194 research outputs found

### 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walk in the $n$-dimensional discrete cube.Comment: This paper is in large part based on the unpublished work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

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