Stochastic and Analytic Methods in Mathematical Physics Yerevan, 2016 Martingale method: applications in the theory of limit theorems and mathematical statistical physics L.A. Khachatryan, B.S. Nahapetian Institute of Mathematics, National Academy of Sciences of Armenia The main goals of the talk are: • to present a new type of limit theorems for random fields with weakly dependent components (particularly, for Gibbs random fields); • to demonstrate the capabilities of the martingale method for studying some models of the mathematical statistical physics (ground states, critical point). Martingale–difference random fields d A collection of random variables (ξt) = (ξt; t 2 Z ), each of which takes value in X will call a random field defined on Zd with phase space X, X ⊂ R (in this talk we consider only the case of finite phase space, i.e. 1 < jXj < 1). The distribution P of the random field (ξt) is the probability measure on d d d (XZ ; BZ ) (BZ is a sigma-algebra generated by the set of cylindric subsets d d d of XZ ) such that P(B) = P (fξt; t 2 Z g 2 B),B 2 BZ . A random field (ξt) is called a martingale–difference random d field if Ejξtj < 1 for any t 2 Z and d d E(ξt/ξs; s 2 Z nftg) = 0 (a.s.) for any t 2 Z : Example. Positive random field with symmetric (with respect to zero) phase space and even finite-dimensional probability distribution, i.e. for all V 2 W and x 2 XV PV (θtxt;t2V )=PV (xt;t2V ) for any θt2f1;−1g: Gibbs random fields A random field P is called a Gibbs random field if it is d positive and for any x 2 X and any t 2 Z there exist strictly d positive uniform (with respect to x¯ 2 XZ nftg) limits P ( ¯ ) x¯ ftg[V xxV qt (x) = lim : V "Zdnftg PV (¯xV ) d (1) x¯ Z nftg d The set Q = qt ; x¯ 2 X ; t 2 Z is called a canonical 1–specification. Representation of Gibbs random fields Theorem 1. Let P be a Gibbs random field. Then corresponding canonical 1–specification Q(1) admits the Gibbs representation with the aid of uniformly convergent potential V Φ = fΦV (x); x 2 X ;V 2 W g; that is expfHx¯(x)g x¯( ) = t qt x P x¯ ; x 2 X; expfHt (z)g z2X where x¯ X Ht (x) = Φftg[J(xx¯J); x 2 X: J⊂W (Zdnftg) Martingale–difference Gibbs random fields Let P be a Gibbs random field with phase space X and let Π = fX1;X2; :::; Xng be a partition of X such that for any k = 1; n X x = 0: x2Xk • If canonical 1–specification Q(1) of the Gibbs random field d dnftg P is such that for any t 2 Z , x¯ 2 XZ , and k = 1; n x¯ x¯ 0 0 qt (x) = qt (x ); x; x 2 Xk; then P is a martingale–difference Gibbs random fields. • If potential Φ corresponding to the Gibbs random field P takes constant values on the elements of a partition Π of d d V phase space X, i.e. for any t 2 Z , V 2 W (Z nftg), x¯ 2 X and k = 1; n 0 0 Φftg[V (xtx¯V ) = Φftg[V (xtx¯V ); x; x 2 Xk; then P is a martingale–difference Gibbs random fields. 1. Martingale method in the theory of limit theorems Classical limit theorems and their refinements Valid for random fields 1. The central limit theorem 1. with independent components; (CLT); 2. with weak dependent 2. Rate of convergence in the components and suitable CLT; decrease of correlations between 3. The law of the iterated components; logarithm; 3. Gibbs r.f. under suitable 4. Asymptotical behavior of conditions on corresponding moments; canonical 1–specification or 5. The local limit theorem potential. (LLT). Let (ξt) be a random field. Denote P SVn = ξt; =S = σ(ξt; t 2 S); t2Vn d d Vn = [−n; n] ; n = 1; 2; :::; S ⊂ Z : Weak dependence conditions A homogenous random field (ξt) is called ergodic if for any I; Λ 2 W 1 X lim P fξt = xt; t 2 Ig \ fξs+a =x ¯s; s 2 Λg = n!1 jVnj a2Vn = P (ξt = xt; t 2 I)P (ξs =x ¯s; s 2 Λ); where x 2 XI, x¯ 2 XΛ. 1. The weakest condition of weak dependence; 2. If Gibbs random field is unique, then it is ergodic; 3. In the general case it is not enough to obtain classical limit theorems. A homogenous random field (ξt) satisfies the uniform strong mixing condition with coefficient 'I if for any fixed I 2 W sup fjP (A=B) − P (A)jg ≤ 'I (ρ(I;V )); A2=I;B2=V ;P (B)>0 where function 'I(ρ), ρ 2 R is such that 'I(ρ) ! 0 as ρ ! 1 and the set I is fixed (ρ(I;V ) stands for distance between sets I and V ). 1. Gibbs random fields satisfy this condition; 2. The CLT and logarithmic rate of convergence in it; law of the iterated logarithm; 3. In general case it is not enough to obtain the LLT. A homogenous random field (ξt) is conditionally independent with coefficient βI if for any I; V; Λ 2 W such that I\V = Ø and I;V ⊂ Λ, and any random variables η1, η2 which are =I- and =V -measurable correspondingly the following relation holds E η1 · η2==ΛnfI[V g − E η1==ΛnfI[V g · E η2==ΛnfI[V g ≤ ≤ βI(ρ(I;V )); d where βI(ρ) ! 0 as ρ ! 1 (and, hence, Λ " Z ) and I is fixed. 1. Gibbs random fields with finite range potentials; 2. The LLT. The Central Limit Theorem Theorem 2 (N., 1995). Let (ξt) be a homogeneous ergodic 2 (Gibbs) martingale–difference random field such that 0 < Eξ0 < 1. Then 0 1 x S 1 Z 2 B Vn C −u =2 lim P q < x = p e du; x 2 R: n!1 @ DS A 2π Vn −∞ Rate of convergence in the CLT. Law of the iterated logarithm Theorem 3 (Kh., N., 2013). Let (ξt) be a homogenous (Gibbs) martingale–difference random field with phase space X, satisfying the uniform strong mixing condition with coefficient 'I such that 1 X d−1 'I(j) ≤ jIj · '(j) and j '(j) < 1; j=1 2 and let Eξ0 > 0. Then 0 1 x S 1 Z 2 B Vn C −u =2 −d=8 sup P @q < xA − p e du ≤ C · n ; x2 DS 2π R Vn −∞ where positive constant C does not depend on n. Theorem 4 (Kh., N., 2013). Under conditions of Theorem 3 0 1 S P Blim sup Vn = 1C = 1: @ n!1 q A 2DSVn ln ln jVnj Exact asymptotic for the moments of sums of components Theorem 5 (Kh., N., 2013). Let (ξt) be a homogenous (Gibbs) martingale–difference random field with phase space X. Then for any k = 1; 2; ::: 2k−1 k−1 E(SVn) = C2k−1 · jVnj ; where constant C2k−1 does not depend on n. If, moreover, the random field (ξt) satisfies the uniform strong mixing condition with coefficient 'I such that 1 X d−1 'I(j) ≤ jIj · '(j) and j '(j) < 1; j=1 then for any k = 1; 2; ::: 2k 2 k k k−1 E(SVn) = (2k − 1)!!(Eξ0) jVnj + C2k jVnj ; where constant C2k does not depend on n. Theorem 6 (Kh., N., 2013). Let (ξt) be a (Gibbs) martingale– difference random field with phase space X, satisfying the uniform strong mixing condition with coefficient 'I such that 1 X d−1 'I(j) ≤ jIj · '(j) and j '(j) < 1; j=1 2 and let Eξ0 > 0. Then for random field (ξt) the CLT is valid and for any k = 1; 2; ::: 0 1k S B Vn C k E @q A ! Eζ as n ! 1; DSVn where ζ is a random variable with standard normal distribution. Local limit theorem Theorem 7 (Kh., N., 2016). Let (ξt) be a (Gibbs) martingale– difference random field with phase space X ⊂ Z, and let there d exists γ > 0 such that for any finite I ⊂ V ⊂ Z P SI = y==V nI ≥ γ for any possible value y of SI: If, in addition, (ξt) is a conditionally independent with coefficient βI such that −3d=2 βI(ρ) ≤ jIjβ(ρ) and β(ρ) = µ(ρ) · ρ ; where µ(ρ) ! 0 arbitrarily slow as ρ ! 1, then for the martingale– difference random field (ξt) the LLT is valid. Associated martingale–difference random fields For any given random field (ξt) by the method of randomization one can construct a martingale–difference random field (ηt) someway associated with (ξt). Advantages: 1. The class of random fields with martingale properties is extended. 2. The class of random fields for which classical limits theorems are valid is extended. 3. The connection between a given random field and associated martingale–difference can be used to study the asymptotical behavior of sums of components of the given random field by using the good properties martingale–differences. 4. Random field, associated with a Gibbs random field, is also Gibbsian. 2. Martingale method in the mathematical statistical physics Martingale model 8 −βh · jy j ;V = ftg <> t Φ~ V (y) = −β jytj · jysj ;V = ft; sg and kt − sk = 1 :> 0; in other cases d P (i) (i) d where kt − sk = t − s ; t; s 2 Z and i=1 yt; ys 2 Y = {−1; 0; 1g Hamiltonian has a form β;h β X X X Hf (y=y¯) = − jy j · jys|−β jy j · jy¯s|−h jy j; V 2 t t t : d t;s2V t2V;s2Z nV : t2V kt−sk=1 kt−sk=1 where h 2 R the external field, β > 0 the inverse temperature.