See e.g. a)J.L. Doob: Stochastic processes, Wiley, New York, 1967. b)W. Feller: An introduction to probability theory and its applications, Vol. 2, Wiley, New York, 1971. c)E.B. Dynkin: Markoflprocesses, Vols. 1,2, Springer, Berlin, 1965. d)R.M. Blumenthal, R.K. Getoor: Markoffprocesses and potential theory, Academic Press, 1968. e) E. Nelson: Dynamical theories of Brownian motion, Princeton University Press, 1967. f) J. Lamperti: Stochastic processes, Springer, Berlin, 1977. g)T. Hida: Stationary stochastic processes, Princeton University Press, 1970. h)D. Wiliams: Diffusion, Markov processes and martingales, J. Wiley, 1979.
PI
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LOCAL AND MARKOFF
FIELDS
245
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246
S. ALBEVERIO
and R. H@EGH-KROHN
[I41 a)R.L. Dobrushin: ‘0. Prob. Appl. 13, (1968) 197-224. b)R.L. Dobrushin: Funct. Anal. and its Appl. 3, 1, (1969) 27-35. c)R.L. Dobrushin: Th. Prob. Appl. 15, 3, (1970) 458. d)M.B. Averintsev: Prob. Per. Inform. 6, (1970) 100-109. e) F. Spitzer: Isr. J. Math. 14, (1973) 92- 103. f) Ch. Preston: Random fields, Lecture Notes in Mathematics 534, Springer, Berlin (1976). [ 151 See e.g. [l] c) d)
[I61 a)See [14] and e.g. b)H.O Georgii: Lecture Notes in Physics 16, Springer, Berlin (1972). c) K. Krickeberg: Processes spatiaux, University Paris V, VI, XI, 1975/76. d)F. Spitzer: Lecture Notes in Maths., 390 Springer, Berlin (1974) 115-186. [I71
[18] [19] [20] [21] [22]
a)K. Urbanik: Tn. Prob. Appl. 1, (1956) 132-134. b)K. Urbanik: Studia Math. 21, (1962) 261-282. c)I.M. Gelfand: Do&. Ak. Nauk SSSR 100, (1955) 853-856. d)K. Ito: Mem. Coil. Sci. Univ. Kyoto 28, (1953) 209-223. E. Nelson: Lecture Notes in Physics 25, Springer, Berlin (1973) 94124. B. Simon: The P(q), Euclidean (quantum)field theory, Princeton University Press, Princeton, F. Guerra, L. Rosen, B. Simon: Ann. Math. 101, (1975) 111-259. See [13] and e.g. [22] S. Kotani: pp. 239 - 250 and Y. Okabe pp. 340- 354 in book of Ref. [3]a)
1974.
[231 a)G.C. Hegerfeldt: Comm. Math. Phys. 35, (1974) 155-171. b)J. Frb;hlich: Nelv. Phys. Acta 47, (1974) 265- 306; Adu. Math. 23, (1977) 119-180 c)J. FrGhlich: Ann. Inst. H. PoincarC A 21, (1974) 271-317. d)V. Glaser: Comm. Math. Phys. 37,(1974) 257-272. e) M.O. Carroll, P. Ottensen: Comm. Math. Phys. 36, (1974) 37-58. f) W. Karwowski: Rep. Math. Phys. 7, (1975) 411416. g)J.L. Challifour: J. Math. Phys. 17, (1976) 1889-1892. h)I. Herbst: J. Math. Phys. 17, (1976) 121&1221. i) H.J. Borchers: pp. 81- 100 in Proc. Ref. [5]a). j) J. Yngvason: Rep. Math. Phys. 13, (1978) 101-115. k)V.P. Gachok: Quantum processes, Izd. Nauk. Dumka, Kiev, 1975. 1) S. Nagamachi, N. Mugibayashi: Pub/. RIMS 12 (Suppl.) (1977) 309-341. [24] S. Albeverio, R. HBegh-Krohn: Canonical relatiuistic quantum fields (to appear in Ann. Inst. H. Poincark). [251
[26] [27] [28] [29] [30]
a)J. Glimm, A. Jaffe, T. Spencer: pp. 175 - 184 in Ref. [ 181; b)J. Glimm, A. Jaffe, T. Spencer: Comm. Math. Phys. 45, (1975) 203-216. S. Albeverio, R. Hoegh-Krohn: J. Funct. Anal. 16, (1974) 39-82. J. FrGhlich, E. Seiler: Helu. Phys. Acta 49, (1976) 889-924. E. Nelson: I. Segal, MIT Prep (1966) 69-73. J. Glimm: Comm. Math. Phys. 8, (1968) 12-25. F. Guerra: Phys. Rev. Lett. 28, (1972) 1213 (and unpublished manuscript
[3ll a)S. Albeverio, R. H#egh-Krohn:
Comm. Math. Phys. 30, (1973) 171-200.
(1971)).
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b)S. Albeverio, R. Hflegh-Krohn: J. Funct. Anal. 19, (1975) 242~272. pp. 35 - 137 in Proc. XII-Winter School Theor. Phys. Karpacz Jancewicz, Wroclaw (1976). [32] J. Friihlich: Comm. Math. Phys. 47, (1976) 233-263. [33] R. Seiler, D. Uhlenbrock: pp. 363-374 in Proc. Ref. [S]aj. [34] S. Albeverio, R. H_@egh-Krohn: Commun. Math. Phys. 68, (1979) 955128. c) S. Albeverio:
1975, Edt. B.
c351 a)See [19] and e.g. b)J. Glimm, A. Jaffe, pp. 1 - 34 and pp. 35 - 66 in New developments in quantumfield theory and statistical mechanics. Proc. Cargese Summer school 1976, Edt. M. Levy, P. Mitter. c) J.P. Eckmann, H. Epstein: IHES - Preprint, 1979, and references therein, for P(q),; d)S. Albeverio: Ref. [31]c) e)J. Friihlich, Y.M. Park: Helu. Phys. Acta 50, (1977) 315-329. f) S. Albeverio, G. Gallavotti, R. HBegh-Krohn: Some results for the exponential interaction in two or more dimensions, Corn. Muth. Phys. 70,(1979) 1877192; see also S. Albeverio, R. Hoegh-Krohn in Proc. Ref. [39]c). g)Ref. [34] for trigonometric interactions and O.A. MC. Bryan: Comm. Math. Phys. 61.(1978) 275-284. Also Y.M. Park, JMP 18, (1977) 2423-2426. h)See for other recent references, H. Englisch: Remarks on MC Bryan’s convergence prooffor (ae4 cos (aq”) - 1 + a2~2/2)2-quanfumfields, Leipzig Preprint Jan. 1979, to appear in Rep. Math. Phys. i) H. Englisch: Wiss. Z. Karl-Marx-Univ. Leipzig, Math. Naturw. R. 27, H3, (1978) 235-243. k)R. Gielerak: Maximally short verijication of the Wghtman axiomsfor the strongly coupled quuntum (Sinew-Gordon), theories, Wroclaw Univ. Preprint, 1979. 1) R. Gielerak: Existence and Wghtrnan uxioms for Sine-Gordon like interaction in arbitrary space-time dimensions, Wroclaw Univ. Preprint Dec. 1979. [361 a)See [18]--[20], [23]b) c). b)F. Guerra, L. Rosen, B. Simon: Ann. Inst. H. Poincare A25, (1976) 231c) J. FrGhlich, B. Simon: Ann. qf Math. 105, (1977) 493-526.
334.
[371 a)Dobrushin, Minlos: Funct. Anal. and its Appl. 7 No. 4, (1973) 81-82. b)Dobrushin, Minlos: pp. 23 -49 in Ref. [31]c). [38] R.L. Dobrushin: see Ref. [14]a), b), c). O.E. Lanford, D. Ruelle: Comm. Math. Phys. 13, (1969) 194215. See also: E.B. Dynkin: pp. 507- 512 in Actes Congres Intern. Math. 1970, Nice, T. 2. H. Fb;llmer, pp. 305 - 317 in Sem. Prob. Strasbourg IX, Lexture Notes in Maths. 465, Springer, Berlin (1975). [391 a)M. Cassandro,
E. Olivieri, A. : Pellegrinotti,
E. Presutti: 2. Wahrscheinlichkeitstheor.
Vizrw. Geb. 41,
(1978) 313-334.
b)G. Royer: J. Math. Pures Appl. 56,(1977) c) J. Bellissard, P. Picco: Lattice quantumfields:
[401 [411
455478.
uniqueness and Markoflproperty, CNRS-CPT Preprint, Dec. 1978, Proc. Bielefeld Encounters in Physics and Mathematics II, Quantum Fields-Algebras, Processes, Ed. L. Streit, Springer, 1980. M.G. Nadkarni, D. Ramachandran, K.P.S. Bhaskara Rao: Coil. Math. 32, (1975) 219-226. L. Schwartz: Radon measures on arbitrary topological spaces and cylindrical measures, Tata, Oxford Univ. Press, 1973.
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and R. HPEGH-KROHN
S. Albeverio, R. Hoegh-Krohn: unpublished manuscript. L. Accardi: Local perturbations of conditional expectations. Marseille Preprint, 1977; J. Math. An. Appl. 72,( 1979) 39-69.
[44]
J. Cannon: Comm. Math. Phys. 36, (1974) 123-132. R. Carmona: Measurable norms and some Banach-space-valued Gaussian processes, Duke M. J. 44, (1977) 109-127. L. Gross, J.P. Eckmann, D. Pearson: J. Funct. Anal. 16, (1974) 388404. See also e.g. D. Sunyach: Ann. Inst. H. Poincart B36, (1974) 233. For a related point of view in the l-dimensional case see Ph. CourrBge, P.-Renouard; P. Priouret, M. Yor: A&risque 22-23, (1976) l-245, 247-290. G. Royer: Ann. ENS 4e Sbr. 8, (1976) 319-338. G. Royer, M. Yor: Ann. Inst. Fourier 26, (1976) 7-24. M. Yor: Etude des measures de probabilitis sur C(R+*, R) quasi-invariantes par /es translations de D(R+* ,R), Ann. Inst. H. Poincare’ B11,(1975)
127-172.
c453
[46]
[47] [48] [49]
a)E. Chen: Markoff systems of imprimitivity, Leuven Preprint, J. Math. Phys. l&(1977) 1832-1837. b)S. Nagamachi, N. Mugibayashi: Progr. 7heor. Phys. 53, (1975) 1812-1826. c) Ph. Blanchard, Ch. Pfister: Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, (1978) 177-190. G. Wentzel: Quantum theory offields, Interscience, 1949. H. Araki: 1. Math. Phys. 1, (1960) 492-504. F. Coester, R. Haag: Phys. Rev. 117, (1960) 1137-1145. A. Klein: Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, (1977), 115-124. S. Albeverio, R. Heegh-Krohn, G. Olsen: The global Markoffpropertyfor lattice systems, CNRS-CPT Marseille, Preprint Nov. 1978, J. Mult. Anal. 11,(1981) 599-607. R. Minlos, Ya. Sinai: 7’heor. and Math. Phys. 2, 2, (1976) 230-243.
c501 a)J. Frahlich, B. Simon, T. Spencer: Comm. Math. Phys. 50, (1976) 79-85. b)F. Dyson, E.H. Lieb, B. Simon: J. Stat. Phys. l&(1978) 335-383. c)R. Schwartz Schor: Common. Math. Phys. 59, (1978) 213-233. d)J. FrGhlich, R. Israel, E. H. Lieb, B. Simon: Comm. Math. Phys. 62,(1978) l-34. e)G. Hegerfeldt, CR. Nappi: Comm. Math. Phys. 53,(1977) 1-7. [Sl] H. Fb;llmer: On the global Markoffproperty, ETH Preprint Dec. 1978, Proc. Ref. [39]c), See also S. Goldstein, Comm. Math. Phys. 74,(1980) 223-234, and recent work quoted in S. Albeverio, R. HeeghKrohn, Adv. Prob., Ed. M. Pinsky, M. Dekker (1983), in particular by S. Albeverio, J. BeIlisard, E. B. Dynkin, J. E. Fenstad, S. Goldstein, R. Hgegh-Krohn, T. Lindstrgm, M. RGckner, B. Simon, H. v. Weizsgcker, W. D. Wick.
REPORTS
ON MATHEMATICAL
No. 2
PHYSICS
LOCAL AND GLOBAL MARKOFF FIELDS S. ALBEVERIOand R. H@EGH-KROHN Fakultlt fur Mathematik, Universitdt Bielefeld, F.R.G. Mathematisches Institut, Ruhr-Universitat Bochum, F.R.G. CNRS - CPT, Marseille, Universite d’Aix-Marseille II, UER de Luminy, France Matematisk Institutt, Universitetet i Oslo, Norway (Received February 12, 1980)
We discuss the relation between local and global Markoff property for processes and (generalized) random fields, both in the case of discrete and continuous “time parameters”. In particular, we give a short account of our proof of the global Markoff property for the Euclidean Sine-Gordon fields in two dimensions. We also give an account of our proof of the uniqueness of the solutions of the DLR equations in this case (extremality of the Gibbs state, implying in particular uniqueness of the phase). We also discuss more generally the relation between the uniqueness and global Markoff properties, and discuss some consequences of the global Markoff property. Finally, we indicate shortly how the global Markoff property is proven (by ourselves together with G. Olsen) in the case of lattice systems.*
1. Markoff processes and Markoff fields In this paper we shall discuss problems connected with the Markoff property of Euclidean (and lattice) random fields. Random fields are an extension to the case of “higher dimensional time” of stochastic processes, so let us start with the latter. Among stochastic processes the class of Markoff processes is a very important one (it contains e.g. the processes describing Brownian motion and more generally it contains the diffusion processes) [l]. Of particular interest are processes and fields which are homogeneous (in the sense that the corresponding measures are invariant with respect to the Euclidean group in Rd). In the case of processes this means that they are given by a Markoff semigroup P, (such that 0
*Modified version of Lectures given at the Winter School in Theoretical 1978 (Dir. W. Karwowski).
v51
Physics, Karpacz, February
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but can also be e.g. a Polish space or, more generally, a Suslin space of distributions like S’(R”) or D’(R”)). The requirement of invariance of the process under the time reversal implies that P, is a symmetric semigroup on L’(dp), i.e. Pf = P,, * meaning adjoint. We also assume the process is conservative i.e. P, 1 = 1. Then we have a stationary process with invariant measure ,u (one has J(P,~~)(x)&fx) = s(P, l)(x)~,(x)@x) = &I)). Observe that since P, maps 1 into 1 it also maps L” into L” and by interpolation we have that P, is a Markoff semigroup on all L’(dp),l < p < co. Let t--t 5, be the corresponding process (which can be set up according to Kolmogorov’s construction, e.g. by identifying the algebra L”(S) with the continuous functions on its spectrum). Then the Markoff property in the sense of the theory of stochastic processes can be expressed e.g. by E(f+f-
I&) = Edf+ IB,)ECf-
I&),
(1.1)
where E(.(B,) means conditional expectation with respect to the o-algebra generated by the time zero variable to(.) and f+, f_ are bounded functions measurable with respect to B, resp. B_, where B, are the a-algebras generated by the variables &, I E( co,O) resp. t E (0, co). In short, “the future is independent of the past given the present” (if instead of E(~IB,) we had (1.1) for E(e) we would have the complete independence of future from past. Thus the Markoff property is an expression of conditional independence). Equivalently (see e.g. [l] a)) EU-+ I&d =
EU-+IB- VW,
(1.2)
where B- VB, is the a-algebra generated by B_ and B,. The Markoff property is actually equivalent with having a Markoff semigroup P,, i.e. a semigroup s.t. 0
LOCAL AND MARKOFF
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227
quantum fields, can also be useful constructive tools (see [8]). Processes with the Osterwalder-Schrader positivity condition ([9]) have in general only weaker “markoflian properties” (this is analyzed completely in the case d = 1 [lo], [47]; note also that generalized free fields are in the class but are not MarkofT). Let us now mention a property which is weaker than the Markoff property as defined above but in another sense than “higher order Markoff’ or “OsterwalderSchrader positive (for processes)“. It is the property ECflB,_.,,,) = ECflB_,V B,) for anyf’bounded measurable with respect to the o-algebra B, _ , ,u,~ B,,, x,j. In analogy with the case of generalized random fields ([ 111, [ 121) this property has been called “local Markoff property” ([ 131 whereas (1.1) itself is then called the “global Markoff property”. Note that the corresponding property to (1.2) E cfl B,,,,,) = ECfl B,v Bb) for anyfB, _ ,u.bl-measurable is what is often understood as Markoff property of a process (not necessarily stationary) indexed by I c [O,~Z). The corresponding property for random fields (with [a.b] replaced by the interior of a smooth closed bounded curve and I replaced by some subset of Rd) is also what is often called in the theory of random fields (Gibbsian fields) (see e.g. [ 141) a “Markoff property”. The global Markoff property (1 .l) is actually the most suitable concept for homogewous processes and fields, but is also difficult to verify; see below. It is however only the global Markoff property that permits to associate to Euclidean random field Markoff semigroups yielding also a complete description of the associated quantum fields by a Schrddinger (canonical, fixed time) representation. Since we are making a sort of a “tour d’horizon” concerning Markoff-type properties, let us mention the “strong Markoff property” [ 151: in the case of homogeneous processes this amounts to (1 .l) with B, replaced by B,. B, being a random time. The above mentioned diffusion processes (associated with Dirichlet diffusion forms) have realizations which have the strong Markoff property (are in fact even Hunt processes [5] .b), [5] c)). For homogeneous random fields a complete extension of this concept has not yet been elaborated, however a partial one is discussed in [5] b), [5] c), [5] d). Let us now discuss in more detail some of the extensions of the concepts discussed above for processes to the case of random fields. These are defined analogously as processes but with the time parameter (which runs over some interval of R) replaced by a point running in some subset of Rd (or Zd, in the discrete case, corresponding to random walk on Z in the one dimensional case). If s(x) (.) for every x E Rd is a random variable on some probability _ space (0,~) with values in a state space S we call x -+ <(X)(C)) a random field with values in S [7], [ 161, [ 171. Then E(x)(u)) is, for fixed U)E Q, an S-valued function of .x. In the case where S = R one speaks of ordinary random field. If S is a distribution space, so that ;(X)(W) is a generalized function in x, for p-a.e.o, then we speak of x -+ c(x)(.) as a generalized random field [17]. In this case (t,(p) (.) = J<(x)(.)cp(x)dx, for cp in a test
228
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function space, e.g. CF(ZP’), and where the integral is understood in the sense of distributions, is an ordinary random variable. For applications in quantum field theory and statistical mechanics one is interested particularly in homogeneous (generalized) random fields, i.e. random fields which are invariant under the Euclidean group E(Rd) in Rd i.e. such that the corresponding measure ~1is invariant under E(Rd). Let us first recall such homogeneous generalized random fields in the Gaussian case. It is well known that ford = 1 the only Gaussian field which is Euclidean invariant and has in addition the Markoff property and continuous paths (stationary symmetric diffusion process) is the Ornstein-Uhlenbeck velocity process t --+&, i.e. the Gaussian (with measure p) process with mean zero and covariance
(-$+m’)-‘,m>O(see
e.g. [l] b). In this case Q can by taken to be C(R) and one has E,cf(<,)) = (P,f)(x) for any fEL2(R,dpo), with E, the conditional expectation given &, =x and with P, = e-‘r’Hmt~ R,H,,,, being the diffusion operator defined as the self-adjoint positive operator uniquely associated with the closed diffusion Dirichlet form g --f +JI Vg(‘dpo, V being the closed gradient operator (closure of the gradient as defined on C,?,(R)) and p,, is the Gaussian measure obtained by restricting p to the a-algebra generated by to (this is the Gauss measure with mean 0 and covariance 1/2m (see [l]b). For d > 2 the corresponding objects are the homogeneous generalized Gaussian random fields with mean zero and covariance (- d + m2)-l in L2(Rd,dx), with A the Laplacian in L2(Rd,dx), m > 0 for d > 3, m > 0 for d 6 2. The probability space of such fields can be taken to be (S’(Rd),dpo) where p” is the Gaussian measure with support within S’(Rd), and is the normal distribution with covariance the scalar product in the real Hilbert space H_ ,(Rd) (i.e. the closure of C;(Rd)) in the norm given by the scalar product given by jq(x)G(x - y)cp(y)dxdy, with G(x-y)=(-A +m2)-’ (x-y)). This measure and the corresponding generalized random field are called the free Markoff lield. Their relevance to quantum fields became clear particularly through the work of Symanzik, Nelson and Guerra connected with the “Markoff property”.’ Let us describe this property. We first remark that the free Markoff field cp --f (5,~)(.), cpES(R~),
‘This property is actually due to the fact that the covariance is the inverse of an elliptic differential operator of second order. Markofian properties of higher order elliptic operators and more generally of pseudodifferential operators are studied e.g. in [7], [8].
LOCAL AND MARKOFF
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229
respect to PO. Define, for any Bore1 subset A c Rd, B(A) as the o-subalgebra of the pmeasurable subsets generated by the linear functions (<,p)(.) with suppp c A. Let now C be a (d - 1)“dimensional piecewise C’ hypersurface which divides Rd - C into two disjoint components Q, and sZ_. Let f+ ,f- be bounded B(S2, ) resp. B&-)-measurable functions. Then we have ECf+f- IC) = Ecf+ IC)Ecf-
IC)>
(2.1)
where E(.JC) is the conditional expectation with respect to the measure p” and the oalgebra B(C). In the present case, where ,u’ is the measure of the free Markoff field, (2.1) holds for all C, even C which extend to infinity, like e.g. hyperplanes ((x~,...,x~)E Rd(x, = O}. We say, because of this, that the free Markoff field has the global Markoff property. Consider now more generally a generalized random field t(x) with corresponding probability space (S’(Rd),p) and suppose 5 has a regularity property in thesensethatE(J(Sr,cp)l2)~C(U~S~(x)G(x-y)cp(y)dxdyforanycpED(U),cp3Oany bounded open set U, with C(U) a constant independent of cp and where E means expectations with respect to p. In this case we can define for any Bore1 subset A of Rd, B(A) in a way corresponding to the one we did before, with ,u’ replaced by p. Then if (2.1) holds for all (d - 1)-dimensional piecewise C’ hypersurface dividing Rd-C into two disjoint components Q+ and Q_ we say that p (and 5) have the global Markof property. Remark. In the case d = 1 this property corresponds to the usual Markoff property for processes (l-dimensional time parameter), in an obvious sense. Moreover the global Markoff property implies the Markoff property with respect to half planes [ 181, [ 191, [20], which is known, see below, to yield a Markoff process in the orthogonal direction. In the case where (2.1) holds only for all (d - 1)-dimensional piecewise C’, bounded hypersurfaces dividing Rd-C into two disjoint components 92, and Q_ we say that p (and 5) have the local Markoff property. Remark. In the case d = 1 the interpretation of local Markoff property, formulated above, is not quite unique. If we take C reduced to a point then the local Markoff property coincides with the global one. However, looking at the statement of the local Markoffproperty for arbitrary d taking C to be spheres and letting Q,, Q_ be the outer resp. inner of these spheres, we find a less trivial l-dimensional analogue in C: the extremes a, h of an interval [a, b] and Sz,_ = ( - x ,a) u (b, x ), 92_ = (a, b). This is the property involving B,, B, discussed above. The local Markoff property understood this way, is then in general not the same as the global one, see [21].”
*For a relation between Markoff property and Mackey’s concept of imprimitivity discussions of concepts related to the Markoff property see also e.g. [45]b), c).
see [45]a). For other
230
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and R. H@EGH-KROHN
Let us illustrate the local versus global Markoff property schematically ford = 2 as the conditional independence of observations within Sz, and Sz- given all the observations on C in the two typical situations
(global Markoff)
(local Markoff)
Remark. The concepts of local and global Markoff properties of Euclidean tields were discussed originally by Nelson and Newman [ 181, [ 121. Markoffian properties of other types, like the n-Markoff property (corresponding to conditional independence when not only the values of j; on C are given but also the derivatives up to order n - 1: a concept which applies to ordinary (not generalized) random fields [7], [S] or the GMarkoff property (in which 5 in a whole neighborhood of C has to be prescribed to get conditional independence) have been discussed in the mathematical literature (e.g. [22]). These other Markoffian random fields do not, however, yield directly relativistic local quantum fields (see however the recent work [S] for a very useful applications of nMarkoflian fields). An intermediate notion of Markoff property between the local and the global one is that of Markoff property with respect to hyperplanes ((2.1) requested to hold for all hyperplanes C). This is actually the Markoff property which is sufficient for yielding relativistic lields from Euclidean ones, as first remarked by Nelson [ 1 l] [ 181. Later Osterwalder and Schrader [9] realized that a weaker positivity condition (which holds e.g. for generalized random fields which are weak limits of generalized random fields t,(x) which are symmetric Markoff processes x1 --, &(xl,xZ,...,xd)) is also in fact sufficient for yielding relativistic fields from Euclidean ones. For related works see also e.g. [23]. Let us now come back to the case of the free Markoff field measure ,n” and the corresponding generalized random field t(x). If we write x = (x1 ,...,x,), x1 E R and look at the process xi -+ {(x1,x 2r...,~d) from xi E R with values in S’(Rdpl), we have (see e.g. [19]) that 5(x1,x2 ,..., xd) -, 5(x1 + s,x 2,...,xd is represented in L2(dpo) by a unitary map U, and if we call E, the conditional expectation with respect to the hyperplane C = {(x1 =0,x, ,..., x,)>,we have
E, USE, = eCsH,
s 3 0,
where eesH is a Markoff semigroup in EoL2(dpo) ~L’(dpz), p” to B(C). Consider now [S] the Dirichlet form
p: being the restriction
of
LOCAL AND MARKOFF
FIELDS
231
first defined in a natural way on cylinder functions
with &C, (Rk,R). This positive quadratic form in L2(d&) is known to be closable ([5]a), b) and its closure, which is ij[ Vf12d& is uniquely associated with the operator H in the sense that (H1’2f;H”2f)
= +JIVjJ’d&
for all f’~D(2?“‘)
= D(V).
As proven in [5] a), [5] b), the physical Hilbert space (Fock space) associated with the free Markoff field is identifiable with EoL2(dpo) rL2(d&) and the physical Hamiltonian is identifiable with H. The time evolution of the relativistic field is given by the one parameter group generated by H. Moreover, the Lorentz-boost is represented by the Dirichlet form
This is the canonical picture of the physics of quantum fields in the interaction free case. Note that much more is known about the process 5, associated with eesH, e.g. it can be realized as a Hunt process [5] b), c) (see also e.g. [5], [24] for other results). Is it possible to have something like this for Euclidean measures different from the Gaussian one? In the cased = 2 the existence of Euclidean measures on S’(R2) is known in several cases, see e.g. [25], [26], [27]. S UCh measures are constructed as follows. One first constructs a family of measures pL, on S’(R’), attached to the family of open bounded regions /1 in R2. The p,, are all absolutely continuous with respect to the free Euclidean field measure 1~’and dp,,/dp’ is of the form dp,,/dpO = e- “,d/je- “.fdp’, with U,, is measurable with respect to the o-algebra B(n) U,, EL’(dp’), e- “4 EL’(d,u’), generated by the fields with support in /1, additive (i.e. U,, 1v,,2 = U,, 1 + U,,, whenever /1, n ,4, = p) and covariant (U,,(T[) = U,,(t) for all Euclidean transformations 7). One then shows that the weak limit p of p,, as /iT R2 (along the net of bounded open subsets of R2) exists (p is then invariant under the Euclidean group in R2, since the limit is taken along arbitrary open bounded sets). We can construct U, as follows. Let x,(x) be the function on Rd whose Fourier transform is the characteristic function Xl,(k) of the ball of radius K > 0 i.e. iK(k) = 1 for (kl < K, iK(k) = 0 for Ikl > K. Let t,(x) = (<*x,)(x), where * denotes convolution of the distribution YES’ with x,. Then t,(x) is a C”-function ofx, hence exp(a<,(x)) is a well defined continuous function of c( and x, for any K > 0 and YES’. Detine:exp(a<,(x)) :- exp(-$a’G,(O)) exp(a&(x)), with G,(O) the value at 0 of xK*G*xK, i.e.
232
S. ALBEVERIO
G,(O)=
and R. H@EGH-KROHN
(2n)Lpk)l’&5.
Then: exp(a&_(x)):EL2(d~o). Let P, be the orthogonal projection in the Hilbert space L*(dp’) onto the subspace generated by C&(X).Then P,,P, = P, for K’ > K i.e. P,, 3 P,. Since :exp(a~,(x)):~P,L*(d~~) and for the o-algebras B, generated by t,(x) we have B, 1 B,, it follows that {exp(a&(x)), K > 0} is a positive martingale with respect to {BJC 2 0). Also j :exp( g<,(x)): dx for any bounded open ,4 is a martingale in the same sense. As K + cc thi orthogonal martingale L*(dp’)
projections
convergence
to a function
P, converge strongly in L*(dp’) to the identity. The
theorem
in L*(dp’)
gives convergence
of U: = j :exp(cc<,(x)):dx
if and only if ~(~:exp(~&(x)):dx)*d~~
in
is uniformly
bounded in K, for fixed A This is satisfied if z2 < 47c since, using the definition of:exp(a<,(x)): and p”, we see that = s ~e’2G~~“-Y~dxdy+ J Jea2Gcx-y)dxdy, AA AA
J( j:exp(&(x)):dx)IdyU A and this integral = -&lnlx-yl
is finite since G(x + 0(1x-yl)
y) is continuous
as x-y-+0
for x - y # 0 and G(x - y)
(and c1*~47~). Let us call U;, with V(S)
E exp(as), the limit for K + co of J:exp(a{,(x)):d
x and write (in the usual suggestive
A
way, based on t,(x) -+ <(x ) as distributions,
as K+ cc) US; = J:exp(N<(x)):dx.
Vi EL*(d,u’) (a fortiori
Moreover
thus U> ELl(d,u’)).
Then
U 1 > G (as L*-limit of non-
negative functions), hence e -‘i E L”,(dp’). Clearly, U y1is additive (since the U > are additive by construction). CJZ is also B(A)-measurable [26], [18]-[20], (see below). Thus we have verified that { U .Y,,A bounded open) is an interaction in the above sense. It is called the “exponential interaction”, v being an exponential function (it was first studied by the authors in [26]; see also [35]f) for a recent investigation of the exponential interaction on Rd,d B 2). Other interactions can be constructed from this. be the projection of J:exp<(x):dx onto P’“‘L2(dpo) = Let j:[“:(x)dx E H(“~6H(“-‘~, where HCk) is the subspace of L2(d,po) generated
by the monomials
LOCAL AND MARKOFF
fi (r,cpi), i=l
qiES(R’), j = l,...,k, and constants.
over, it is B( /l)-measurable,
Clearly,
as limit of B( Il)-measurable
= L2(dpo) -
limS:exp(&,(x)):dx,
exp([,(x)):dx
and%“):exp<,(x):
Then
FIELDS
thus
233
j:<“:(x)dx A
is additive. More-
quantities (since j:expt(x):dx A
P’“) j:exp&(x): =
lim
S P?exp(<,(x)):dx A
is, for XE /1, &&measurable).
U);, with v(s) now defined as V(S)= 3, $ a,$,
a,,;l E R, defined by U f; =
k=O x Id x is also B(/i)-measurable, additive and in L2(dpo). From the type of A estimates originally due to Nelson [28], Glimm [29] and Guerra [30] one proves also =
i.kg?kJ:g’:(
that e-“~~~p(d~o) for all 1
+ j:exp(A
i(at(x)
+ 0):dx - iok
the
+ B)):dx} with
,~~(5)P”‘Bi!:i”j:(x)dx. ‘J
Such interactions were introduced in a regularized version by us in [31] and the regularization was removed by Frohlich in [32]. Again it follows easily that US; is additive, B(n)-measurable and in [32] it is shown that e - ‘: E L’ @PO), hence U: is an interaction, “the trigonometric interaction” (“Sine-Gordon interaction”). For a very interesting study of this interaction see also [33], [27]. Let now U, be any of the interactions U: described above. It is easy to show that the corresponding measure dp, = e-“~d,u”Ije -“Ad,u” has the global Markoff property. This was proved first (for the case of polynomial interactions) by Nelson [18]. The observation is, as formulated in [34], that due to the global Markoff property of p” and the additivity of U,, U, = Vi + U; + U:, where Uj are B(Q,)-measurable, UC, is B(C)-measurable, R, being the components of R2 into which R2 - C is divided by C, hence
E,,cf+f- ICI = QJ.f+ I’3EJ.L
IO
However p” is not Euclidean invariant, hence the Markoff generalized random field (4,~“) is not homogeneous. The interesting question is now whether one can use pu, to
S. ALBEVERIO
234
and R. H@EGH-KROHN
construct a homogeneous Markoff field. As mentioned above one can indeed show that there exist Euclidean invariant measures p on S’(R2) which are weak limits of pu, as Al R2. This is so in particular for u of one of the forms u(s) = Ilchasdv(a), u(s) = 2 ;
a,s j , a2n > 0, i/a,
suppv c (-2&,2&),
i~R[26],
small [25] a), L;(S)= Acos(as + Q), a2 < 2&,i
small
j=O
[27], (and suitable superpositions thereof). For further references concerning such (and related) Euclidean measures see e.g. [ 191. In particular, they are known to be non-Gaussian. Other Euclidean measures are obtained replacing p” 1B(A) in the definition of p,, by fi?no, where dpop,, is the Gaussian measure with covariance (- A?,, + m2)- ‘, where d;,, is the Laplacian with self-adjoint boundary conditions on 8 /1 (e.g. Dirichlet ones, see e.g. [36]). More generally, one can consider a probability measure p (Gibbs state for the interaction U,) on S’(R’) such that pIR(n,) coincides with ,u~ for any /b open in /i. The question of uniqueness of ,U (for given v, hence p,,) arises. A general uniqueness result of this kind is described by Dobrushin and Minlos in [37] for the case of polynomial interactions (with a sketch of proof). A complete proof for the case of trigonometric interactions is provided by us in [34]. Moreover, we show in this case that p also has the global Markoff property [34]. Let us now describe these results, starting from the one about uniqueness. 2. The proof of uniqueness of the Gibbs state and the global Markoff property for Euclidean fields with trigonometric interaction To prove uniqueness of the Gibbs state we first make the following elementary remarks. Let G(pn) be the set of Gibbs states (probability measures) on {pcl,,/1 bounded open} and call ALEG(p.,,) extreme if it has the property that VEG(pCln)and absolute continuity of v with respect to p implies p = v. Let B, = n B( A), A bounded AEB,
b-p@)
open,
2 = R2 - A_ Call
B,
trivial
for p if
:O,l.
1. We have: (B,
trivialfor
p)q(p
extreme).
Proof If p is not extreme then there exists a function h 2 0, hEL’(dp), h $1, such that hpEG(p,,). But by the definition of conditional expectations, E,(hl &fp I B(A) = h,u I B(A), but PLEG(~~), hence E,(hl A) = E,,(hl A), thus hp I B(n) = E,,(hl n)p I I{( A), thus h = E,,(hl
?i) ,u-a.e.,
i.e. h is B( &measurable,
hence h is B,( = n B( 2?))A
LOCAL AND MARKOFF
FIELDS
235
measurable. By assumption h + 1,~a.e., i.e. 3 E > 0, AEB, s.t. p(A) # O,l, h(r) 2 E, for
E,(W)
’ g conditional expectations, b em
with respect to B,.
Proof: Using EPA = E, on BAo, A0 c A (since ,UEG&J,
we see that E,,(.j ;i) is a monotone decreasing family of projections in L’(dp) (the family A being partially ordered by inclusion). Thus ENA(*I;i) converges strongly as operators in L2(dy) to a projection F. Then E (-1CD) d E_,(./ A), thus E,(.l “c) 6 F. On the other hand, for any gEFL2(dp) we have SEE,,” (.I A,,)L2(d,u) for all n sufficiently large thus g is B(a)measurable, thus B,-mea&able i.e. in the range of E,(.lco), which then implies F d E,,(.lco), so that with the former inequality in the other direction we get F = E,(*l a):q.e.d. 3. For arbitrary subalgebras E,(.lBo)
B, and measures
trivial for v.
= E,(.)==Bo
Proof: =c-: If v(A) # 0,l for some AEB,
v one has:
then E&)
# 0,l. But E,(x~(B,)
= xA
= OJ, thus x,., #E&J&, contrary to the assumption. =:v(A) = 0,l for all AEB, implies E,(xAIBo) = xA = E&J. By taking limits of sums of characteristic function this implies E,(.IB,) = E,(s). 4. E,A(.ln)JE,(*)oB,
trivialfor
p.
Proof: 3 : Immediate from 3. =: From 3 we have E,(.lx) = E,(.) 5. p extreme
= B,
and then, by 2,E,,,(.l;i)lE,(.).
trivial for p.
Proofi If B * is non-trivial for p then 3 A E B 1 , y(A) # 0,l. Thenx, is B , -measurable, xA $ 1 and xAp <
6. p extreme o E,(.l,$JE,(*). Proof: By 1, 4, 5. 7.
E,(~l~)lE,(9.
In order to prove this the following main observation is used. By the fact that p is a Gibbs measure we have Efl(fl A) = EPA (JI 2) for any B,,-measurable function, A, c A (“the local Markoff property” of p [ll], [ 121). Now observe that
~%,(fl~) = EpO(fe-“AI~)l~,de-“nl A). From the local Markoff property
of ~1’ we have
E,,(ff-““I-$
= Epo(fe-““IdA).
236
S. ALBEVERIO
and R. HGEGH-KROHN
We now use the following basic observation ([34]; related ideas are in [37] b), [S]) E,o(g18A)(p) = Ep;A(g + $dn), for p-lo-a.e. y and any gEL’(dp’), where
the integration being in the sense of generalized functions, where P,(x,dy) is the Poisson kernel for the Dirichlet problem (- A + m’)h(x) = 0 in Ah(x) = h,(x), h,~CF(ii A), for x~dA (so that h(x) = j Pc(x,dy)h,(y)). Using this we get PA
where for any g EL’ (dp’) we define g”“V (0 = g(
- UC;:,,)) we
= Ep,,,(8,“’ ).
Thus to show E,,,(*[ n)J E,(.) it is sufficient to prove that ~5:~ converges weakly as AtR2 for p-a.e. q andpVAconverges tofin the sense of distributions, for p - a.e. q, as A tR2. These results have been indeed proven by us [34] in the case where v is a trigonometric interaction with A2 < 27c and i sufficiently small. The proof of the weak convergence of &fV as At RZ is rather technical and involves an extension of the cluster expansion method of Frohlich and Seiler [27] (the reason being that py,, is the measure for the interaction UyV(<) EEU,(< + $v), which is not U, itself but its translate by $i;n) (Frohlich’s and Seiler’s method is based on the fundamental paper by Glimm, Jaffe and Spencer [25]a) on the cluster expansion for polynomial interactions). The proof thatp$A
for some subsequence A,,,t R2 such that exp( - m’d(A,,84,)) A,,? R2, a, c A,,, by using the inequality E,
s I$,““(x)12dx *0
< ue-‘n’df’@n)l~j >
--, 0 for some m’ > 0, as
LOCAL AND MARKOFF
FIELDS
237
for any & c /1, /lo n being open bounded regular a some constant, which is proven by using the regularity property of p together with potential theoretical methods, see [34] for details. Note that this basic ingredient, namely that tj”;’ -+ 0,is model independent, in the sense that all what is used is that one has a regular random field measure p. We have thus the following result [34]. THEOREM 2.1. Let tx2 < 271then there is a 1, > 0 depending on CI(and m, > 0) such thatfor 111 < 1, ifn~S’(R~) is such that $,8”(x) --+0 locally uniformly as AT R2 (and this
is the casefor p-a.e. n~s’(R~)) then d,u$ z e-“ncT)dpOan(~l~)/E(e-“~l8A)(~) converges weakly as AT R2, where ,tt is the Euclidean measure for the trigonometric interactions constructed in [27] (Gibbs measurefor the interaction given below). Here ,t.P”(<1n) is the free Euclidean field with condition 4 = n on 8A (such that jf(t)dpO”“([(q) = = jf(e + ti”,”)dpouoa”(5), dcl0~ being the free field measure with Dirichlet boundary condition on 8 A). Moreover,
there is one and only one regular measure p, regular in the
above sense, which is a Gibbs statefor
U, = Lj:cos(aS(x)
state (equivalently: the o-algebra at infinity*B, EJ.1 A) -+ E,(o) as nt R’).
+ 0):dx. ,u is an extreme Gibbs
- n Bx is trivial for p; equivalently:
Remark. This proof gives in particular the uniqueness of solutions of the DLR equations [38], [20]. Uniqueness results of this type in models of statistical mechanics have been proven particularly by Dobrushin, see e.g. [14] and [39]. Independence of particular boundary conditions, (obviously properly contained in our class) has been proven for Pi models in [36] b), c). We now turn to the proof of the global Markoff property for the Euclidean measure p. Again we restrict ourselves to the above case of trigonometric interactions. We want to show that
EJf+f-
IC) = E,cf,
IC)E,(f-
I C)
for arbitrary fk which are bounded and B,* -measurable and arbitrary (piecewise C ‘) curves C (not necessarily bounded), separating R2 into two regions A+ u C and /i_ u C with /1+ n A_ = Q. Using now that p is a Gibbs state (measure) we see that E,(f 1Av C) = Eqn(fl AvC) for any f which is BAO-measurable, no c /1/1 bounded open, hence using that ,u,, is global Markoff, so that E,,(f+ f- 1/iv C) = E,,(f+ ) AvC) E,“Cf_ I Av C) whenever fk are B, O-measurable, with AC = A, n A’, we see that it is enough to prove that EWn(fl AV C) converges as AT R2, for any f which is Bno-measurable for arbitrary open bounded A’, to E,(f 1C). Now one uses the fundamental formula E,,(f
I Av C)(v)= E,o(fe~uAIAv Ch)IEpo(_UAl Av C)(v) = Eflo(f ::,“A e_‘C,il””)/EWo(e-“%bd ) = E,,,9CU
aA(f,,c”an)
S. ALBEVERIO
238
and R. H@EGH-KROHN
(similar to the one derived before for Au C replaced by /1, and proven in the same way), with UcUaA >f f;UJ” , pC,“f” defined as the corresponding quantities defined before for the case% = $9,i.e. with t+QUaA replacing now I@. As for the proof the uniqueness it is thus enough to prove that ~5%~~ converges weakly as /It R2, for p - a.e. q (which we have proven [34] for the weak trigonometric interactions we are considering by the cluster expansion method for the interaction UzU,“” instead of LJaA n,rt ) and, on the other hand, that i@Jafx) --*$:(x) as /iT R2, locally uniformly in x for all q ES’(R’) (which is proven similarly as 1+9? 4 0 by potential theoretical methods, for any regular random field measure p). See [34] for details. Hence we arrive at the following result:
THEOREM
2.2. Let ~1~< 2n. Then there is a I, > 0 depending only on CI(and m, > 0)
such that, for IAl < 3L1, the unique Euclidean interactions
(of Theorem
Gibbs measure
2.1) has the global Markoff
u of the trigonometric
property
in the sense
that
E,(f+fIC) = E,(f+ IW,cf- I’2 f or any piecewise Cl-curve (which can extend to infinity) dividing R2 into two regions Q, UC and (;2_ UC with Sz, n& = $4 and uny integrable functions f+ which are B,,-measurable, BA, being the o-algebras generated by p -+ ([,p),p being any measure such that suppp c A, and Jdp(x)dp(y)G(x -y)
Let us now make some remarks concerning the proof of the global Markoff property. It might be tempting to think that once the uniqueness result (Theorem 2.1) has been proven the global Markoff property would follow immediately. This is however not so in general (what is true is that the proof of the global Markoff property is similar to that of the uniqueness). The difficulty consists in the following. The uniqueness theorem implies that the o-algebra at infinity B, is trivial for p. However this does not imply (unless the Bore1 o-algebra at infinity is countably generated, which is too strong an assumption (counterexamples to similar situations are known, e.g. [40])) that B, is trivial for the conditioned measure pc(t Iq) (which exists, incidentally, by the fact that S’(Rd) is a Suslin space [41]) for ,u - a.e.q. In fact for each A E B, one has p&4 1~) = 0 for all q 4 S,, p(S,) = 0, but the exceptional set S, depends on A, hence one can not deduce in general ,u&*(~) = 0,l for all ~ES,S some set, with p(S) = 0 (unless B, is countably generated). The global Markoff property is however equivalent to EJ.1 Au C) +E,(*IC), hence to the triviality of B, for pc. Note that in general one has, by monotonicity, convergence in the sense of L2(dp)-projections, of Ep(.l Au C) to some projections in L’(dp) as ArR*. The identification of the limit with E,(.jC) needs however further work (like that done above), unless B, is countably generated. If the Bore1 o-algebra at infinity were countably generated we would have the following much simpler situation:
239
LOCAL AND MARKOFF FIELDS
LEMMA. Let (X, B,p) be a measurable space. Let B, be a net of sub-a-algebras, directed so that $X < a’, B,, c B,, then there exists a”, X’ < ci’ such that B,.. c B,, c B,. Let B” be afixed sub-o-algebra
such that the regular disintegration pso of p with respect
to B” exists (i.e. ECfl B’)(q) = Jf(S)dpao(<[q),for p 1B” - a.e.q, with pL,o(
,LL-
generated.
IThen, for
i.s. and strongly in L’(d,u) to E,cfjB”),
any fE L’(dp),
E,,(flB,V
B”)
as c1-+ x.
Proof:
1. For any sub- o-algebra B’ and any measure v one has, as observed above, E,(.lB’) = E,(e) o v(A) = 0,l for all A E B’. as a -+ co, as projections in L*(dv) (by monotonicity). 2. E,(.lB,) 1 U.lB,) 3. E,(.(B,)JE,(.)
as projections
4. For any AEB&,&I) disintegration).
in L*(dv)ov(A)
= jp,o(Alq)dpBO(q)
5. p(A) = 0,l for all AEB,
= 0,l for all AEB,.
with d,uBo = d,u 1B” (by the regular
=pLBO(Alq) = 0,l for all YES, with $‘(S,)
= 0.
6. B, countably generated implies pBo(A 1~) = 0,l for all q E S and all q ES and all AEB,, with $‘(S) = 0, S independent of A. 7. By 1 for pBOinstead of v we get EpgO(.IB,) = E,,, (.I 8,) 8. Identify (using the definition) 9. Conclude
E,,,(.Iq)(.IB,)
with E(.(B,v
= EpSO(.). B’).
the proof of the Lemma using 6, 7, 3 and 8.
This lemma then shows, that if the Bore1 a-algebra at infinity were countably generated, E,,(.( /iv C) -+ E,(.(C), which by what we saw above would imply the global Markoff property. It can also be proven that, for generalized random fields on Rd, the local Markoff property and “the property of short range correlations in d independent directions” (in the sense that E(.l/1;)1 E(-) as 14 co, as projections in L*(dp), with J 3 {x~ Rdllxal 2 l} for d independent vectors c( in Rd) together with the countable generation of the Bore1 o-algebra at infinity would imply already the global Markoff property [42]. In turn any measure which is locally Markoff, Euclidean invariant, satisfies the Osterwalder-Schrader positivity condition [9] and the cluster property (in the sense of the intimum of the spectrum of the Hamiltonian being a simple eigenvalue) would already imply the property of short range correlations in d independent directions, hence, by the above, it would also imply the global Markoff property [42]. Remark. 3. If we take C = (x’,x*)E R*)x’ = 0}, i.e. C is the “time zero” axis in R*, in Theorem 2.2, then we see that “the global Markoff property for hyperplanes” [ 111, [18]-[20] is a particular case of the global Markoff property proven above, for trigonometric interactions [34]. The weaker property E + E _ = E, = E _ E +, with E f
240
S. ALBEVERIO
and R. H@EGH-KROHN
defined as E(.ISZ”,), Sz, = {(x’,x2)~R21x1 >
S’(R)
R
where V is the closure of the gradient FC’
s
dxd/Jo(O = 3 Ff 12dPO>
t
in L2(dpo)! L’(R)
operator
defined as a map from
(i.e. (Bf)(c,)
df
=
md~). s
This Dirichlet form can be identified with a regular Dirichlet diffusion form in the sense of [5]a) on C(X), where X is the spectrum of the uniformly closed algebra F C2 in L”(dpu,). For these results see [5] b), [5]c) (part of these results is written for the case of to the case of trigonometric polynomial interactions; however the adaptation interaction is easy). We have dx= R
-$A-p.V,
LOCAL AND MARKOFF
(this is the mathematical
241
sense in which formally Hz_
s +--_
a2
Ghd2
R
s
dx + &,(x))dx R
over the “flat space” (S’(R),l?d<,,(x)) (See [5] for this point). x the infinitesimal generator of the Lorentz boost is given on FC2 by
is a Schrodinger Analogously,
FIELDS
operator
(4X50) = -
+x
& +B(C,(x))&)dx.
R
For more details see [5], [24]. 4. p,, is S(R)-quasi invariant, abelian group S’(R) in L2(dpo)
hence there exist unitary representations of the by multiplication cp+ ei(e’~@)and translation cp
+(e”(@“f)(&J = [do&, + GQGJ11’2f(~0 + cp) and (5~)~ I satisfy the Weyl commutation relations. rc((p) is essentially self-adjoint on {ei(cb@), $ES(R)) and coincides with the physical momentum operator. This gives the whole “Schrodinger interactions. The representation” of canonical field theory3 with trigonometric equation of motion is i[n(qo),H] = :0’:(q) + (.,( - d + m2)q) with:v’:(q) = - a;lJ: sin(ax + O):q(x)dx ([5], [24], [34]). 5. HP0 is a.n elliptic operator in infinite dimensions, one expects the ground state to have the usual properties of ground states of elliptic finite dimensional operators (Frobenius theorem, regularity properties). In fact one has that the inlimum of the spectrum of HP0 in L’(dp,) is zero, simple, isolated, with eigenvector Q(tO) = 1. Szis an analytic vector for 7t((p).~~(5~ + TV)is analytic in t and CL,,is strictly positive (in the sense that the conditional measures with respect to arbitrary finite dimensional subspaces have strictly positive density with respect to Lebesgue measure, on compacts) (for these results see [5] b)). 6. The process t + qt associated with the Dirichlet form iJ( yf12dpo can be realized as a strong Markoff, in fact a Hunt process with state space differing from X by a polar set. Both 5, and qt satisfy the stochastic equation d<, = /&,)dt + dw,, where w, is the Brownian motion on S(R) c L2(R) c S’(Rd), and have the same infinitesimal generator on FC2.([5]a)-c)).
3The trigonometric interaction models are the first class of models for which the ideas of canonical quantum field theory (like those expressed e.g. in [46]) are verified.
242
S. ALBEVERIO
and R. H@EGH-KROHN
Remark. We shall end with a remark concerning the global Markoff property for lattice systems. In [48] we have shown, in collaboration with G. Olsen, that for a class of lattice systems for which Dobrushin’s method [ 141 a)-c) yields uniqueness of Gibbs states one can also prove the global Markoff property, adapting the method used by us for Euclidean fields. Let us shortly describe our result. Consider a lattice Zd in d-dimensions. Let @ be a (finite range) interaction on Zd i.e. a real-valued function defined on the subsets of Zd, with support on finitely many finite subsets of Zd and such that for A c Zd, @(A) = @(- A). Define, for any subset A c Zd, 11@(A) (1= J.JcP((/i)l(lAI- l), where the sum is over all finite subsets of Zd. Let oi be the function oi(A) = 1 if i is a point in A and ~i( n) = 0 if i$ A and let B, be the a-algebra generated by the oi,ie A. Define the interaction measure (note the analogy with the measures pn for the
Euclidean dpA(a)
fields
in the
E exp(-
bounded
C OiC @(A’) II itA
j-
A’
itA*
open
where da(a)
GE ll {O,l}(i, by
iEz*
oj))d~(o)
C OiC@(A’)
itA
A c Rd)dpn(a),
region
A’
fl
j- ic,l’
= II dcci( cri), d cli being a probability
oj)da(a)
1 -l,
measure on {O,l},i,, the i-th copy of
i&
the 2-point space {OJ}. Note that oic @(A’) n
,
crj is the interaction
between the
lattice point i and the rest of the la&e (on&;i;‘,se points j give a non-vanishing contribution which are such that they can be obtained by adding to i the vector corresponding to a point in some of the finite sets on which @has support. E.g. if, for d = 2, @ has only support on the finite set {(O,O),(- LO), (OJ), (1,O)) then only the nearest neighbors ofj contribute. ,Uis called a Gibbs measure to (plno, A,, bounded} if E,Cfl Jo) = &I,(fl A) for any bounded shin’s results about
existence
lows that if 11@/I< ie-’ limE(fl$ d
= E(f)
Bore1 function f on II {O,l},,. From Dobru-
and uniqueness
of Gibbs measures
then p is unique, p I B,
(this is the corresponding
[14]a)-c)
it fol-
is trivial, where B,, z n Bn, and
statement
for lattice systezs
to our
%iqueness result Theorem 2.1). We shall now see that analogous considerations as in the case of Euclidean fields lead to the global Markoff property. To state the latter (and the corresponding local Markoff property) we consider decompositions Zd -= Q + u C v ii_, where Q +, C, Q _ are subsets of Zd which are pairwise disjoint
LOCAL AND MARKOFF
FIELDS
243
and D +, O_ are such that one cannot reach any finite subset of R, from Q_ by translation by vectors corresponding to a set in the support of @(i.e. there is no nonvanishing interaction between the random variables of the regions Sz, and a_, and C separates Q+ and Q_ in this sense. E.g., if @is a nearest neighbors interaction, C must be such that between points of Sz+ and L?_ there must be at least one of C, if @is a next nearest neighbors interaction, between points of Q_ and Sz, there must be a least 2 of C etc.). We say then that C insulates Q + and s2 _ . As for Euclidean fields it follows easily, see [48] for details, that the local Markoff property E,(f+f_ IC) = E,(f+ jC)E,(f_ IC) holds, where Q + is finite andf, are Q k -measurable, and ,Uis any Gibbs measure, and C insulates Q + and Q _. Moreover, the application of the ideas used for proving the global Markoff property for Euclidean fields, described above in the case of trigonometric interactions, yields first Ercc_,V,Cf+f-I JoI = &d_f+
I ‘JoKL,dL
Ia,
where ,~+(.j~) is the conditional probability measure defined by /J and the o-algebra B,. By the uniqueness result of Dobrushin, applied this time to the Gibbs measure &.lvl), under the assumption 11@I(< fe- ‘, we get that E,cc+,,(~l&) + EPc&.) as /i,, r Zd, and then we get fi,~+JJ+f- 1 = Epd.Iqd_f+)E,,tl,df), CL- a.s., i.e. E,cf,fI WI) = E,(f+ I C)(n)E,(f_ IC)(q) for p - a.e. q, which is the global Markoff property, s2 +, C, Sz- being arbitrary (not necessarily finite) such that C insulates Q, and Q_. We thus have the following THEOREM.
< $e-I.
Let @ be any interaction offinite range on the lattice Zd, such that //@II Then to the interaction measures
{P~,~}, dP~(~)=Z~‘exP(- C Oix @3(A) n k,l
,4'
oj)do(aL
j- isA’
where Z, is the number such that dpA is a probability measure, and do is any product of probability measures on {O,l), there is one and only one Gibbs measure ,tt,as proven by Dobrushin. Moreover,for s2, from
any disjoint partition Zd = Q+ v C v Q _ such that C insulates
52_ with respect
H iO-'I,i,eft BQ+-measurable,
to the interaction
and any bounded
we have the global Murkof
Bore1 function
on
property
isz"
EJf+fE,(.IC)
IC) = EJY+ IC)E,,cf-IC),
being the conditional expectation
with respect to p and B,.
The global Markoff property can be used to deduce consequences corresponding to the one we derived for the case of Euclidean fields. In particular, one has, for any splitting Zd = Z x Zd- ’ with i = (k,l), ke Z, 1E Zd- ’ a symmetric stationary Markoff chain k + & with values in {O,l} and invariant distribution p 1B,, C being the
244
S. ALBEVERIO
and R. H@EGH-KROHN
hyperplane k = 0. The corresponding symmetric transition matrix e-H, H > 0 on L2(d,u 1B,) is the “Onsagerian” (“transfer matrix”) postulated in many papers of classical statistical mechanics (see especially [49]). The relation of this transfer matrix and the one constructed using an adaptation [50] of Osterwalder-Schrader method is entirely similar to the one discussed above concerning the relation of the global Markoff property of Euclidean fields and Osterwalder-Schrader construction of the physical Hilbert space and the Hamilton operator. Remark. Recently J. Bellissard and P. Picco have extended our results on the global Markoff property to more general lattice systems, using their uniqueness results for such systems [39] c) (as for uniqueness results see also [39] a), [39] b). H. Follmer also remarked [51] that the global Markoff property for lattice systems follows whenever Dobrushin’s uniqueness conditions are satisfied using similar ideas as in our work [48]. Moreover, he shows that if the interaction is attractive and admits a “high density state” and a “low density state” then each of the states has the global Markoff property. Added in proof: The global Markoff property has been proven recently for exponential interactions by R. Gielerak (J.M.P. 24 (1983), 347-355). For further recent references see the authors’ contribution to Adu. Prob., ed. M. Pinsky, M. Dekker (1983).
Acknowledgement )JJe are very grateful to Professor Witold Karwowski and the Organizing Committee of the Karpacz 1978 Winter School of Theoretical Physics for a very kind invitation, where the first version of this work was presented in lectures, and for doing everything to arrange an extremely pleasant stay in Karpacz and Wroclaw. We are also grateful to the Centre de Physique Theorique, CNRS, Marseille and the UER, Universite d’Aix-Marseille II, Luminy, for their hospitality and financial support.
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