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Laguerre entire functions and the Lee–Yang property
Applied Mathematics and Computation 141 (2003) 103–112 www.elsevier.com/locate/amc
Laguerre entire functions and the Lee–Yang property q Yuri Kozitsky Institute of Mathematics, Marie Curie-Sklodowska University, PL 20-031 Lublin, Poland
Abstract Laguerre entire functions are the polynomials of a single complex variable possessing real nonpositive zeros only or their limits on compact subsets of C. These functions are employed to establish a property of isotropic (i.e., OðNÞ-invariant) probability measures on RN , N 2 N. It is called the Lee–Yang property since, in the case N ¼ 1, it corresponds to the property of the partition function of certain models of statistical physics, first established by T.D. Lee and C.N. Yang. A class of measures possessing this property is described. Certain connections of the Lee–Yang property with other aspects of the analytic theory of probability measures are discussed. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Exponential type entire functions; Laguerre entire functions, Nonpositive zeros; Cauchy problem
1. Introduction Discovered in the fifties by the famous physicists Lee and Yang [11,23], a property of the partition functions of certain models of statistical physics later was called the Lee–Yang property. Since that time methods based on this property appeared in statistical physics and quantum field theory (see e.g., [4,19]). The original variant of the Lee–Yang theorem was refined and extended in a number of papers (see e.g., [1,6,9,13,17,21,22] and the references therein).
q
Supported in part by KBN Grant 2P03A 02915. E-mail address: [email protected] (Y. Kozitsky).
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00324-7
104
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
As examples of the application of methods based on this theory we mention [6– 8,16]. A simplest version of the Lee–Yang theorem may be described as follows. Obviously, ! n n X X 1X UðzÞ ¼ exp z sj þ Jij si sj ; z 2 C ð1:1Þ 2 i;j¼1 s2N j¼1 is an exponential type entire function. Summation in (1.1) is taken over 2n n configurations s ¼ ðs1 ; . . . ; sn Þ 2 N ¼ f1; þ1g . The Lee–Yang theorem states that all zeros of U are purely imaginary provided all Jij are nonnegative. In statistical physics this function is known as a partition function of a ferromagnetic Ising model in an external magnetic field z, which describes a system of spins fs1 ; . . . ; sn g interacting via ferromagnetic coupling Jij P 0. Thus, for a symmetric probability measure l on R, the Lee–Yang property may be said to occur if Z Ul ðzÞ ¼ expðzxÞ dlðxÞ; z 2 C ð1:2Þ R
is an entire function, which possesses imaginary zeros only or does not vanish at all. Hence the measure l which P describes the probability distribution of the total magnetic moment mðsÞ ¼ nj¼1 sj of the ferromagnetic Ising model, defined in the following way: ! n 1 X 1X lðBÞ ¼ exp Jij si sj ; Z s:mðsÞ2B 2 i;j¼1 where B is a Borel subset of R and ! n X 1X Z¼ exp Jij si sj 2 i;j¼1 s2N possesses the Lee–Yang property. By means of the so-called Ising approximation, in [20] it was proved that the above fact implies the Lee–Yang property for the probability measure dmðxÞ ¼ C expðax2 x4 Þ dx with arbitrary a 2 R. Later, it was shown [4,19], that there exist even polynomials /, limx!þ1 /ðxÞ ¼ þ1, deg / ¼ 6, such that the probability measure dm/ ðxÞ ¼ C expð/ðxÞÞ dx;
ð1:3Þ
does not possess the Lee–Yang property. The search of the polynomials / for which the measures m/ possess the Lee–Yang property was stimulated by the attempts to construct polynomial models of quantum field theory [19], which
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
105
took place in the seventies and eighties. In a more general setting this problem may be formulated as follows. Problem 1.1. To find necessary and sufficient conditions for an even function / : R ! R, under which the probability measure m/ (1.3) possesses the Lee– Yang property, i.e., its function (1.2) is entire and has imaginary zeros only or has them none. Obviously, this is a particular case of a much more older problem of description of probability measures m on R, for which the characteristic functions um ðzÞ ¼ Um ðizÞ are entire functions possessing real zeros only. A partial answer to this question was given by P olya in his article [18] on the problem of zeros of Riemannian entire functions (for a detailed discussion of this problem, see [2]). Namely, he showed that the characteristic function of the measure (1.3) with /ðxÞ ¼ a cosh x, a > 0 is an entire function possessing real zeros only. In their brilliant paper, GolÕdberg and OsrtovsÕkii [5] have proved that a ridge entire function u, which is of finite order, (ridge means juðx þ iyÞj 6 juðiyÞj, all characteristic functions are ridge) possesses real zeros only if and only if it has the following infinite-product representation: uðzÞ ¼ C expðjz2 þ idzÞ
1 Y ð1 ck z2 Þ;
j P 0;
d 2 R;
k¼1 1 X
ck > 0;
ck < 1
ð1:4Þ
k¼1
which means, in particular, that its order is at most two. This establishes the form of the function Ul (1.2) for l possessing the Lee–Yang property. But the functions obeying the representations of this type are known as Laguerre entire functions [2,10]. Putting all these facts together one concludes that the Laguerre entire functions could constitute a proper setting for developing the notion of the Lee–Yang property. Another problem of this kind is how to generalize the notion of the Lee– Yang property to the measures on RN , N > 1. Of course, one can choose a direction in RN (i.e., a unit vector e 2 RN ) and set Z expðzðe; xÞÞdlðxÞ; z 2 C; ð1:5Þ Ul ðzÞ ¼ RN
where ð ; Þ is the scalar product in RN . Then, the Lee–Yang property of l could be said to occur if Ul would be an even entire function possessing imaginary zeros only or not vanishing at all. This approach was used in [3] where it was proved that for the probability measure dmðxÞ ¼ C expðaðx; xÞ ðx; xÞ2 Þ dx;
ð1:6Þ
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Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
the function Um (1.5) is entire and has real zeros only, that occurs for all a 2 R and N ¼ 2; 3. Let r stand for the measure which is concentrated on the unit sphere S N 1 RN and, being restricted to it, coincides with the uniform measure. For this measure, the function Ur may be obtained explicitly (see below). It appears, this entire function has imaginary zeros only, hence r possesses the Lee–Yang property as well. One observes that all these measures are isotropic, that is they are invariant with respect to the orthogonal group OðN Þ. For isotropic measures, the above property is independent of the choice of the unit vector e in (1.5). For nonisotropic measures, it would be not the case, therefore, the notion of this kind may be really useful first of all for isotropic measures. In this article, we describe a construction based on the use of the Laguerre entire function, which establishes the Lee–Yang property for isotropic measures and gives some description of such measures. Partially, we review our results already published in [6,9,10], but such statements as Theorems 3.1, 3.4 are new. One more problem connected to the analytic properties of characteristic functions of probability measures was communicated by Lukacs in [14]. Let D be the set of characteristic functions u, such that the functions wðzÞ ¼ 1=uðizÞ are again characteristic functions. Thus, according to [14] the van Danzig problem consists in the description of D. Let DE stand for the subset of D consisting of entire functions. Lukacs gave some examples of the elements of DE , namely, the functions cos z;
sin z ; z
expðz2 =2Þ
belong to this class. One observes that all the three corresponding measures possess the Lee–Yang property. Further, in [14] it was shown that the operations T1 : uðzÞ 7!
u0 ðzÞ ; zu00 ð0Þ
T2 : uðzÞ 7!
u00 ðzÞ u00 ð0Þ
preserve DE . In what follows, we have one more problem. Problem 1.2. What would be another examples of operations which preserve DE ?
2. Definitions and preliminaria In the sequel, F, P, Pþ will stand for the set of all entire functions of a single complex variable, for the set of all such polynomials, and for the set of all polynomials which have real nonpositive zeros only, respectively.
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
107
Definition 2.1. The closure of the set Pþ in the topology of uniform convergence on compact subsets of the complex plane C is called the set of Laguerre entire functions L. Due to Laguerre and P olya we have the following characterization of L. Proposition 2.1. Every element of L may be written as f ðzÞ ¼ C expðjzÞzm
1 Y ð1 þ ck zÞ;
C 2 C;
k¼1
j P 0;
m 2 N0 ¼ N [ f0g;
ck P 0;
1 X
ck < 1:
ð2:1Þ
k¼1
More information about such functions may be found in [2,10]. In particular, for every f 2 L, the sequence of its derivatives at zero ff ðnÞ ð0Þgn2N0 is a sequence of P olya numbers [2,12]. The sequence of complex numbers fak gk2N0 is called a sequence of P olya numbers if, for every polynomial pðzÞ ¼ p0 þ p1 z þ þ pm zm possessing real zeros only, the polynomial ðapÞðzÞ ¼ p0 a0 þ p1 a1 z þ þ pm am zm possesses real zeros only as well. Given N 2 N, let MN stand for the set of all isotropic (i.e., OðN Þ-invariant) probability measures on RN , such that Z expðaðx; xÞÞ dlðxÞ < 1; ð2:2Þ RN
for some a > 0. A function F : RN ! C is called isotropic if, for every U 2 OðN Þ, FU ¼ F . Here def
FU ðxÞ ¼ F ðUxÞ;
x 2 RN :
An infinitely differentiable function F : RN ! C is called real analytic if its Taylor series expansion, centred at a given x 2 RN , converges absolutely at any point and for any such x. Clearly, each a polynomial P : RN ! C is a real analytic function. Let PN stand for the set of all isotropic polynomials. There exists a classical statement characterizing this set. Proposition 2.2 (Study–Weyl Theorem). There exists a bijection between P and PN , which has the following form P 3 pððx; xÞÞ ¼ P ðxÞ 2 PN : Generalizations of this theorem to real analytic functions may be found in [15]. Let FN stand for the set of all isotropic real analytic functions. Obviously, for every f 2 F, the function F defined by f as follows:
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Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
8x 2 RN
F ðxÞ ¼ f ððx; xÞÞ;
ð2:3Þ
belongs to FN . We set FN ðLÞ ¼ fF 2 FN jð9f 2 LÞF ðxÞ ¼ f ððx; xÞÞg:
ð2:4Þ
It is not hard to show that, for every l 2 MN , the function Z Fl ðxÞ ¼ expððx; yÞÞ dlðyÞ
ð2:5Þ
RN
belongs to FN . Definition 2.2. A measure l 2 MN is said to possess the Lee–Yang property if Fl 2 FN ðLÞ. The set of all such measures will be denoted by MN ðLÞ.
3. The results In the sequel for a measure l 2 MN ðLÞ, Fl will stand for the function (2.5), fl will stand for the functions connected with Fl by (2.3). By Definition 2.2, each such fl possesses the representation fl ðzÞ ¼ expðjzÞ
1 Y ð1 þ ck zÞ;
j P 0;
k¼1
ck P 0;
1 X
ck < 1:
ð3:1Þ
k¼1
In view of the GolÕdberg–OstrovsÕkii theorem, for the parameters j and ck , one has the following possibilities: (a) j ¼ c1 ¼ ¼ ck ¼ 0, i.e., fl ðzÞ 1 or, equivalently, l is the Dirac measure concentrated at zero; (b) j > 0, c1 ¼ ¼ ck ¼ 0, which corresponds to a nondegenerate isotropic Gaussian measure; (c) j ¼ 0, all ck are positive, in this case l is a sub-Gaussian isotropic measure, certain examples of such measures are given below; (d) j > 0 and all ck are positive, which may correspond to the convolution of the measures of the above kind. The paramters j, ck , k 2 N determine the following numerical characteristics of l: m1 ðlÞ ¼ j þ
1 X k¼1
ck ;
ms ðlÞ ¼
1 X
csk ;
s 2 N n f1g
ð3:2Þ
k¼1
which in turn determine the moments and semi-invariants of l. From the representation (3.1) one concludes that, for l 2 MN ðLÞ, the function def
gl ðzÞ ¼ log fl ðzÞ
ð3:3Þ
is holomorphic in a neighborhood of the origin, where it can be expanded
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
gl ðzÞ ¼
1 X 1 ðsÞ g ð0Þzs ; s! l s¼1
s1
glðsÞ ð0Þ ¼ ð1Þ
ðs 1Þ!ms ðlÞ
109
ð3:4Þ
which, amongst others, establishes a sign rule for its derivatives. Now let us give some examples of the measures possessing the Lee–Yang property. First of all, it is an isotropic Gaussian measure, for which one has gl ðzÞ ¼ jz. Another example is the above mentioned measure r, concentrated on the unit sphere, for which fr ðzÞ ¼ wN =2 ðzÞ;
Ur ðzÞ ¼ wN =2 ðz2 Þ;
def
wh ðzÞ ¼
1 X n¼1
CðhÞ n z: Cðh þ nÞ
The function wh ðzÞ is a Wright function, it may be written also with the help of the Bessel functions, pffiffi ip wh ðzÞ ¼ CðhÞ exp ð1 hÞ zð1hÞ=2 Jh1 ð2i zÞ; ð3:5Þ 2 its order is one half (hence its j ¼ 0), and all its zeros are negative. The latter fact may be established either by direct methods given in [2] or on the base of (3.5). For the Gaussian measure, glð1Þ ð0Þ ¼ j, glðsÞ ð0Þ ¼ 0; for all s P 2. On the other hand, for the measure r, one has jgrð2Þ ð0Þj ð1Þ 2 ½gr ð0Þ
¼
2 : N þ2
These two examples are, in a sense, extreme, which is stated in our first theorem. Theorem 3.1. For every l 2 MN ðLÞ the following bounds hold: 06
jglð2Þ ð0Þj ð1Þ 2 ½gl ð0Þ
6
2 : N þ2
ð3:6Þ
These bounds are achieved only for the Gaussian measures and the measure r, which is discussed above. As it follows from [20] and [3], the measure (1.6) possesses the Lee–Yang property for all a 2 R and for N ¼ 1; 2; 3. Our next theorem essentially extends this result. Theorem 3.2. Given a function / : C ! C, which maps Rþ ¼ ½0; þ1Þ into itself, let there exist b 2 Rþ , such that the derivative /0 obeys the condition /0 ðzÞ þ b 2 L. Then, for all N 2 N and for arbitrary h 2 L, which maps Rþ into itself, the probability measure
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Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
dlðxÞ ¼ Chððx; xÞÞ expð/ððx; xÞÞÞ dx;
ð3:7Þ
possesses the Lee–Yang property. This theorem includes all the results pffiffi of this kind mentioned above. Thus, for N ¼ 1, h 1, and /ðzÞ ¼ a coshð zÞ, a > 0, it gives P olyaÕs result [18]. For N ¼ 1; 2; 3, h 1, and /ðzÞ ¼ az þ z2 , a 2 R, it gives the results established in [20] ðN ¼ 1Þ and in [3] ðN ¼ 2; 3Þ. For the measures (3.7) with h 1, we have the following statement. Theorem 3.3. Let / : C ! C satisfy the conditions of the previous theorem. Then, for the probability measure dlðxÞ ¼ C exp ð /ððx; xÞÞÞ dx;
ð3:8Þ
the function fl (2.3) obeys the following differential equation ð4D/ðKN Þfl ÞðzÞ ¼ fl ðzÞ:
ð3:9Þ
Here KN ¼ 2ND þ 4zD2 ;
D¼
d dz
ð3:10Þ
and the differential operator /ðKN Þ (possibly of infinite order) acts on an entire function f as follows ð/ðKN Þf ÞðzÞ ¼
1 X 1 ðnÞ / ð0ÞðKnN f ÞðzÞ; n! n¼0
where the series converges uniformly on compact subsets of C.
1
For the measures (3.8) with /ðzÞ ¼ az þ z2 , we have one more result of the type of (3.6). Theorem 3.4. Let the measure l be given by (3.8) with /ðzÞ ¼ az þ z2 . Then, for all N 2 N, one has the following estimate for the derivatives defined in (3.3), jglð2Þ ð0Þj ð1Þ 2 ½gl ð0Þ
6
2 : a2
Finally, let us say some words about the proof of these results. Given an entire function f and a positive b, we set kf kb ¼ supfjf ðzÞj expðbjzjÞg:
ð3:11Þ
z2C
1
A detailed consideration of infinite order differential operators of this kind is given in [10].
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
111
Set also def
La ¼ f/ 2 Ljk/kb < 1; 8b > ag;
aP0
ð3:12Þ
and, for h P 0, Dh ¼ hD þ zD2 ;
D¼
d : dz
ð3:13Þ
Given a P 0, the set La defined in (3.12) may be equipped in the topology Ta generated by the family of norms fk kb ; b > ag. For /; f 2 F, we set 1 X 1 ðnÞ / ð0ÞðDnh f ÞðzÞ: n! n¼0
ð/ðDh Þf ÞðzÞ ¼
ð3:14Þ
In [10], we proved the following statement. Proposition 3.1. For every h P 0, for any nonnegative numbers a and b obeying the condition ab < 1, and for arbitrary / 2 La and f 2 Lb , the 1 function /ðDh Þf belongs to Lc with c ¼ b½1 ab . The map ð/; f Þ 7! /ðDh Þf , acting from La Lb into Lc , is continuous. Theorem 3.3 may be proved, roughly speaking, by integration by parts. Comparing Dh with KN given by (3.10), one concludes that, by Proposition 3.1, the operator /ðKN Þ in (3.9) preserves the class L. Then one constructs a sequence of functions un ðzÞ ¼ ð4D/ðKN Þun1 ÞðzÞ;
n 2 N;
u0 2 L
which converges, uniformly on compact subsets of C, to fl . This proves Theorem 3.2 for h 1. To complete the proof one writes for the measure (3.7) fl ðzÞ ¼ ðhðKN ÞuÞðzÞ; where uððx; xÞÞ ¼ C
Z
expððx; yÞ /ðy; yÞÞ dy
RN
which means u 2 L. Since hðKN Þ preserves the latter set, one has fl 2 L.
References [1] T. Asano, Lee–Yang theorem and the Griffiths inequality for the anisotropic Heisenberg ferromagnet, Phys. Rev. Lett. 24 (1970) 1409–1411. [2] L. Iliev, Laguerre Entire Functions, Bulgarian Academy of Sciences, Sofia, 1987. [3] F. Dunlop, C. Newman, Multicomponent field theories and classical rotators, Comm. Math. Phys. 44 (1975) 223–235.
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[4] J. Glimm, A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer-Verlag, New York, Berlin, 1987. [5] A.A. GolÕdberg, I.V. OstrovsÕkii, The growth of entire ridge functions with real zeros, Math. Phys. Funct. Anal., No. V, 3–10, 156 Akad. Nauk. Ukrain. SSR Fiz.-Tehn. Inst. Nizkih Temperatur Kharkov, 1974 (in Russian). [6] Yu.V. Kozitsky, Hierarchical vector model of a ferromagnet in the method of collective variables. The Lee–Yang theorem, Theor. Math. Phys. 58 (1984) 96–108. [7] Yu.V. Kozitsky, Hierarchical model of a ferromagnet. Self-similar block-spin distributions and the Lee–Yang property, Rep. Math. Phys. 26 (1988) 429–445. [8] Yu. Kozitsky, Hierarchical ferromagnetic vector spin model possessing the Lee–Yang property. Thermodynamic limit at the critical point and above, J. Stat. Phys. 87 (1997) 799– 820. [9] Yu.V. Kozitsky, N.O. Melnik, Random vectors possessing the Lee–Yang property, Theor. Math. Phys. 78 (1989) 177–186. [10] Yu. Kozitsky, L. Wołowski, Laguerre entire functions and related locally convex spaces, Complex Variables Theory Appl. 44 (2001) 225–244. [11] T.D. Lee, C.N. Yang, Statistical theory of equation of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87 (1952) 410–419. [12] B.J. Levin, Distribution of zeros of entire functions, Am. Math. Soc. (1964). [13] E.H. Lieb, A.D. Sokal, A general Lee–Yang theorem for one-component and multicomponent ferromagnets, Comm. Math. Phys. 80 (1981) 153–179. [14] E. Lukacs, Contributions to a problem of D. van Danzig, Probab. Theory Appl. 13 (1968) 114–125. [15] D. Luna, Fonctions differentiables invariantes sous lÕop erations dÕun groupe reductif, Ann. Inst. Fourier, Grenoble 26 (1976) 33–49. [16] C.M. Newman, Inequalities for Ising models and field theories which obey the Lee–Yang theorem, Comm. Math. Phys. 41 (1975) 1–9. [17] A.I. Ovseevich, A generalization of the Lee–Yang theorem, Russ. J. Math. Phys. 2 (1994) 261– 264. €ber die Integraldarstellung der Riemannschen funktion, Acta Math. 48 [18] G. P olya, Bemerkung u (1926) 305–317. [19] B. Simon, The P ð/Þ2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, 1974. [20] B. Simon, R. Griffiths, The ð/4 Þ2 field theory as a classical Ising model, Comm. Math. Phys. 33 (1973) 145–164. [21] G.S. Sylvester, A note on the Lee–Yang theorem, J. Math. Anal. Appl. 81 (1981) 88–91. [22] K. Yamada, H. Nakano, M. Yamashita, Distribution of zeros of the partition function in the first and second phase transitions, Prog. Theor. Phys. 65 (1981) 1603–1612. [23] C.N. Yang, T.D. Lee, Statistical theory of equation of state and phase transitions. I. Theory of condensation, Phys. Rev. 87 (1952) 404–409.
Laguerre entire functions and the Lee–Yang property q Yuri Kozitsky Institute of Mathematics, Marie Curie-Sklodowska University, PL 20-031 Lublin, Poland
Abstract Laguerre entire functions are the polynomials of a single complex variable possessing real nonpositive zeros only or their limits on compact subsets of C. These functions are employed to establish a property of isotropic (i.e., OðNÞ-invariant) probability measures on RN , N 2 N. It is called the Lee–Yang property since, in the case N ¼ 1, it corresponds to the property of the partition function of certain models of statistical physics, first established by T.D. Lee and C.N. Yang. A class of measures possessing this property is described. Certain connections of the Lee–Yang property with other aspects of the analytic theory of probability measures are discussed. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Exponential type entire functions; Laguerre entire functions, Nonpositive zeros; Cauchy problem
1. Introduction Discovered in the fifties by the famous physicists Lee and Yang [11,23], a property of the partition functions of certain models of statistical physics later was called the Lee–Yang property. Since that time methods based on this property appeared in statistical physics and quantum field theory (see e.g., [4,19]). The original variant of the Lee–Yang theorem was refined and extended in a number of papers (see e.g., [1,6,9,13,17,21,22] and the references therein).
q
Supported in part by KBN Grant 2P03A 02915. E-mail address: [email protected] (Y. Kozitsky).
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00324-7
104
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
As examples of the application of methods based on this theory we mention [6– 8,16]. A simplest version of the Lee–Yang theorem may be described as follows. Obviously, ! n n X X 1X UðzÞ ¼ exp z sj þ Jij si sj ; z 2 C ð1:1Þ 2 i;j¼1 s2N j¼1 is an exponential type entire function. Summation in (1.1) is taken over 2n n configurations s ¼ ðs1 ; . . . ; sn Þ 2 N ¼ f1; þ1g . The Lee–Yang theorem states that all zeros of U are purely imaginary provided all Jij are nonnegative. In statistical physics this function is known as a partition function of a ferromagnetic Ising model in an external magnetic field z, which describes a system of spins fs1 ; . . . ; sn g interacting via ferromagnetic coupling Jij P 0. Thus, for a symmetric probability measure l on R, the Lee–Yang property may be said to occur if Z Ul ðzÞ ¼ expðzxÞ dlðxÞ; z 2 C ð1:2Þ R
is an entire function, which possesses imaginary zeros only or does not vanish at all. Hence the measure l which P describes the probability distribution of the total magnetic moment mðsÞ ¼ nj¼1 sj of the ferromagnetic Ising model, defined in the following way: ! n 1 X 1X lðBÞ ¼ exp Jij si sj ; Z s:mðsÞ2B 2 i;j¼1 where B is a Borel subset of R and ! n X 1X Z¼ exp Jij si sj 2 i;j¼1 s2N possesses the Lee–Yang property. By means of the so-called Ising approximation, in [20] it was proved that the above fact implies the Lee–Yang property for the probability measure dmðxÞ ¼ C expðax2 x4 Þ dx with arbitrary a 2 R. Later, it was shown [4,19], that there exist even polynomials /, limx!þ1 /ðxÞ ¼ þ1, deg / ¼ 6, such that the probability measure dm/ ðxÞ ¼ C expð/ðxÞÞ dx;
ð1:3Þ
does not possess the Lee–Yang property. The search of the polynomials / for which the measures m/ possess the Lee–Yang property was stimulated by the attempts to construct polynomial models of quantum field theory [19], which
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
105
took place in the seventies and eighties. In a more general setting this problem may be formulated as follows. Problem 1.1. To find necessary and sufficient conditions for an even function / : R ! R, under which the probability measure m/ (1.3) possesses the Lee– Yang property, i.e., its function (1.2) is entire and has imaginary zeros only or has them none. Obviously, this is a particular case of a much more older problem of description of probability measures m on R, for which the characteristic functions um ðzÞ ¼ Um ðizÞ are entire functions possessing real zeros only. A partial answer to this question was given by P olya in his article [18] on the problem of zeros of Riemannian entire functions (for a detailed discussion of this problem, see [2]). Namely, he showed that the characteristic function of the measure (1.3) with /ðxÞ ¼ a cosh x, a > 0 is an entire function possessing real zeros only. In their brilliant paper, GolÕdberg and OsrtovsÕkii [5] have proved that a ridge entire function u, which is of finite order, (ridge means juðx þ iyÞj 6 juðiyÞj, all characteristic functions are ridge) possesses real zeros only if and only if it has the following infinite-product representation: uðzÞ ¼ C expðjz2 þ idzÞ
1 Y ð1 ck z2 Þ;
j P 0;
d 2 R;
k¼1 1 X
ck > 0;
ck < 1
ð1:4Þ
k¼1
which means, in particular, that its order is at most two. This establishes the form of the function Ul (1.2) for l possessing the Lee–Yang property. But the functions obeying the representations of this type are known as Laguerre entire functions [2,10]. Putting all these facts together one concludes that the Laguerre entire functions could constitute a proper setting for developing the notion of the Lee–Yang property. Another problem of this kind is how to generalize the notion of the Lee– Yang property to the measures on RN , N > 1. Of course, one can choose a direction in RN (i.e., a unit vector e 2 RN ) and set Z expðzðe; xÞÞdlðxÞ; z 2 C; ð1:5Þ Ul ðzÞ ¼ RN
where ð ; Þ is the scalar product in RN . Then, the Lee–Yang property of l could be said to occur if Ul would be an even entire function possessing imaginary zeros only or not vanishing at all. This approach was used in [3] where it was proved that for the probability measure dmðxÞ ¼ C expðaðx; xÞ ðx; xÞ2 Þ dx;
ð1:6Þ
106
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
the function Um (1.5) is entire and has real zeros only, that occurs for all a 2 R and N ¼ 2; 3. Let r stand for the measure which is concentrated on the unit sphere S N 1 RN and, being restricted to it, coincides with the uniform measure. For this measure, the function Ur may be obtained explicitly (see below). It appears, this entire function has imaginary zeros only, hence r possesses the Lee–Yang property as well. One observes that all these measures are isotropic, that is they are invariant with respect to the orthogonal group OðN Þ. For isotropic measures, the above property is independent of the choice of the unit vector e in (1.5). For nonisotropic measures, it would be not the case, therefore, the notion of this kind may be really useful first of all for isotropic measures. In this article, we describe a construction based on the use of the Laguerre entire function, which establishes the Lee–Yang property for isotropic measures and gives some description of such measures. Partially, we review our results already published in [6,9,10], but such statements as Theorems 3.1, 3.4 are new. One more problem connected to the analytic properties of characteristic functions of probability measures was communicated by Lukacs in [14]. Let D be the set of characteristic functions u, such that the functions wðzÞ ¼ 1=uðizÞ are again characteristic functions. Thus, according to [14] the van Danzig problem consists in the description of D. Let DE stand for the subset of D consisting of entire functions. Lukacs gave some examples of the elements of DE , namely, the functions cos z;
sin z ; z
expðz2 =2Þ
belong to this class. One observes that all the three corresponding measures possess the Lee–Yang property. Further, in [14] it was shown that the operations T1 : uðzÞ 7!
u0 ðzÞ ; zu00 ð0Þ
T2 : uðzÞ 7!
u00 ðzÞ u00 ð0Þ
preserve DE . In what follows, we have one more problem. Problem 1.2. What would be another examples of operations which preserve DE ?
2. Definitions and preliminaria In the sequel, F, P, Pþ will stand for the set of all entire functions of a single complex variable, for the set of all such polynomials, and for the set of all polynomials which have real nonpositive zeros only, respectively.
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
107
Definition 2.1. The closure of the set Pþ in the topology of uniform convergence on compact subsets of the complex plane C is called the set of Laguerre entire functions L. Due to Laguerre and P olya we have the following characterization of L. Proposition 2.1. Every element of L may be written as f ðzÞ ¼ C expðjzÞzm
1 Y ð1 þ ck zÞ;
C 2 C;
k¼1
j P 0;
m 2 N0 ¼ N [ f0g;
ck P 0;
1 X
ck < 1:
ð2:1Þ
k¼1
More information about such functions may be found in [2,10]. In particular, for every f 2 L, the sequence of its derivatives at zero ff ðnÞ ð0Þgn2N0 is a sequence of P olya numbers [2,12]. The sequence of complex numbers fak gk2N0 is called a sequence of P olya numbers if, for every polynomial pðzÞ ¼ p0 þ p1 z þ þ pm zm possessing real zeros only, the polynomial ðapÞðzÞ ¼ p0 a0 þ p1 a1 z þ þ pm am zm possesses real zeros only as well. Given N 2 N, let MN stand for the set of all isotropic (i.e., OðN Þ-invariant) probability measures on RN , such that Z expðaðx; xÞÞ dlðxÞ < 1; ð2:2Þ RN
for some a > 0. A function F : RN ! C is called isotropic if, for every U 2 OðN Þ, FU ¼ F . Here def
FU ðxÞ ¼ F ðUxÞ;
x 2 RN :
An infinitely differentiable function F : RN ! C is called real analytic if its Taylor series expansion, centred at a given x 2 RN , converges absolutely at any point and for any such x. Clearly, each a polynomial P : RN ! C is a real analytic function. Let PN stand for the set of all isotropic polynomials. There exists a classical statement characterizing this set. Proposition 2.2 (Study–Weyl Theorem). There exists a bijection between P and PN , which has the following form P 3 pððx; xÞÞ ¼ P ðxÞ 2 PN : Generalizations of this theorem to real analytic functions may be found in [15]. Let FN stand for the set of all isotropic real analytic functions. Obviously, for every f 2 F, the function F defined by f as follows:
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Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
8x 2 RN
F ðxÞ ¼ f ððx; xÞÞ;
ð2:3Þ
belongs to FN . We set FN ðLÞ ¼ fF 2 FN jð9f 2 LÞF ðxÞ ¼ f ððx; xÞÞg:
ð2:4Þ
It is not hard to show that, for every l 2 MN , the function Z Fl ðxÞ ¼ expððx; yÞÞ dlðyÞ
ð2:5Þ
RN
belongs to FN . Definition 2.2. A measure l 2 MN is said to possess the Lee–Yang property if Fl 2 FN ðLÞ. The set of all such measures will be denoted by MN ðLÞ.
3. The results In the sequel for a measure l 2 MN ðLÞ, Fl will stand for the function (2.5), fl will stand for the functions connected with Fl by (2.3). By Definition 2.2, each such fl possesses the representation fl ðzÞ ¼ expðjzÞ
1 Y ð1 þ ck zÞ;
j P 0;
k¼1
ck P 0;
1 X
ck < 1:
ð3:1Þ
k¼1
In view of the GolÕdberg–OstrovsÕkii theorem, for the parameters j and ck , one has the following possibilities: (a) j ¼ c1 ¼ ¼ ck ¼ 0, i.e., fl ðzÞ 1 or, equivalently, l is the Dirac measure concentrated at zero; (b) j > 0, c1 ¼ ¼ ck ¼ 0, which corresponds to a nondegenerate isotropic Gaussian measure; (c) j ¼ 0, all ck are positive, in this case l is a sub-Gaussian isotropic measure, certain examples of such measures are given below; (d) j > 0 and all ck are positive, which may correspond to the convolution of the measures of the above kind. The paramters j, ck , k 2 N determine the following numerical characteristics of l: m1 ðlÞ ¼ j þ
1 X k¼1
ck ;
ms ðlÞ ¼
1 X
csk ;
s 2 N n f1g
ð3:2Þ
k¼1
which in turn determine the moments and semi-invariants of l. From the representation (3.1) one concludes that, for l 2 MN ðLÞ, the function def
gl ðzÞ ¼ log fl ðzÞ
ð3:3Þ
is holomorphic in a neighborhood of the origin, where it can be expanded
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
gl ðzÞ ¼
1 X 1 ðsÞ g ð0Þzs ; s! l s¼1
s1
glðsÞ ð0Þ ¼ ð1Þ
ðs 1Þ!ms ðlÞ
109
ð3:4Þ
which, amongst others, establishes a sign rule for its derivatives. Now let us give some examples of the measures possessing the Lee–Yang property. First of all, it is an isotropic Gaussian measure, for which one has gl ðzÞ ¼ jz. Another example is the above mentioned measure r, concentrated on the unit sphere, for which fr ðzÞ ¼ wN =2 ðzÞ;
Ur ðzÞ ¼ wN =2 ðz2 Þ;
def
wh ðzÞ ¼
1 X n¼1
CðhÞ n z: Cðh þ nÞ
The function wh ðzÞ is a Wright function, it may be written also with the help of the Bessel functions, pffiffi ip wh ðzÞ ¼ CðhÞ exp ð1 hÞ zð1hÞ=2 Jh1 ð2i zÞ; ð3:5Þ 2 its order is one half (hence its j ¼ 0), and all its zeros are negative. The latter fact may be established either by direct methods given in [2] or on the base of (3.5). For the Gaussian measure, glð1Þ ð0Þ ¼ j, glðsÞ ð0Þ ¼ 0; for all s P 2. On the other hand, for the measure r, one has jgrð2Þ ð0Þj ð1Þ 2 ½gr ð0Þ
¼
2 : N þ2
These two examples are, in a sense, extreme, which is stated in our first theorem. Theorem 3.1. For every l 2 MN ðLÞ the following bounds hold: 06
jglð2Þ ð0Þj ð1Þ 2 ½gl ð0Þ
6
2 : N þ2
ð3:6Þ
These bounds are achieved only for the Gaussian measures and the measure r, which is discussed above. As it follows from [20] and [3], the measure (1.6) possesses the Lee–Yang property for all a 2 R and for N ¼ 1; 2; 3. Our next theorem essentially extends this result. Theorem 3.2. Given a function / : C ! C, which maps Rþ ¼ ½0; þ1Þ into itself, let there exist b 2 Rþ , such that the derivative /0 obeys the condition /0 ðzÞ þ b 2 L. Then, for all N 2 N and for arbitrary h 2 L, which maps Rþ into itself, the probability measure
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Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
dlðxÞ ¼ Chððx; xÞÞ expð/ððx; xÞÞÞ dx;
ð3:7Þ
possesses the Lee–Yang property. This theorem includes all the results pffiffi of this kind mentioned above. Thus, for N ¼ 1, h 1, and /ðzÞ ¼ a coshð zÞ, a > 0, it gives P olyaÕs result [18]. For N ¼ 1; 2; 3, h 1, and /ðzÞ ¼ az þ z2 , a 2 R, it gives the results established in [20] ðN ¼ 1Þ and in [3] ðN ¼ 2; 3Þ. For the measures (3.7) with h 1, we have the following statement. Theorem 3.3. Let / : C ! C satisfy the conditions of the previous theorem. Then, for the probability measure dlðxÞ ¼ C exp ð /ððx; xÞÞÞ dx;
ð3:8Þ
the function fl (2.3) obeys the following differential equation ð4D/ðKN Þfl ÞðzÞ ¼ fl ðzÞ:
ð3:9Þ
Here KN ¼ 2ND þ 4zD2 ;
D¼
d dz
ð3:10Þ
and the differential operator /ðKN Þ (possibly of infinite order) acts on an entire function f as follows ð/ðKN Þf ÞðzÞ ¼
1 X 1 ðnÞ / ð0ÞðKnN f ÞðzÞ; n! n¼0
where the series converges uniformly on compact subsets of C.
1
For the measures (3.8) with /ðzÞ ¼ az þ z2 , we have one more result of the type of (3.6). Theorem 3.4. Let the measure l be given by (3.8) with /ðzÞ ¼ az þ z2 . Then, for all N 2 N, one has the following estimate for the derivatives defined in (3.3), jglð2Þ ð0Þj ð1Þ 2 ½gl ð0Þ
6
2 : a2
Finally, let us say some words about the proof of these results. Given an entire function f and a positive b, we set kf kb ¼ supfjf ðzÞj expðbjzjÞg:
ð3:11Þ
z2C
1
A detailed consideration of infinite order differential operators of this kind is given in [10].
Y. Kozitsky / Appl. Math. Comput. 141 (2003) 103–112
111
Set also def
La ¼ f/ 2 Ljk/kb < 1; 8b > ag;
aP0
ð3:12Þ
and, for h P 0, Dh ¼ hD þ zD2 ;
D¼
d : dz
ð3:13Þ
Given a P 0, the set La defined in (3.12) may be equipped in the topology Ta generated by the family of norms fk kb ; b > ag. For /; f 2 F, we set 1 X 1 ðnÞ / ð0ÞðDnh f ÞðzÞ: n! n¼0
ð/ðDh Þf ÞðzÞ ¼
ð3:14Þ
In [10], we proved the following statement. Proposition 3.1. For every h P 0, for any nonnegative numbers a and b obeying the condition ab < 1, and for arbitrary / 2 La and f 2 Lb , the 1 function /ðDh Þf belongs to Lc with c ¼ b½1 ab . The map ð/; f Þ 7! /ðDh Þf , acting from La Lb into Lc , is continuous. Theorem 3.3 may be proved, roughly speaking, by integration by parts. Comparing Dh with KN given by (3.10), one concludes that, by Proposition 3.1, the operator /ðKN Þ in (3.9) preserves the class L. Then one constructs a sequence of functions un ðzÞ ¼ ð4D/ðKN Þun1 ÞðzÞ;
n 2 N;
u0 2 L
which converges, uniformly on compact subsets of C, to fl . This proves Theorem 3.2 for h 1. To complete the proof one writes for the measure (3.7) fl ðzÞ ¼ ðhðKN ÞuÞðzÞ; where uððx; xÞÞ ¼ C
Z
expððx; yÞ /ðy; yÞÞ dy
RN
which means u 2 L. Since hðKN Þ preserves the latter set, one has fl 2 L.
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