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

The Parisi formula for mixed pp-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].Comment: Published in at http://dx.doi.org/10.1214/12-AOP800 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A connection between the Ghirlanda--Guerra identities and ultrametricity

We consider a symmetric positive definite weakly exchangeable infinite random matrix and show that, under the technical condition that its elements take a finite number of values, the Ghirlanda--Guerra identities imply ultrametricity.Comment: Published in at http://dx.doi.org/10.1214/09-AOP484 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A deletion-invariance property for random measures satisfying the Ghirlanda-Guerra identities

We show that if a discrete random measure on the unit ball of a separable Hilbert space satisfies the Ghirlanda-Guerra identities then by randomly deleting half of the points and renormalizing the weights of the remaining points we obtain the same random measure in distribution up to rotations

On the Dovbysh-Sudakov representation result

We present a detailed proof of the Dovbysh-Sudakov representation for symmetric positive definite weakly exchangeable infinite random arrays, called Gram-de Finetti matrices, which is based on the representation result of Aldous and Hoover for arbitrary (not necessarily positive definite) symmetric weakly exchangeable arrays

A note on Talagrand's positivity principle

Talagrand's positivity principle states that one can slightly perturb a Hamiltonian in the Sherrington-Kirkpatrick model in such a way that the overlap of two configurations under the perturbed Gibbs' measure will become typically nonnegative. In this note we observe that abstracting from the setting of the SK model only improves the result and does not require any modifications in Talagrand's argument. In this version, for example, positivity principle immediately applies to the setting of Aizenman-Sims-Starr interpolation. Also, abstracting from the SK model improves the conditions in the Ghirlanda-Guerra identities and as a consequence results in a perturbation of smaller order necessary to ensure positivity of the overlap