Space–time large deviation lower bounds for spin particle systems

5 November 2001

Physics Letters A 290 (2001) 65–71 www.elsevier.com/locate/pla

Space–time large deviation lower bounds for spin particle systems Jinwen Chen Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Received 27 June 2000; received in revised form 9 August 2001; accepted 27 September 2001 Communicated by A.P. Fordy

Abstract In this Letter we study large deviation lower bounds for space–time empirical processes of spin particle systems with possibly vanishing spin flip rates. We present an approach for obtaining such lower bounds for systems starting from almost all configurations w.r.t. an translation invariant and extremal invariant measure of the systems, so that in such cases we have full large deviation principle.  2001 Elsevier Science B.V. All rights reserved. Keywords: Large deviation; Interacting particle system; Ergodicity

1. Introduction In [1–4], full large deviation principle (LDP) for space–time empirical processes of spin flip particle systems with strictly positive, translation invariant and finite range spin flip rates were obtained. A corresponding variational principle gives that the stationary Markov measures of such systems are just the zeros of the LD rate functions. The similar problems were also studied for systems with vanishing flip rates. But in this situation, we have not shown that the lower and upper bounds can be governed by a same rate function. We also explained that we may not have full LDP starting from some initial distributions. Thus an interesting problem is to consider that starting from what initial distributions we can obtain full LDP, since this will more explicitly reflect ergodic and fluctuation behavior of the systems. In this Letter we present an approach that implies an application in obtaining such a LDP. Since in [2] and [3] we already had very general upper bounds, thus what we really need to do is to give the accompanying lower bounds. The initial distributions we will consider are relevant to some extremal invariant measures of the systems. Since an approximation procedure shows that almost all the results in [1–5] can be extended to the long range cases, we will work for general spin systems. d Now we define the systems and objects we will consider. Let E = {0, 1}Z and {c(i, ·), i ∈ Z d } be a family of nonnegative continuous functions on E. In this Letter we assume this family be translation invariant, i.e., there is a nonnegative continuous function c0 (·) on E such that c(i, η) = c0 (θi η),

∀i ∈ Z d , η ∈ E,

E-mail address: [email protected] (J. Chen). 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 6 4 7 - 8

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J. Chen / Physics Letters A 290 (2001) 65–71

where θi is the shift operator on E defined by (θi η)(j ) = η(i + j ) for j ∈ Z d . We also assume that
    supc0 ηi − c0 (η) < ∞, i∈Z d

(1.1)

η

where ηi ∈ E is obtained from η by flipping the spin at site i, i.e., ηi satisfies ηi (j ) = η(j ) if j = i and 1 − η(i) if j = i. Then it is well known [6] that there corresponds a unique Feller–Markov process {Pη∞ , η ∈ E} on the path space Ω+ = D([0, ∞), E), the space of cadlag functions from [0, ∞) to E, equipped with the Skorohod topology and the corresponding Borel σ -algebra. The generator L of the process acts on a local function f on E (i.e., there is a finite subset Λ of Z d such that f (η) depends only on {η(i), i ∈ Λ}) as
    Lf (η) = c(i, η) f ηi − f (η) . i

This intuitively means that the probability rate for flipping the spin at site i in configuration η is c(i, η), and that at any time, such spin flip is allowed at at most one site. This Markov process is called a spin system with spin flip rates {c(i, ·), i ∈ Z d }. The system is said to be of finite range, if there is a finite subset Λ of Z d , such that c0 (η) depends only on {η(i), i ∈ Λ}. The system is said to be ergodic, if there is a unique invariant probability measure ν and for each initial distribution µ the law µt = Pµ∞ (ωt ∈ ·) of the system at time t converges weakly to ν as  t → ∞, where Pµ∞ = Pη∞ µ(dη). One of the main purpose for studying the LDP is to get better understanding of the fluctuation behavior of the system around its stationary states. To do this, we can use the empirical processes defined as follows. For t > 0 and i ∈ Z d , we first define the space–time shift operator θt,i on Ω+ by (θt,i ω)s (j ) = ωt +s (i + j ), s
0, j ∈ Z d . For n
1, define Λn = {i ∈ Z d , −n < ij  n, 1  j  d}. Then for ω ∈ Ω+ and m, n
1, define the space–time (m, n)-periodic element ω(m,n) of ω as (m,n)

ωt +ms (i + nj ) = ωt (i)

for 0  t < m, s
0, i ∈ Λn , j ∈ Z d ,

where nj = (nj1 , . . . , njd ). Now for m, n
1 and ω ∈ Ω+ , the empirical process is defined by
 1 Rm,n = Rm,n (ω) = δθt,i ω(m,n) dt, m(2n)d m

i∈Λn 0

where δω is the usual Dirac measure. It is easy to check that Rm,n ∈ Ms (Ω+ ), the space of probability measures on Ω+ which are shift invariant under each θt,i . Ms (Ω+ ) is equipped with the weak topology. In the finite range cases, the large deviation principle for {Pη∞ (Rm,n ∈ ·), m, n
1} and the corresponding variational principle are studies in [1–4]. The main results obtained are summarized as follows. Let {Pη , η ∈ E} be the spin system with c0 ≡ 1. This is a special noninteracting spin system which were used as a standard reference system. Denote Fp = σ {ωt , t  0}, the σ -algebra generated by history before time 0. For Q ∈ Ms (Ω+ ), let Qω be the regular conditional law of Q given Fp . Then we have Theorem A. If a translation invariant spin flip system {c(i, ·), i, η} is of finite range, then there is an affine and lower semi-continuous (lsc) function H∞ from Ms (Ω+ ) to [0, ∞] with compact level sets (i.e., ∀a
0, {Q, H∞ (Q)  a} is a compact subset of Ms (Ω)), such that for every closed subset F of Ms (Ω+ ), 1 log sup Pη∞ (Rm,n ∈ F )  − inf H∞ (Q). d Q∈F m(2n) m,n→∞ η lim sup

If, in addition, c0 is strictly positive, then for every open subset G of Ms (Ω+ ), lim inf

m,n→∞

1 log inf Pη∞ (Rm,n ∈ G)
− inf H∞ (Q). η Q∈G m(2n)d

J. Chen / Physics Letters A 290 (2001) 65–71

Furthermore,

67



1 H (Q) − E Q [1 − c0 (ω0 ) + 0 log c0 (ωt − )N0 (dt)], if H (Q) < ∞, (1.2) +∞, if H (Q) = ∞,

where H is used for H∞ when c0 ≡ 1, ωs− is the left limit in time, Ni (t) = k
1 I{τk (i)t } is the total jump times at site i up to time t, and 0 ≡ τ0 (i) < τ1 (i) < · · · < τk (i) < · · · are the successive jump times of ω at site i. Moreover, H∞ (Q) = 0 iff Qω = Pω∞0 Q-a.s. In other words, H∞ (Q) = 0 iff Q is a stationary Markov measure on Ω+ with {Pη∞ , η ∈ E} as its regular conditional laws. H∞ (Q) =

As one can see, the results in the case when the spin flip rates are strictly positive, i.e., when inf c0 > 0, are quite satisfactory. The function H∞ turns out to be a natural candidate for a LD rate function even for systems with possibly vanishing spin flip rates. The upper bounds are included in Theorem A. Our aim of the present Letter is to give the accompanying LD lower bounds. In these cases, conditions on the initial configurations are reasonably required. We will consider those initial configurations, which are relevant to some extremal invariant measures of the systems. As we stated in the first paragraph, an approximation procedure using a sequence of systems with strictly positive and finite range flip rates shows that almost all the above results in Theorem A can be extended to long range cases, so we will work for all translation invariant systems satisfying (1.1). Given a spin flip systems satisfying (1.1). Denote by m1 (E) the set of all probability measures on E, mi (E) the set of all invariant probability measures of the system and ms (E) the set of all translation invariant probability measures on E. (mi (E) ∩ ms (E))e is the set of extremal points in mi (E) ∩ ms (E). For ν ∈ m1 (E), let mcν (E) be the set of measures in m1 (E) which are absolutely continuous w.r.t. ν. For Q ∈ Ms (Ω+ ), let µQ be its single time marginal, i.e., µQ (A) = Q(ωt ∈ A) for any t
0, A measurable in E. Then our main result is the following Theorem 1. Given a translation invariant spin system which satisfies (1.1), and define H∞ by (1.2). Let ν ∈ (mi (E) ∩ ms (E))e . If H∞ (Q) < ∞ implies that is a sequence Qn ⇒ Q with µQn ∈ mcν (E), then for ν almost every η ∈ E, for every open G ⊂ Ms (Ω+ ), we have 1 log Pη∞ (Rm,n ∈ G)
− inf H∞ (Q). Q∈G m(2n)d ∞ In particular, (1.3) holds with Pη replaced by Pν∞ . lim inf

m,n→∞

(1.3)

Remark 1. A condition stronger than the one used in Theorem 1 is that H∞ (Q) < ∞ implies µQ  ν. This condition maybe more practical when one wants to check it. A possible way to verify this condition is to show that H∞ (Q) < ∞ implies I (µQ ) < ∞ and that I (µ) < ∞ implies µ  ν, where I is the usual Donsker–Varadhan functional defined by  Lf , I (µ) = − inf f ∈Cb (E), inf f >0 f and L is the generator of the system. Furthermore, since I (µ) < ∞ implies µ  νρ , for any ρ ∈ (0, 1), where νρ is the product probability measure on E with marginal density ρ, we see that if H∞ (Q) < ∞ implies I (µQ ) < ∞ and νρ  ν for some ρ ∈ (0, 1), then the condition of Theorem 1 is satisfied. 2. Proof of Theorem 1 In this section we prove Theorem 1. Obviously we only need to prove that for Q ∈ Ms (Ω+ ) with H∞ (Q) < ∞ and µQ  ν, and for any open G containing Q, for ν almost all η’s in E, lim inf

m,n→∞

1 log Pη∞ (Rm,n ∈ G)
−H∞ (Q). m(2n)d

(2.1)

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J. Chen / Physics Letters A 290 (2001) 65–71

Then since E is a Polish space, it is easily seen that we can determine a measurable subset E0 of E with ν full measure, such that for each η in E0 , (2.1) holds for any Q with H∞ (Q) < ∞ and all open G  Q, i.e., we have ν almost sure full LDP. To make the notations simple, we assume d = 1 and will prove (2.1) in the case m = n → ∞. The proofs in general cases are the same. Now we prove (2.1) by several lemmas. Lemma 1. Let ν ∈ (mi (E) ∩ ms (E))e and µ is not singular w.r.t. ν. Then for every nonnegative and bounded measurable function f on E, there is c > 0 such that m n
1 ∞ lim E Pµ f (θi ωt ) dt
cν(f ). m,n→∞ 2mn i=−n+1 0

Proof. Since ν ∈ (mi (E) ∩ ms (E))e implies Pν∞ is space–time ergodic (see [2, Theorem 2]), there is E0 ⊂ E with ν(E0 ) = 1, such that ∀η ∈ E0 , m n
1 f (θi ωt ) dt = ν(f ) lim m,n→∞ 2mn

Pη∞ -a.s.

i=−n+1 0

Now since µ is not singular w.r.t. ν, we have µ(E0 ) > 0 and m n
1 ∞ E Pµ f (θi ωt ) dt
µ(E0 )ν(f ). m,n→∞ 2mn lim

✷

i=−n+1 0

Lemma 2. Let ν ∈ (mi (E) ∩ ms (E))e , Q ∈ Ms (Ω+ ) be space–time ergodic with H∞ (Q) < ∞ and µQ being not singular w.r.t. ν. Given an open G containing Q and a measurable A with ν(A) > 0. Then for fixed / and σ > 0, there is an integer T
1 such that ∀δ > 0, there exists A0 ⊂ E with µQ (A0 ) > 1 − δ, such that 1 1 log inf lim inf 2 m,n→∞ 2mn η∈A0 2σ mn

σ m




−σ n
1 du 2 2T

T T


dv Pη∞ (Rm,n ∈ G, θi+j ωm+u+v ∈ A)

j =−T +1 0


−H∞ (Q) − /. Proof. For ω ∈ Ω+ , by the Markov property and the translation invariance of the system we have 1 ∆(m, n, T ; ω0 ) ≡ 2 2σ mn 




−σ n
dPω∞0

= Rm,n ∈G

σ m 1 du 2 2T

1 2σ 2 mn




T T
j =−T +1 0

j =−T +1 0

σ m 1 du 2 T

−σ n
Thus we see that if we denote 1 gT (η) = 2 T

T T


dv Pη∞ (θj ωv ∈ A),

   dv Pω∞0 Rm,n ∈ G, θi+j ωm+u+v ∈A T T
j =−T +1 0

dv Pθ∞  iω

m+u

  θj ωv ∈ A .

J. Chen / Physics Letters A 290 (2001) 65–71

69

then it follows that for constant c > 0,  ∆(m, n, T ; ω0 ) = Rm,n ∈G


where

Γm,n =

1 2 2σ mn

dPω∞0

1 dQω dQω 2σ 2 mn




σ m    du gT θi ωm+u

−σ n
   cν(A) exp −2(1 + σ )2 mn H∞ (Q) + //2 Qω (Γm,n ), 2

(2.2)

σ m    cν(A) , du
gT θi ωm+u 2




−σ n
      dQω   2 log (ω )  2(1 + σ ) mn H∞ (Q) + //2 . dPω∞0 F(1+σ )m,(1+σ )n Since Q is space–time ergodic, it follows that 1 lim 2 m,n→∞ 2σ mn




σ m 1 gT (θi ωm+u ) du = 2 T

−σ n
T T


Pµ∞Q (θj ωv ∈ A) dv

Q-a.s.

j =−T +1 0

Since ν ∈ (mi (E) ∩ ms (E))e and µQ is not singular w.r.t. ν, from Lemma 1 we see that for some c > 0, if T is chosen to be sufficiently large, then 1 T2

T T
j =−T +1 0

Pµ∞Q (θj ωv ∈ A) dv
cν(A).

Furthermore, since Q is ergodic and H∞ (Q) < ∞, the proof of Proposition 4.1 in [4] implies that    dQω  1 log lim = H∞ (Q) Q-a.s. m,n→∞ 2mn dPω∞0 Fm,n Combining these we see that for Q almost all ω’s, lim Qω (Γm,n ) = 1.

m,n→∞

This together with (2.2) gives that 1 / log ∆(m, n, T ; ω0 )
−H∞ (Q) − 2mn 2 The desired conclusion follows from this easily. ✷ lim

m,n→∞

Q-a.s.



Lemma 3. Given ν ∈ (mi (E) ∩ ms (E))e . Let Q = kl=1 λl Ql with λl > 0, kl=1 λl = 1 and Ql ∈ Ms (Ω+ ) being space–time ergodic and satisfying that H∞ (Ql ) < ∞ and that µQl is not singular w.r.t. ν. Then for every open set G containing Q, (2.1) holds for ν almost all η’s. Proof. The cases k  2 are simpler than the cases k > 2. Thus without loss of generality, we prove the lemma Ql and integers in the case k = 3, this will keep the notations simpler. First we choose open sets Gl containing

M, N
1, such that if Ql ∈ Gl and ||Ql − Ql ||FM,N is sufficiently small for 1  l  3, then 3l=1 λl Ql ∈ G, where || · ||FM,N denotes the total variation norm of probability measures on the σ -algebra FM,N = σ {ωt (i), 0 

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J. Chen / Physics Letters A 290 (2001) 65–71

t  M, −N < i  N} generated by the states in the 2N × M space–time box. Now write nl = λl n. Then by the definition of Rm,n we know that if n is sufficiently large, then 3   Rnl ,n (θnl−1 ,0 ω) ∈ Gl ⊂ {Rn,n ∈ G}, l=1

where n0 is defined to 0. We further note that for fixed T
1 and sufficiently small σ > 0, if n is sufficiently large, 0  u  σ n, 0  v  T , |i|  σ n and |j |  T , then   Rn ,n (θn ,0 ω) − Rn ,n (θn +u+v,i+j ω) l

l

l−1

FM,N

l−1

is sufficiently small. Thus for η ∈ E and measurable Al ⊂ E, 1  l  3 (to be chosen later), if we denote  Ω1 = Rn1 ,n (θu,i ω) ∈ G1 , θi1 +j1 ωn1 +u+u1 +v1 ∈ A2 ,  Ω2 = Rn2 ,n (θn1 +u+u1 +v1 ,i1 +j1 ω) ∈ G2 , θi1 +j1 +i2 +j2 ωn1 +n2 +u+u1 +v1 +u2 +v2 ∈ A3 ,  Ω3 = Rn3 ,n (θn1 +n2 +u+u1 +v1 +u2 +v2 ,i1 +j1 +i2 +j2 ω) ∈ G3 , then for any 0  u  σ n1 , |i|  σ n1 , 0  ul  σ nl , 0  vl  T , |il |  σ nl and |jl |  T , l = 1, 2, we have 3 

Ωl ⊂ {Rn,n ∈ G}.

l=1

From this, the Markov property and the translation invariance of the system we obtain Pη∞ (Rn,n

σ n1
 1 1 ∈ G)
du 2(σ n1 )2 2(σ n1 )2 |i|σ n1 0

×

1 2(σ n2 )2




σ n2 du2

|i2 |σ n2 0

σ n1 T 1
du1 dv1 2T 2




|i1 |σ n1 0

1
2T 2

T

|j1 |T 0

dv2 Pη∞ (Ω1 ∩ Ω2 ∩ Ω3 )

|j2 |T 0

σ n1
 1
Γn1 Γn2 Γn3 Pη∞ (θi ωu ∈ A1 ) du, 2(σ n1 )2 |i|σ n1 0

where Γn1

Γn2

1 = inf x∈A1 2(σ n1 )2 1 = inf y∈A2 2(σ n2 )2

Γn3 = inf

z∈A3




σ n1 T 1
du1 dv1 Px∞ (Rn1 ,n ∈ G1 , θi1 +j1 ωn1 +u1 +v1 ∈ A2 ), 2T 2

|i1 |σ n1 0




σ n2 T 1
du2 dv2 Py∞ (Rn2 ,n ∈ G2 , θi2 +j2 ωn2 +u2 +v2 ∈ A3 ), 2T 2

|i2 |σ n2 0

Pz∞ (Rn3 ,n

|j1 |T 0

|j2 |T 0

∈ G3 ).

∀/ > 0, according to Lemma 2, we can first choose A3 with µQ3 (A3 ) close to 1, such that lim

n→∞

1 log Γn3
−H∞ (Q3 ) − /. n3 n

(2.3)

J. Chen / Physics Letters A 290 (2001) 65–71

71

Since µQ3 is not singular w.r.t. ν, this A3 can be chosen to satisfy ν(A3 ) > 0. Then still by Lemma 2, we can choose A2 with µQ2 (A2 ) close to 1, such that ν(A2 ) > 0 and lim

n→∞

1 log Γn2
−H∞ (Q2 ) − /. n2 n

Applying Lemma 2 once again we can choose A1 with µQ1 (A1 ) close to 1, such that ν(A1 ) > 0 and 1 log Γn1
−H∞ (Q1 ) − /. n→∞ n1 n lim

For this A1 , by the space–time ergodicity of Pν∞ , we have σ n1
 1 lim IA1 (θi ωu ) du = ν(A1 ) > 0 n→∞ 2(σ n1 )2

Pν∞ -a.s.

|i|σ n1 0

Thus for ν almost all η’s, σ n1
 1 lim Pη∞ (θi ωu ∈ A1 ) du = ν(A1 ). n1 →∞ 2(σ n1 )2 |i|σ n1 0

Combining these with (2.3) we obtain that for ν almost all η’s,
1 lim inf 2 log Pη∞ (Rn,n ∈ G)
− λl H∞ (Ql ) − 3/ = −H∞ (Q) − 3/, n→∞ 2n 3

l=1

since H∞ is affine. This gives (2.1). ✷ Proof of Theorem 1. Now the proof of Theorem 1 is a direct

consequence of Lemma 3. Let Q satisfy H∞ (Q) < ∞ and µQ  ν. Then as usually done, we can assume Q = kl=1 λl Ql , where each Ql is space–time ergodic, λl > 0

with kl=1 λl = 1 and µQl ∈ mcν (E). Furthermore, H∞ (Ql ) < ∞. Thus applying Lemma 3 we obtain the desired conclusion. ✷

Acknowledgement Thanks are given to the referee for helpful suggestions.

References [1] [2] [3] [4] [5] [6]

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