Limiting dynamics for spherical models of spin glasses at high temperature

We analyze the coupled non-linear integro-differential equations whose solutions is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p-spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds

Dynamic phase diagram of the REM

By studying the two-time overlap correlation function, we give a comprehensive analysis of the phase diagram of the Random Hopping Dynamics of the Random Energy Model (REM) on time-scales that are exponential in the volume. These results are derived from the convergence properties of the clock process associated to the dynamics and fine properties of the simple random walk in the $n$-dimensional discrete cube.Comment: This paper is in large part based on the unpublished work arXiv:1008.3849. In particular, the analysis of the overlap correlation function is new as well as the study of the high temperature and short time-scale transition line between aging and stationarit

Convergence to extremal processes in random environments and extremal ageing in SK models

This paper extends recent results on aging in mean field spin glasses on short time scales, obtained by Ben Arous and Gun [2] in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in Gayrard [8,9] and naturally complements Bovier and Gayrard [6].Comment: Revised version contains minor change

Slow relaxation, dynamic transitions and extreme value statistics in disordered systems

We show that the dynamics of simple disordered models, like the directed Trap Model and the Random Energy Model, takes place at a coexistence point between active and inactive dynamical phases. We relate the presence of a dynamic phase transition in these models to the extreme value statistics of the associated random energy landscape

Universality of REM-like aging in mean field spin glasses

Aging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, aging occurs precisely as predicted by the REM-like trap model of Bouchaud and Dean. This is the first rigorous result about aging in mean field models except for the REM and the spherical model.Comment: 4 page

Quantitative Theory of a Relaxation Function in a Glass-Forming System

We present a quantitative theory for a relaxation function in a simple glass-forming model (binary mixture of particles with different interaction parameters). It is shown that the slowing down is caused by the competition between locally favored regions (clusters) which are long lived but each of which relaxes as a simple function of time. Without the clusters the relaxation of the background is simply determined by one typical length which we deduce from an elementary statistical mechanical argument. The total relaxation function (which depends on time in a nontrivial manner) is quantitatively determined as a weighted sum over the clusters and the background. The `fragility' in this system can be understood quantitatively since it is determined by the temperature dependence of the number fractions of the locally favored regions.Comment: 4 pages, 5 figure

Spectral measure of heavy tailed band and covariance random matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $\mu$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^\sigma$ whose (i,j) entry is $\sigma(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $\alpha$-stable law, $0<\alpha<2$, and $\sigma$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^\sigma$ and with appropriate rescaling $a_N$, we prove that the distribution $\mu$ of $a_N^{-1}Y_N^\sigma + D_N$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.Comment: 31 pages, minor modifications, mainly in the regularity argument for Theorem 1.3. To appear in Communications in Mathematical Physic