Universality in Sherrington-Kirkpatrick s Spin Glass Model

Transcription

1 Universality in Sherrington-Kirkpatrick s Spin Glass Model Philippe CARMOA, Yueyun HU 5th ovember 08 arxiv:math/ v mathpr May 004 Abstract We show that the limiting free energy in Sherrington-Kirkpatrick s Spin Glass Model does not depend on the environment Introduction The physical system is an -spin configuration σ = σ,,σ ) {,} Each configuration σ is given a Boltzmann weight e β H σ)+h i σ i where β = > 0 is the T inverse of the temperature, h is the intensity of the magnetic interaction, H σ) is the random Hamiltonian H σ) = H σ,ξ) = ξ ij σ i σ j, i,j and ξ ij ) i,j is an iid family of random variables, admitting order three moments, which we normalize: Eξ = 0, E ξ =, E ξ 3 < + ) The object of interest is the random Gibbs measure fσ) = Z σ fσ)e β H σ,ξ)+h i σ i, and in particular the partition function Z = Z β,ξ) = σ e β H σ,ξ)+h i σ i We shall denote by g = g ij ) i,j an environment of iid Gaussian standard random variables 0, )) PCarmona: Laboratoire Jean Leray, UMR 669, Université de antes, 908, F-443, antes cedex 03, philippecarmona@mathuniv-nantesfr YHu: Laboratoire de Probabilités et Modèles Aléatoires CRS UMR-7599), Université Paris VI, 4 Place Jussieu, F-755 Paris cedex 05, hu@ccrjussieufr

2 Recently, F Guerra and FL Toninelli, gave a rigorous proof, at the mathematical level, of the convergence of free energy to a deterministic limit, in a Gaussian environment, logz β,g) α β) as and in average Talagrand 4 then proved that one can replace the Gaussian environment by a Bernoulli environmentη ij, Pη ij = ±) =, and obtain the same limit: α β) We shall generalize this result Theorem Assume the environmentξ satisfies ) Then, logz β,ξ) α β) as and in average Furthermore, the averagesα β,ξ) def = ElogZ β,ξ) satisfy α β,ξ) α β,g) 9E ξ 3 β 3 Therefore the limiting free energy α β) does not depend on the environment, hence the Universality in the title of this paper : this independence from the particular disorder was already clear to Sherrington and Kirkpatrick 3 although they had no mathematical proof of this fact Guerra and Toninelli provided a physical proof in the case the environment is symmetric with a finite fourth moment) otice eventually that α β) can be determined in a Gaussian framework where Talagrand 5 recently proved that it is the solution of G Parisi s variational formula The universality property can be mechanically extended to the ground states, that is the supremum of the families of random variables: S ξ) = sup σ i,j σ i σ j ξ ij = lim β + β logz β,ξ) F Guerra and FL Toninelli, proved that 3/ S g) converges as and in average to a deterministic limite Here is the generalization : Theorem Assume the environmentξ satisfies ) Then, 3/ S ξ) e as and in average Furthermore, the averages satisfy, for a universal constantc > 0, 3/ ES ξ) ES g) C +E ξ 3) /6 We end this introduction by observing that we do not need the random variables ξ ij to share the same distribution They only need to be independent, to satisfy ) and such that sup ij E ξ ij 3 < +

3 Comparison of free energies Let us begin with an Integration by parts Lemma Lemma 3 Let ξ be a real random variable such that E ξ 3 < + and Eξ = 0 Let F : R R be twice continuously differentiable with F = sup x R F x) < + Then EξFξ) E ξ EF ξ) 3 F E ξ 3 Proof Observe first, that by Taylor s formula, Fξ) F0) ξf 0) ξ F, F ξ) F 0) ξ F Therefore, EξFξ) E ξ EF ξ) = EξFξ) E ξ EF ξ) F0)Eξ = EξFξ) F0) ξf 0)) E ξ EF 0) F ξ) F E ξ 3 +E ξ E ξ ) F E ξ 3 +E ξ 3 3 E ξ 3 ) 3 3 F E ξ 3 In the general framework,x = X,,X d ) is a random vector defined on a probability space Ω,F,P) such that for any i : X i The environment is an iid family of random variables ξ,,ξ d ) defined on Ω ξ),f ξ),p), distributed as a fixed random variableξ satisfying ) The Gibbs measure, partition function and averaged free energy are thus fx) = Zβ,ξ) E fx)e β d i= X iξ i Zβ,ξ) = E e β d i= X iξ i, αβ,ξ) = ElogZβ,ξ) Observe that to defineαβ,ξ) we do not need to assume exponential moments for the random variable ξ, since logzβ,ξ) β d i= ξ i We now approximate the derivative of the averaged free energy: Lemma 4 αβ, ξ) β d = βe Xi Xi ) +9dE ξ 3 Oβ ), i= where Oβ ) β 3

4 Remark 5 In a Gaussian random environment, the integration by parts formula is an exact formula, therefore the remainder9d E ξ 3 Oβ ) vanishes Proof We have αβ, ξ) β d = E Zβ,ξ) E i= X i ξ i e β d i= X iξ i = E d ξ i F i ξ i ), withf i z) = EX ie βx i z+ψ i X) and ψ Ee βx i z+ψ i X) i X) = β j i X jξ j independent ofξ i If we define H z) = EHeβX i z+ψ i X) Ee βx i z+ψ i X), then Hence, z H z) = β HX i z) H z) X i z)) X F i z) = X i z), F iz) z) = β i X i z)) ) X F i z) = β 3 z) i 3 Xi z) X i z) + X i z)) 3 Since X i, we have F i 6β,0 F i z) β and F i ξ i ) = X i, F iξ i ) = β X i Xi ) We infer from Lemma 3 that sinceeξ =, E X i ξ i = Eξ i F i ξ i ) = βe X i Xi +9E ξ 3 Oβ ), with Oβ ) β Therefore, d αβ, ξ) = βe X β i Xi ) +9dE ξ 3 Oβ ) i= The next step is the comparison of the averaged free energies for the environments ξ and g standard normal) Proposition 6 For any β R, αβ,ξ) αβ,g) 9dE ξ 3 β 3 Proof The interpolation technique of F Guerra relies on the introduction of a two parameter Hamiltonian: Zt,x) = E e t d i= X ig i + x d i= X iξ i and averaged free energy αt,x) = ElogZt,x) where the environments g and ξ are assumed to be independent of each other, g being standard normal By Lemma 4, d t α = E X i Xi i= d x α = E X i Xi +9dE ξ 3 O x), i= 4 i=

5 with O x) x We follow the path xs) = t 0 s,0 s t 0 Then, d ds αs,t 0 s) 9dE ξ 3 t 0, and thus, integrating on 0,t 0 α0,t 0 ) αt 0,0) 9dE ξ 3 t 3/ 0 This is the desired result for β > 0 take β = t 0 ) For negative β, we consider the environment ξ instead We shall now estimate the fluctuations of free energy, the environment is still constructed with iid random variablesξ,,ξ d ) satisfying ) Lemma 7 There exists some universal constantc > 0 such that Consequently, we have E sup E logzβ,ξ) αβ,ξ) 3 ce ξ 3 β 3 d 3/ d X i ) i= X i ξ i E sup X i ) d i= ) 3 X i ξ i ce ξ 3 d 3/ Proof We shall use a martingale decomposition Let F k = σ{ξ,ξ k },k, be the natural filtration generated by ξ k ) Consider the sequence of martingale difference j := E logzβ,ξ) Fj E logzβ,ξ) Fj withf 0 the trivialσ-field Then logzβ,ξ) αβ,ξ) = d j j= j d, Burkholder s martingale inequality says that for some universal constantc > 0, E d 3 d 3/ j c E j) j= To estimate j, we definez j) := E measureq j) by Then j= e β d i=,i j X iξ i and an auxiliary random probability Q j) FX,,X d )) := Z j) E FX,,X d )e β d i=,i j X iξ i, F ) 0 Zβ,ξ) = Z j) Q j) e βx j ξ j ) SinceZ j) is independent ofξ j,logz j) has the same conditional expectation with respect tof j as tof j It follows that j = E logq j) e βx j ξ j ) Fj ) E logq j) e βx j ξ j ) Fj ) 5

6 Using the fact that X j, we get logq j) e βx j ξ j ) β ξj This implies that j β ξ j +E ξ j ) It follows that d E logzβ,ξ) αβ,ξ) 3 c E j= j ) 3/ d c β 3 E ξj +E ξ j ) c β 3 d j= d E ξ j +E ξ j ) 3 j= ce ξ 3 β 3 d 3/, ) 3/ where we used the convexity of the function x x 3/ in the third inequality Finally, considering logzβ,ξ) and lettingβ, we obtain the second estimate and end the β proof 3 Application to Sherrington-Kirkpatrick s model of spin glass Observe that Z β,ξ) = σ e β H σ,ξ)+h i σ i = E e β H τ,ξ)+h i τ i, where τ i ) i are iid with distribution Pτ i = ) = We get rid of the magnetic field by introducing tilted laws: P τ i = ±) = With these notations we have e±h, so that coshh) Ef τ Z β,ξ) = coshh) E Convergence of free energy : Theorem i) = E fτi )e i hτ Ee hτ i e β H τ,ξ) Applying Proposition 6 tox ij = τ i τ j,β β andd = yields α β,ξ) α β,g) = α β,ξ) α β,g) Furthermore, the fluctuations can be controlled by Lemma 7: 9 E ξ 3 β ) 3 = 9E ξ 3 β 3 ) E logz 3 β,ξ) α β,ξ) ce ξ 3 β 3 3/, this gives the as convergence by Borel-Cantelli s Lemma 6

7 Convergence of ground state : Theorem We have, restricting the sum to a configuration yielding a maximum Hamiltonian to get the lower bound, e β S ξ) Z β,ξ) = σ e β H σ,ξ) e β S ξ) Therefore, ES ξ) β α β,ξ) ES ξ) log β Combining with inequality ) yields, by takingβ = /6 3/ ES g) ES ξ) log + β β α β,ξ) α β,g) log +CE ξ 3 β β C +E ξ 3) /6 The almost sure convergence follows in the same way from the control of fluctuations and Borel-Cantelli s Lemma 4 Some Extensions and Generalizations 4 The p-spin model of spin glasses The partition function is Z β,ξ) = σ e β p H σ,ξ)+h i σ i = E e β p H τ,ξ)+h i τ i, where τ i ) i are iid with distribution Pτ i = ) = we get rid of the magnetic field by introducing tilted laws so we assume, without loss in generality, that h = 0) The Hamiltonian is H σ,ξ) = σ i σ ip ξ i i p i,,i p wherξ i i p is an iid family of random variables with common distribution satisfying ) Applying Proposition 6 tox i i p = τ i τ ip, β β p andd = yields α β,ξ) α β,g) 9E ξ 3 β 3 p 4 Integration by parts and comparison of free energies The more information we get on the random media, the more precise our comparison results can be In particular, the more gaussian the environment looks like, the closer the 7

8 free energy is to the gaussian free energy For example, we shall assume here that the random variableξ satisfies E ξ 4 < +, Eξ = E ξ 3 = 0, E ξ = 3) A typical variable in this class is the BernoulliPη = ± = We get the approximate integration by parts formula Lemma 8 Assume that the real random variable ξ satisfies 3) and that the function F : R R is of class C 3 with bounded third derivative F 3) < + Then, Proof This is again Taylor s formula: EξFξ) E ξ EF ξ) F 3) E ξ 4 Fξ) = F0)+ξF 0)+ ξ F 0)+O ξ 3 F 3) ) F ξ) = F 0)+ξF 0)+Oξ F 3) ) Repeating, mutatis mutandis, the proof of Proposition 6 we obtain Proposition 9 There exists a constant C > 0 such that for any environment ξ satisfying 3), and for a Gaussian environmentg, αβ,ξ) αβ,g) CE ξ 4 dβ 4 4) In the framework of Sherrington-Kirkpatrick model of spin glass, this yields The ground state comparison is now α β,ξ) α β,g) CE ξ 4 β 4 3/ ES ξ) ES g) C +E ξ 4) /4 This is of the same order than Talagrand s result Corollary of 4) established for Bernoulli random variables Acknowledgements We gratefully acknowledge fruitful conversations with Francesco Guerra, held during his stay at Université de antes as Professeur Invité in 003 and 004 References Francesco Guerra, Broken replica symmetry bounds in the mean field spin glass model, Commun Math Phys ), no, English) Francesco Guerra and Fabio Lucio Toninelli, The thermodynamic limit in mean field spin glass models, Commun Math Phys 30 00), no, 7 79 English) 8

9 3 David Sherrington and Scott Kirkpatrick, Infinite-ranged model of spin-glasses, Phys Rev B 7 978), Michel Talagrand, Gaussian averages, Bernoulli averages, and Gibbs measures, Random Struct Algorithms 00), no 3-4, English) 5, The generalized Parisi formula, C R, Math, Acad Sci Paris ), no, 4 English) 9