arXiv subjects

Wei-Kuo Chen

Publications and source records attributed to Wei-Kuo Chen.

49 records · Page 3Linked to original sources

arXiv2014

Universality of chaos and ultrametricity in mixed p-spin models

We prove disorder universality of chaos phenomena and ultrametricity in the mixed p-spin model under mild moment assumptions on the environment. This establishes the long-standing belief among physicists that the Parisi solution in mean-field models is universal. Our results extend to universal properties of other physical observables in the mixed p-spin model as well as in different spin glass models. These include universality of quenched disorder chaos in the Edwards-Anderson (EA) model and quenched concentration for the magnetization in both EA and mixed p-spin models under non-Gaussian environments. In addition, we show quenched self-averaging for the overlap in the random field Ising model under small perturbation of the external field.

math.PR↗

arXiv2014

Some examples of quenched self-averaging in models with Gaussian disorder

In this paper we give an elementary approach to several results of Chatterjee in arXiv:0907.3381 and arXiv:1404.7178, as well as some generalizations. First, we prove quenched disorder chaos for the bond overlap in the Edwards-Anderson type models with Gaussian disorder. The proof extends to systems at different temperatures and covers a number of other models, such as the mixed $p$-spin model, Sherrington-Kirkpatrick model with multi-dimensional spins and diluted $p$-spin model. Next, we adapt the same idea to prove quenched self-averaging of the bond magnetization for one system and use it to show quenched self-averaging of the site overlap for random field models with positively correlated spins. Finally, we show self-averaging for certain modifications of the random field itself.

math.PR↗

arXiv2014

The Parisi formula has a unique minimizer

In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand and later generalized to the mixed p-spin models by Panchenko. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi functional.

math.PR↗

arXiv2014

Free energy and complexity of spherical bipartite models

We investigate both free energy and complexity of the spherical bipartite spin glass model. We first prove a variational formula in high temperature for the limiting free energy based on the well-known Crisanti-Sommers representation of the mixed p-spin spherical model. Next, we show that the mean number of local minima at low levels of energy is exponentially large in the size of the system and we derive a bound on the location of the ground state energy.

math.PR↗

arXiv2013

Chaos in the mixed even-spin models

We consider a disordered system obtained by coupling two mixed even-spin models together. The chaos problem is concerned with the behavior of the coupled system when the external parameters in the two models, such as, temperature, disorder, or external field, are slightly different. It is conjectured that the overlap between two independently sampled spin configurations from, respectively, the Gibbs measures of the two models is essentially concentrated around a constant under the coupled Gibbs measure. Using the extended Guerra replica symmetry breaking bound together with a recent development of controlling the overlap using the Ghirlanda-Guerra identities as well as a new family of identities, we present rigorous results on chaos in temperature. In addition, chaos in disorder and in external field are addressed.

math.PR↗

arXiv2013

Partial results on the convexity of the Parisi functional with PDE approach

We investigate the convexity problem for the Parisi functional defined on the space of the so-called functional ordered parameters in the Sherrington-Kirkpatrick model. In the recent work of Panchenko [3], he proved that this functional is convex along one-sided directions with a probabilistic method. In this paper, we will study this problem with a PDE approach that simplifies his original proof and presents more general results.

math.PR↗

arXiv2013

Improved Hölder and reverse Hölder inequalities for correlated Gaussian random vectors

We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp-Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian Hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in literatures including the Prékopa-Leindler and Barthe inequalities.

math.PR↗

arXiv2013

Disorder chaos in the Sherrington-Kirkpatrick model with external field

We consider a spin system obtained by coupling two distinct Sherrington-Kirkpatrick (SK) models with the same temperature and external field whose Hamiltonians are correlated. The disorder chaos conjecture for the SK model states that the overlap under the corresponding Gibbs measure is essentially concentrated at a single value. In the absence of external field, this statement was first confirmed by Chatterjee [Disorder chaos and multiple valleys in spin glasses (2009) Preprint]. In the present paper, using Guerra's replica symmetry breaking bound, we prove that the SK model is also chaotic in the presence of the external field and the position of the overlap is determined by an equation related to Guerra's bound and the Parisi measure.

math.PR↗

arXiv2013

On properties of Parisi measures

We investigate the structure of Parisi measures, the functional order parameters of mixed p-spin models in mean field spin glasses. In the absence of external field, we prove that a Parisi measure satisfies the following properties. First, at all temperatures, the support of any Parisi measure contains the origin. If it contains an open interval, then the measure has a smooth density on this interval. Next, we give a criterion on temperature parameters for which a Parisi measure is neither Replica Symmetric nor One Replica Symmetry Breaking. Finally, we show that in the Sherrington-Kirkpatrick model, slightly above the critical temperature, the largest number in the support of a Parisi measure is a jump discontinuity. An analogue of these results is discussed in the spherical mixed p-spin models. As a tool to establish these facts and of independent interest, we study functionals of the associated Parisi PDEs and derive regularity properties of their solutions.

math.PR↗

arXiv2012

On the Mixed Even-Spin Sherrington-Kirkpatrick Model with Ferromagnetic Interaction

We study a spin system with both mixed even-spin Sherrington-Kirkpatrick (SK) couplings and Curie-Weiss (CW) interaction. Our main results are: (i) The thermodynamic limit of the free energy is given by a variational formula involving the free energy of the SK model with a change in the external field. (ii) In the presence of a centered Gaussian external field, the positivity of the overlap and the extended Ghirlanda-Guerra identities hold on a dense subset of the temperature parameters. (iii) We establish a general inequality between the magnetization and overlap. (iv) We construct a temperature region in which the magnetization can be quantitatively controlled and deduce different senses of convergence for the magnetization depending on whether the external field is present or not. Our approach is based on techniques from the study of the CW and SK models and results in convex analysis.

math.PR↗

arXiv2012

The Aizenman-Sims-Starr scheme and Parisi formula for mixed p-spin spherical models

The Parisi formula for the free energy in the spherical models with mixed even p-spin interactions was proven in Michel Talagrand [16]. In this paper we study the general mixed p-spin spherical models including p-spin interactions for odd p. We establish the Aizenman-Sims-Starr scheme and from this together with many well-known results and Dmitry Panchenko's recent proof on the Parisi ultrametricity conjecture [11], we prove the Parisi formula.

math.PR↗

arXiv2012

An approach to chaos in some mixed p-spin models

We consider the problems of chaos in disorder and temperature for coupled copies of the mixed p-spin models. Under certain assumptions on the parameters of the models we will first prove a weak form of chaos by showing that the overlap is concentrated around its Gibbs average depending on the disorder and then obtain several results toward strong chaos by providing control of the overlap between two systems in terms of their Parisi measures.

math.PR↗

arXiv2011

Central Limit Theorems for Cavity and Local Fields of the Sherrington-Kirkpatrick Model

One of the remarkable applications of the cavity method is to prove the Thouless-Anderson-Palmer (TAP) system of equations in the high temperature regime of the Sherrington-Kirkpatrick (SK) model. This naturally leads us to the important study of the limit laws for cavity and local fields. The first quantitative results for both fields based on Stein's method were studied by Chatterjee. Although Stein's method provides us an efficient approach for obtaining the limiting distributions, the nature of this method restricts the derivation of optimal and general results. In this paper, our study based on Gaussian interpolation obtains the CLT for the cavity field. With the help of this result, we conclude the CLT for local fields. In both cases, more refined moment estimates are given.

math.PR↗