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Dmitry Panchenko

Publications and source records attributed to Dmitry Panchenko.

At least 37 records · Page 2Linked to original sources
arXiv2015

The free energy in a multi-species Sherrington-Kirkpatrick model

The authors of [Ann. Henri Poincaré 16 (2015) 691-708] introduced a multi-species version of the Sherrington-Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra's replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species form of the Ghirlanda-Guerra identities and showing that they force the overlaps within species to be completely determined by the overlaps of the whole system.

math.PR↗
arXiv2015

Fluctuations of the free energy in the mixed $p$-spin models with external field

We show that the free energy in the mixed $p$-spin models of spin glasses does not superconcentrate in the presence of external field, which means that its variance is of the order suggested by the Poincaré inequality. This complements the result of Chatterjee who showed that the free energy superconcentrates when there is no external field. For models without odd $p$-spin interactions for $p\geq 3$, we prove the central limit theorem for the free energy at any temperature and give an explicit formula for the limiting variance. Although we only deal with the case of Ising spins, all our results can be extended to the spherical models as well.

math.PR↗
arXiv2015

The Parisi ultrametricity conjecture

In this paper we prove that the support of a random measure on the unit ball of a separable Hilbert space that satisfies the Ghirlanda-Guerra identities must be ultrametric with probability one. This implies the Parisi ultrametricity conjecture in mean-field spin glass models, such as the Sherrington-Kirkpatrick and mixed $p$-spin models, for which Gibbs' measures are known to satisfy the Ghirlanda-Guerra identities in the thermodynamic limit.

math.PR↗
arXiv2015

Structure of finite-RSB asymptotic Gibbs measures in the diluted spin glass models

We suggest a possible approach to proving the Mézard-Parisi formula for the free energy in the diluted spin glass models, such as diluted K-spin or random K-sat model at any positive temperature. In the main contribution of the paper, we show that a certain small modification of the Hamiltonian in any of these models forces all finite-RSB asymptotic Gibbs measures in the sense of the overlaps to satisfy the Mézard-Parisi ansatz for the distribution of spins. Unfortunately, what is still missing is a description of the general full-RSB asymptotic Gibbs measures. If one could show that the general case can be approximated by finite-RSB case in the right sense then one could a posteriori remove the small modification of the Hamiltonian to recover the Mézard-Parisi formula for the original model.

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 for mixed $p$-spin models

The Parisi formula for the free energy in the Sherrington-Kirkpatrick and mixed $p$-spin models for even $p\geq2$ was proved in the seminal work of Michel Talagrand [Ann. of Math. (2) 163 (2006) 221-263]. In this paper we prove the Parisi formula for general mixed $p$-spin models which also include $p$-spin interactions for odd $p$. Most of the ideas used in the paper are well known and can now be combined following a recent proof of the Parisi ultrametricity conjecture in [Ann. of Math. (2) 177 (2013) 383-393].

math.PR↗
arXiv2013

Structure of 1-RSB asymptotic Gibbs measures in the diluted p-spin models

In this paper we study asymptotic Gibbs measures in the diluted p-spin models in the so called 1-RSB case, when the overlap takes two values $q_*, q^*\in [0,1].$ When the external field is not present and the overlap is not equal to zero, we prove that such asymptotic Gibbs measures are described by the Mézard-Parisi ansatz conjectured in [MP]. When the external field is present, we prove that the overlap can not be equal to zero and all 1-RSB asymptotic Gibbs measures are described by the Mézard-Parisi ansatz. Finally, we give a characterization of the exceptional case when there is no external field and the smallest overlap value is equal to zero, although it does not go as far as the Mézard-Parisi ansatz. Our approach is based on the cavity computations combined with the hierarchical exchangeability of pure states.

math.PR↗
arXiv2013

A hierarchical version of the de Finetti and Aldous-Hoover representations

We consider random arrays indexed by the leaves of an infinitary rooted tree of finite depth, with the distribution invariant under the rearrangements that preserve the tree structure. We call such arrays hierarchically exchangeable and prove that they satisfy an analogue of de Finetti's theorem. We also prove a more general result for arrays indexed by several trees, which includes a hierarchical version of the Aldous-Hoover representation.

math.PR↗
arXiv2013

Hierarchical exchangeability of pure states in mean field spin glass models

The main result in this paper is motivated by the Mézard-Parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models, such as the random K-sat model. Using the fact that one can safely assume the validity of the Ghirlanda-Guerra identities in these models, we prove hierarchical exchangeability of pure states for the asymptotic Gibbs measures, which allows us to apply a representation result for hierarchically exchangeable arrays recently proved in arXiv:1301.1259. Comparing this representation with the predictions of the Mézard-Parisi ansatz, one can see that the key property still missing is that the multi-overlaps between pure states depend only on their overlaps.

math.PR↗
arXiv2013

Spin glass models from the point of view of spin distributions

In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function $σ:[0,1]^4\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington-Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $σ$. In the setting of the Sherrington-Kirkpatrick model, the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps, and we give an explicit formula for $σ$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda-Guerra identities and stochastic stability and describe the expected Parisi ansatz in the diluted models in terms of $σ$.

math.PR↗
arXiv2013

On the replica symmetric solution of the K-sat model

In this paper we translate Talagrand's solution of the K-sat model at high temperature into the language of asymptotic Gibbs measures. Using exact cavity equations in the infinite volume limit allows us to remove many technicalities of the inductions on the system size, which clarifies the main ideas of the proof. This approach also yields a larger region of parameters where the system is in a pure state and, in particular, for small connectivity parameter we prove the replica symmetric formula for the free energy at any temperature.

math.PR↗
arXiv2012

The Sherrington-Kirkpatrick model: an overview

The goal of this paper is to review some of the main ideas that emerged from the attempts to confirm mathematically the predictions of the celebrated Parisi ansatz in the Sherrington-Kirkpatrick model. We try to focus on the big picture while sketching the proofs of only a few selected results, but an interested reader can find most of the missing details in [31] and [44].

math-ph↗
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

A new representation of the Ghirlanda-Guerra identities with applications

In this paper we obtain a new family of identities for random measures on the unit ball of a separable Hilbert space which arise as the asymptotic analogues of the Gibbs measures in the Sherrington-Kirkpatrick and $p$-spin models and which are known to satisfy the Ghirlanda-Guerra identities. We give several applications of the new identities to structural results for such measures.

math.PR↗
arXiv2011

A unified stability property in spin glasses

Gibbs' measures in the Sherrington-Kirkpatrick type models satisfy two asymptotic stability properties, the Aizenman-Contucci stochastic stability and the Ghirlanda-Guerra identities, which play a fundamental role in our current understanding of these models. In this paper we show that one can combine these two properties very naturally into one unified stability property.

math.PR↗
arXiv2011

Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case

In this paper we give another proof of the fact that a random overlap array, which satisfies the Ghirlanda-Guerra identities and whose elements take values in a finite set, is ultrametric with probability one. The new proof bypasses random change of density invariance principles for directing measures of such arrays and, in addition to the Dobvysh-Sudakov representation, is based only on elementary algebraic consequences of the Ghirlanda-Guerra identities.

math.PR↗