We consider a finite range spin glass model in arbitrary dimension, where the
strength of the two-body coupling decays to zero over some distance
$\gamma^{-1}$. We show that, under mild assumptions on the interaction
potential, the infinite-volume free energy of the system converges to that of
the Sherrington-Kirkpatrick one, in the Kac limit $\gamma\to0$. This could be a
first step toward an expansion around mean field theory, for spin glass
systems.Comment: 4 pages; references updated, typos correcte

Franz, Silvio

Toninelli, Fabio Lucio

English

arXiv

We consider a finite-range spin glass model in arbitrary dimension, where the strength of the two-body coupling decays to zero over some distance ¿-1. We show that, under mild assumptions on the interaction potential, the infinite-volume free energy of the system converges to that of the Sherrington-Kirkpatrick one, in the Kac limit ¿¿0. This could be a first step toward an expansion around mean-field theory, for spin glass systems

Franz, S.

Toninelli, F.L.

Pure OAI Repository

 Kac limit for finite-range spin glassesCitation for published version (APA):Franz, S., & Toninelli, F. L. (2004). Kac limit for finite-range spin glasses. Physical Review Letters, 92(3), Article030602. https://doi.org/10.1103/PhysRevLett.92.030602DOI:10.1103/PhysRevLett.92.030602Document status and date:Published: 01/01/2004Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3Kac Limit for Finite-Range Spin GlassesSilvio Franz1,* and Fabio Lucio Toninelli2,†1The Abdus Salam International Center for Theoretical Physics, Condensed Matter Group, Strada Costiera 11,P.O. Box 586, I-34100 Trieste, Italy2EURANDOM, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands(Received 16 October 2003; published 23 January 2004)030602-1We consider a finite-range spin glass model in arbitrary dimension, where the strength of the two-body coupling decays to zero over some distance 1. We show that, under mild assumptions on theinteraction potential, the infinite-volume free energy of the system converges to that of the Sherrington-Kirkpatrick one, in the Kac limit  ! 0. This could be a first step toward an expansion around mean-field theory, for spin glass systems.DOI: 10.1103/PhysRevLett.92.030602 PACS numbers: 05.20.–y, 75.10.Nrmathematical physics, where one considers variables in- i;j2 i2Despite years of debate, the nature of the spin glassphase of the finite dimensional systems remains a ma-jor open problem in statistical physics. Two competingtheories have been proposed as candidates to explain spinglass physics at low temperature: the theory of replicasymmetry breaking [1,2] and the droplet theory [3,4].The former, based on the analysis of the long rangeSherrington-Kirkpatrick (SK) spin glass, predicts a richphenomenology with ergodicity breaking not related toany physical symmetry breaking and susceptibilityanomalies related to the presence on many pure states.The latter assimilates spin glasses to some kind of ‘‘dis-guised ferromagnet’’— albeit with complex phenomen-ology — where the transition appears as a conventionalsymmetry breaking phenomenon. Both theories beingnonrigorous in the applications to finite dimensional sys-tems, it appears very difficult to solve the question on apurely theoretical ground. On the other hand, experimentsin 3D and numerical simulations in 3D and 4D fail to givecompelling evidence in favor of one or the other of thetwo theories: The times probed in the experiments are tooshort to settle the question of the presence or absence ofreplica symmetry breaking and the related issue ofasymptotic existence of response anomalies during agingdynamics, and the length scales probed in the simulationsare too small to infer the behavior of the thermodynamiclimit. Rigorous analysis of finite dimensional systemsturns out to be very hard, and thus far has not been ableto exclude either scenario, although it has produced [5]considerable conceptual clarification, and shown some ofthe subtleties hidden even in the definition of the infinite-volume limit of these models. Even at the mean-fieldlevel, only very recently, simple interpolation methodshave been introduced [6–8] which have allowed one toprove [9] the Parisi solution for the SK model. Inter-polation methods have subsequently been applied alsoin the context of finite-range spin glasses, e.g., in [10].In this Letter, we focus our attention on the Kac limit offinite-range spin glasses as first considered in [11], andlater studied in [12,13]. Kac models are a classical tool of0031-9007=04=92(3)=030602(3)$22.50 teracting via a potential with finite range   1, whichtends to infinity after the thermodynamic limit is taken.In a classical paper [14], Penrose and Lebowitz provedthat, for conventional nondisordered systems, the free en-ergy tends (modulo the Maxwell construction) to the oneof the corresponding mean-field system where the inter-actions do not decay with distance and scale with the sizeof the system. We combine here the idea of the interpolat-ing model with the idea [14] of dividing the system intoboxes of suitable size to prove the same property in spinglasses.Other disordered models with Kac-type interactionshave been studied in previous literature. For instance,see [15] and references therein for the case of theHopfield model.The model we consider is defined on the d-dimensionallattice Zd, with Ising spin degrees of freedom i 1; i 2 Zd. Given a finite hypercube  of side L, onedefines the finite volume Hamiltonian asH ; h; J  Xi;j2wi j;2WsJijij  hXi2i;(1)where W Pi2Zdwi; and wr;  dr forsome smooth, non-negative function r, decaying suf-ficiently fast for jrj ! 1 to have W<1. The parame-ter   1 is the inverse range of the interaction. Thequenched couplings Jij are independent identically dis-tributed Gaussian N0; 1 variables, and we denote by Ethe corresponding averages. As is well known [16,17], theinfinite-volume limit of the quenched free energyf; h   limL!11jjE lnZ ; h; J (2)exists.On the other hand, the Hamiltonian of the SK spinglass mean-field model is defined as [18]HSKjj; h; J  12jjp X Jijij  hXi; (3)2004 The American Physical Society 030602-1P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3where jj  Ld is the number of lattice sites in .Subadditivity of the corresponding free energy and exis-tence of its infinite-volume limitfSK; h   limL!11jjE lnZSKjj; h; J; (4)has been proven in [6].It was recently shown in [13] that the free energy ofmodel (1) is bounded below by that of SK:f; h  fSK; h; (5)for any value of d, , h, and , provided that the potentiali j is non-negative definite, i.e., its Fourier trans-form is non-negative. For instance, it is immediate tocheck this condition for i j  ePd1jijj; whichfor d  1 is just the potential considered originally byKac in [19]. In the present paper, we provide the comple-mentary bound, which allows to fully characterize thequenched free energy in the Kac limit  ! 0: AssumethatPi2Zdi<1. Then, for any  and h, one haslim!0f; h  fSK; h: (6)If, in addition, all the Fourier components of  are non-negative, thenlim!0f; h  fSK; h: (7)Together with Talagrand’s recently established proof [9]of the Parisi ansatz for the SK model, this shows that theParisi theory [1] gives the correct free energy for finitedimensional spin glasses in the Kac limit.The idea of the proof is to interpolate between the Kacmodel in a volume jj and a system made of a collectionof many independent SK subsystems of volume M  ‘d.The crucial point, as in [14], is to choose‘    L; (8)and to let the three lengths diverge in this order. Let usdivide the box  into subcubes n of volume M, n 1; . . . ; jj=M, and introduce the interpolating partitionfunction,ZtXexp1 tp XnXi;j2nJij2Mp ijexptp Xi;j2wi j;2WsJ0ijijhXi2i;where the Gaussian variables J0 are independent of the J.Note that1jjE lnZ0 1ME lnZSKM ; h; J; (9)1jjE lnZ1 1jjE lnZ ; h; J: (10)As we show below, one has030602-2lim!0limL!1ddt1jjE lnZt  0 (11)uniformly for 0  t  1. After integration on t between 0and 1, and taking the large M limit, one finds thereforethe desired result:lim!0f; h  limM!11ME lnZSKM ; h; J fSK; h: (12)Denoting as hi the Gibbs average, the computation of thet derivative gives, up to terms negligible for large L,ddt1jjE lnZt 24jjEXnXi;j2n1Mhiji2Xi;j2wi j;Whiji2	;(13)where we have used integration by parts on the Gaussiandisorder and the propertylimL!11jjXi;j2wi j;W 1: (14)Introducing two replicas with identical quenched cou-plings and spin configurations 1; 2, we can write(13) asddt1jjElnZt24jjEXn1MXi;j2nh1i 2i 1j2j iXi;j2wij;Wh1i 2i 1j2j i	:(15)Denoting the partial overlap in the nth subcube as qn12 1=MPi2n1i 2i , the first term of the right-hand side (rhs)can be rewritten as2M4jjXnEhqn12 2i: (16)As for the second term, definingwmn  supi2m;j2nwi j;W; (17)and using the straightforward inequality 2xy  x2  y2,one has1jjXi;j2wi j;WEh1i 2i 1j2j i M22jjXn;mwmnEhqn12 2 qm12 2i:(18)In the Kac limit  ! 0, the diagonal terms n  m give a030602-2P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3vanishing contribution. As for the nondiagonal ones, oneobserves thatlim!0Xmnwmn 1M; (19)where the summation runs only on one of the two indices,so that finally the rhs of (18) is bounded above byMjjXnEhqn12 2i; (20)apart from a negligible error term. Together with Eqs. (15)and (16), this proves (11) and therefore the theorem.As a side remark, it is easy to employ this method,together with that of [13], to obtain a new proof of theexistence of the thermodynamic limit for the SK model,independent of the convexity argument developed in [6].It is possible to generalize this theorem to the ‘‘dilutedKac spin glass’’ case [13], where each given spin iinteracts with a finite random number of other spins j,which are chosen randomly according to a probabilitydistribution that decays to zero on the scale , as ji jjdiverges. In the Kac limit  ! 1, one can prove that thefree energy of the model converges to that of its mean-field counterpart, which in that case is the Viana-Braymodel [20]. Full details of the proof are given in [21].A second generalization of our result is to consider tworeplicas of the system, coupled via a term depending ontheir mutual overlap. This problem has been considered,for instance, in [22] and is relevant for the study of glassydynamics, especially if applied to models which exhibit‘‘one-step replica symmetry breaking’’ [1]. The new fea-ture here is that, at the mean-field level, the free energy ofthe coupled system can be expressed [22] in terms of aneffective potential depending on the overlap, which turnsout to be nonconvex. It was argued in [23] that a minimalmodification of the theory in finite dimension requiresrestoration of the convexity through the Maxwell con-struction. This, analogously to the ordered case [14],emerges naturally in the Kac limit of finite-range models.We plan to report on this soon [24].The main interest of the result presented in this Letteris that it could represent, for spin glasses, a first steptoward an expansion around the mean-field case, whichwould hopefully shed some light on the nature of the spinglass phase for models with finite —albeit large—inter-action range. This hope is supported by the fact that asimilar program has been successfully carried on re-cently for nonrandom ferromagnetic spin systems [25–27] and continuous particle systems [28], showing that indimension d  2 it is possible to write a controlled ex-pansion around the   0 point, and to prove rigorouslythat for large but finite  the system has a phase transition(broken spin flip or liquid vapor, respectively) with coex-isting phases.We would like to thank Francesco Guerra for manyenlightening conversations. F. L.T. is grateful to the030602-3Condensed Matter Group of the ICTP for kind hospitalityduring the preparation of this work. This work was sup-ported in part by the European Community’s HumanPotential program under Contract No. HPRN-CT-2002-00319 STIPCO.*Electronic address: franz@ictp.trieste.it†Electronic address: ftoninel@euridice.tue.nl[1] M. Mézard, G. Parisi, and M. A. Virasoro, Spin GlassTheory and Beyond (World Scientific, Singapore, 1987).[2] E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. Ruiz-Lorenzo, and F. Zuliani, J. Stat. Phys. 98, 973 (2000).[3] D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601(1986).[4] A. J. Bray and M. A. Moore, in Heidelberg Colloquium onGlassy Dynamics, edited by J. L. Van Hemmen andI. Morgenstern (Springer-Verlag, Berlin, 1986), p. 121.[5] C. M. Newman and D. L. Stein, J. Phys. Condens. Matter15, R1319 (2003).[6] F. Guerra and F. L. Toninelli, Commun. Math. Phys. 230,71 (2002).[7] F. Guerra, Commun. Math. Phys. 233, 1 (2003).[8] M. Aizenman, R. Sims, and S. L. Starr, cond-mat/0306386 (to be published).[9] M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337, 111(2003).[10] P. Contucci and S. Graffi, math-ph/0302013 (to bepublished).[11] J. Fröhlich and B. Zegarlinski, Commun. Math. Phys.112, 553 (1987).[12] A. Bovier, J. Stat. Phys. 91, 459 (1998).[13] F. Guerra and F. L. Toninelli, J. Phys. A 36, 10 987 (2003).[14] J. L. Lebowitz and O. Penrose, J. Math. Phys. (N.Y.) 7, 98(1966).[15] A. Bovier, V. Gayrard, and P. Picco, in MathematicalAspects of Spin Glasses and Neuronal Networks, editedby A. Bovier and P. Picco, Progress in Probability(Birkhäuser, Basel, 1998), Vol. 41, pp. 3.[16] P. A. Vuillermot, J. Phys. A 10, 1319 (1977).[17] A. C. D. van Enter and J. L. van Hemmen, J. Stat. Phys.32, 141 (1983).[18] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35,1792 (1975).[19] M. Kac, Phys. Fluids 2, 8 (1959).[20] L. Viana and A. J. Bray, J. Phys. C 18, 3037 (1985).[21] S. Franz and F. L. Toninelli (to be published).[22] S. Franz and G. Parisi, J. Phys. I (France) 5, 1401 (1995).[23] S. Franz and G. Parisi, Phys. Rev. Lett. 79, 2486 (1997);Physica (Amsterdam) 261A, 317 (1998).[24] S. Franz and F. L. Toninelli (unpublished).[25] M. Cassandro and E. Presutti, Markov Proc. Rel. Fields2, 241 (1996).[26] A. Bovier and M. Zahradnı́k, J. Stat. Phys. 87, 311 (1997).[27] T. Bodineau and E. Presutti, Commun. Math. Phys. 189,287 (1997).[28] J. L. Lebowitz, A. E. Mazel, and E. Presutti, Phys. Rev.Lett. 80, 4701 (1998); J. Stat. Phys. 94, 955 (1999).030602-3

Kac limit for finite-range spin glasses

 Kac limit for finite-range spin glassesCitation for published version (APA):Franz, S., & Toninelli, F. L. (2004). Kac limit for finite-range spin glasses. Physical Review Letters, 92(3), Article030602. https://doi.org/10.1103/PhysRevLett.92.030602DOI:10.1103/PhysRevLett.92.030602Document status and date:Published: 01/01/2004Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3Kac Limit for Finite-Range Spin GlassesSilvio Franz1,* and Fabio Lucio Toninelli2,†1The Abdus Salam International Center for Theoretical Physics, Condensed Matter Group, Strada Costiera 11,P.O. Box 586, I-34100 Trieste, Italy2EURANDOM, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands(Received 16 October 2003; published 23 January 2004)030602-1We consider a finite-range spin glass model in arbitrary dimension, where the strength of the two-body coupling decays to zero over some distance 1. We show that, under mild assumptions on theinteraction potential, the infinite-volume free energy of the system converges to that of the Sherrington-Kirkpatrick one, in the Kac limit  ! 0. This could be a first step toward an expansion around mean-field theory, for spin glass systems.DOI: 10.1103/PhysRevLett.92.030602 PACS numbers: 05.20.–y, 75.10.Nrmathematical physics, where one considers variables in- i;j2 i2Despite years of debate, the nature of the spin glassphase of the finite dimensional systems remains a ma-jor open problem in statistical physics. Two competingtheories have been proposed as candidates to explain spinglass physics at low temperature: the theory of replicasymmetry breaking [1,2] and the droplet theory [3,4].The former, based on the analysis of the long rangeSherrington-Kirkpatrick (SK) spin glass, predicts a richphenomenology with ergodicity breaking not related toany physical symmetry breaking and susceptibilityanomalies related to the presence on many pure states.The latter assimilates spin glasses to some kind of ‘‘dis-guised ferromagnet’’— albeit with complex phenomen-ology — where the transition appears as a conventionalsymmetry breaking phenomenon. Both theories beingnonrigorous in the applications to finite dimensional sys-tems, it appears very difficult to solve the question on apurely theoretical ground. On the other hand, experimentsin 3D and numerical simulations in 3D and 4D fail to givecompelling evidence in favor of one or the other of thetwo theories: The times probed in the experiments are tooshort to settle the question of the presence or absence ofreplica symmetry breaking and the related issue ofasymptotic existence of response anomalies during agingdynamics, and the length scales probed in the simulationsare too small to infer the behavior of the thermodynamiclimit. Rigorous analysis of finite dimensional systemsturns out to be very hard, and thus far has not been ableto exclude either scenario, although it has produced [5]considerable conceptual clarification, and shown some ofthe subtleties hidden even in the definition of the infinite-volume limit of these models. Even at the mean-fieldlevel, only very recently, simple interpolation methodshave been introduced [6–8] which have allowed one toprove [9] the Parisi solution for the SK model. Inter-polation methods have subsequently been applied alsoin the context of finite-range spin glasses, e.g., in [10].In this Letter, we focus our attention on the Kac limit offinite-range spin glasses as first considered in [11], andlater studied in [12,13]. Kac models are a classical tool of0031-9007=04=92(3)=030602(3)$22.50 teracting via a potential with finite range   1, whichtends to infinity after the thermodynamic limit is taken.In a classical paper [14], Penrose and Lebowitz provedthat, for conventional nondisordered systems, the free en-ergy tends (modulo the Maxwell construction) to the oneof the corresponding mean-field system where the inter-actions do not decay with distance and scale with the sizeof the system. We combine here the idea of the interpolat-ing model with the idea [14] of dividing the system intoboxes of suitable size to prove the same property in spinglasses.Other disordered models with Kac-type interactionshave been studied in previous literature. For instance,see [15] and references therein for the case of theHopfield model.The model we consider is defined on the d-dimensionallattice Zd, with Ising spin degrees of freedom i 1; i 2 Zd. Given a finite hypercube  of side L, onedefines the finite volume Hamiltonian asH ; h; J  Xi;j2wi j;2WsJijij  hXi2i;(1)where W Pi2Zdwi; and wr;  dr forsome smooth, non-negative function r, decaying suf-ficiently fast for jrj ! 1 to have W<1. The parame-ter   1 is the inverse range of the interaction. Thequenched couplings Jij are independent identically dis-tributed Gaussian N0; 1 variables, and we denote by Ethe corresponding averages. As is well known [16,17], theinfinite-volume limit of the quenched free energyf; h   limL!11jjE lnZ ; h; J (2)exists.On the other hand, the Hamiltonian of the SK spinglass mean-field model is defined as [18]HSKjj; h; J  12jjp X Jijij  hXi; (3)2004 The American Physical Society 030602-1P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3where jj  Ld is the number of lattice sites in .Subadditivity of the corresponding free energy and exis-tence of its infinite-volume limitfSK; h   limL!11jjE lnZSKjj; h; J; (4)has been proven in [6].It was recently shown in [13] that the free energy ofmodel (1) is bounded below by that of SK:f; h  fSK; h; (5)for any value of d, , h, and , provided that the potentiali j is non-negative definite, i.e., its Fourier trans-form is non-negative. For instance, it is immediate tocheck this condition for i j  ePd1jijj; whichfor d  1 is just the potential considered originally byKac in [19]. In the present paper, we provide the comple-mentary bound, which allows to fully characterize thequenched free energy in the Kac limit  ! 0: AssumethatPi2Zdi<1. Then, for any  and h, one haslim!0f; h  fSK; h: (6)If, in addition, all the Fourier components of  are non-negative, thenlim!0f; h  fSK; h: (7)Together with Talagrand’s recently established proof [9]of the Parisi ansatz for the SK model, this shows that theParisi theory [1] gives the correct free energy for finitedimensional spin glasses in the Kac limit.The idea of the proof is to interpolate between the Kacmodel in a volume jj and a system made of a collectionof many independent SK subsystems of volume M  ‘d.The crucial point, as in [14], is to choose‘    L; (8)and to let the three lengths diverge in this order. Let usdivide the box  into subcubes n of volume M, n 1; . . . ; jj=M, and introduce the interpolating partitionfunction,ZtXexp1 tp XnXi;j2nJij2Mp ijexptp Xi;j2wi j;2WsJ0ijijhXi2i;where the Gaussian variables J0 are independent of the J.Note that1jjE lnZ0 1ME lnZSKM ; h; J; (9)1jjE lnZ1 1jjE lnZ ; h; J: (10)As we show below, one has030602-2lim!0limL!1ddt1jjE lnZt  0 (11)uniformly for 0  t  1. After integration on t between 0and 1, and taking the large M limit, one finds thereforethe desired result:lim!0f; h  limM!11ME lnZSKM ; h; J fSK; h: (12)Denoting as hi the Gibbs average, the computation of thet derivative gives, up to terms negligible for large L,ddt1jjE lnZt 24jjEXnXi;j2n1Mhiji2Xi;j2wi j;Whiji2	;(13)where we have used integration by parts on the Gaussiandisorder and the propertylimL!11jjXi;j2wi j;W 1: (14)Introducing two replicas with identical quenched cou-plings and spin configurations 1; 2, we can write(13) asddt1jjElnZt24jjEXn1MXi;j2nh1i 2i 1j2j iXi;j2wij;Wh1i 2i 1j2j i	:(15)Denoting the partial overlap in the nth subcube as qn12 1=MPi2n1i 2i , the first term of the right-hand side (rhs)can be rewritten as2M4jjXnEhqn12 2i: (16)As for the second term, definingwmn  supi2m;j2nwi j;W; (17)and using the straightforward inequality 2xy  x2  y2,one has1jjXi;j2wi j;WEh1i 2i 1j2j i M22jjXn;mwmnEhqn12 2 qm12 2i:(18)In the Kac limit  ! 0, the diagonal terms n  m give a030602-2P H Y S I C A L R E V I E W L E T T E R S week ending23 JANUARY 2004VOLUME 92, NUMBER 3vanishing contribution. As for the nondiagonal ones, oneobserves thatlim!0Xmnwmn 1M; (19)where the summation runs only on one of the two indices,so that finally the rhs of (18) is bounded above byMjjXnEhqn12 2i; (20)apart from a negligible error term. Together with Eqs. (15)and (16), this proves (11) and therefore the theorem.As a side remark, it is easy to employ this method,together with that of [13], to obtain a new proof of theexistence of the thermodynamic limit for the SK model,independent of the convexity argument developed in [6].It is possible to generalize this theorem to the ‘‘dilutedKac spin glass’’ case [13], where each given spin iinteracts with a finite random number of other spins j,which are chosen randomly according to a probabilitydistribution that decays to zero on the scale , as ji jjdiverges. In the Kac limit  ! 1, one can prove that thefree energy of the model converges to that of its mean-field counterpart, which in that case is the Viana-Braymodel [20]. Full details of the proof are given in [21].A second generalization of our result is to consider tworeplicas of the system, coupled via a term depending ontheir mutual overlap. This problem has been considered,for instance, in [22] and is relevant for the study of glassydynamics, especially if applied to models which exhibit‘‘one-step replica symmetry breaking’’ [1]. The new fea-ture here is that, at the mean-field level, the free energy ofthe coupled system can be expressed [22] in terms of aneffective potential depending on the overlap, which turnsout to be nonconvex. It was argued in [23] that a minimalmodification of the theory in finite dimension requiresrestoration of the convexity through the Maxwell con-struction. This, analogously to the ordered case [14],emerges naturally in the Kac limit of finite-range models.We plan to report on this soon [24].The main interest of the result presented in this Letteris that it could represent, for spin glasses, a first steptoward an expansion around the mean-field case, whichwould hopefully shed some light on the nature of the spinglass phase for models with finite —albeit large—inter-action range. This hope is supported by the fact that asimilar program has been successfully carried on re-cently for nonrandom ferromagnetic spin systems [25–27] and continuous particle systems [28], showing that indimension d  2 it is possible to write a controlled ex-pansion around the   0 point, and to prove rigorouslythat for large but finite  the system has a phase transition(broken spin flip or liquid vapor, respectively) with coex-isting phases.We would like to thank Francesco Guerra for manyenlightening conversations. F. L.T. is grateful to the030602-3Condensed Matter Group of the ICTP for kind hospitalityduring the preparation of this work. This work was sup-ported in part by the European Community’s HumanPotential program under Contract No. HPRN-CT-2002-00319 STIPCO.*Electronic address: franz@ictp.trieste.it†Electronic address: ftoninel@euridice.tue.nl[1] M. Mézard, G. Parisi, and M. A. Virasoro, Spin GlassTheory and Beyond (World Scientific, Singapore, 1987).[2] E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. Ruiz-Lorenzo, and F. Zuliani, J. Stat. Phys. 98, 973 (2000).[3] D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601(1986).[4] A. J. Bray and M. A. Moore, in Heidelberg Colloquium onGlassy Dynamics, edited by J. L. Van Hemmen andI. Morgenstern (Springer-Verlag, Berlin, 1986), p. 121.[5] C. M. Newman and D. L. Stein, J. Phys. Condens. Matter15, R1319 (2003).[6] F. Guerra and F. L. Toninelli, Commun. Math. Phys. 230,71 (2002).[7] F. Guerra, Commun. Math. Phys. 233, 1 (2003).[8] M. Aizenman, R. Sims, and S. L. Starr, cond-mat/0306386 (to be published).[9] M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337, 111(2003).[10] P. Contucci and S. Graffi, math-ph/0302013 (to bepublished).[11] J. Fröhlich and B. Zegarlinski, Commun. Math. Phys.112, 553 (1987).[12] A. Bovier, J. Stat. Phys. 91, 459 (1998).[13] F. Guerra and F. L. Toninelli, J. Phys. A 36, 10 987 (2003).[14] J. L. Lebowitz and O. Penrose, J. Math. Phys. (N.Y.) 7, 98(1966).[15] A. Bovier, V. Gayrard, and P. Picco, in MathematicalAspects of Spin Glasses and Neuronal Networks, editedby A. Bovier and P. Picco, Progress in Probability(Birkhäuser, Basel, 1998), Vol. 41, pp. 3.[16] P. A. Vuillermot, J. Phys. A 10, 1319 (1977).[17] A. C. D. van Enter and J. L. van Hemmen, J. Stat. Phys.32, 141 (1983).[18] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35,1792 (1975).[19] M. Kac, Phys. Fluids 2, 8 (1959).[20] L. Viana and A. J. Bray, J. Phys. C 18, 3037 (1985).[21] S. Franz and F. L. Toninelli (to be published).[22] S. Franz and G. Parisi, J. Phys. I (France) 5, 1401 (1995).[23] S. Franz and G. Parisi, Phys. Rev. Lett. 79, 2486 (1997);Physica (Amsterdam) 261A, 317 (1998).[24] S. Franz and F. L. Toninelli (unpublished).[25] M. Cassandro and E. Presutti, Markov Proc. Rel. Fields2, 241 (1996).[26] A. Bovier and M. Zahradnı́k, J. Stat. Phys. 87, 311 (1997).[27] T. Bodineau and E. Presutti, Commun. Math. Phys. 189,287 (1997).[28] J. L. Lebowitz, A. E. Mazel, and E. Presutti, Phys. Rev.Lett. 80, 4701 (1998); J. Stat. Phys. 94, 955 (1999).030602-3

The Kac limit for finite-range spin glasses

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