273 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walk in the -dimensional discrete cube.Comment: This paper is in large part based on the unpublished work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
Aging in the random energy model
In this letter we announce rigorous results on the phenomenon of aging in the
Glauber dynamics of the random energy model and their relation to Bouchaud's
'REM-like' trap model. We show that, below the critical temperature, if we
consider a time-scale that diverges with the system size in such a way that
equilibrium is almost, but not quite reached on that scale, a suitably defined
autocorrelation function has the same asymptotic behaviour than its analog in
the trap model.Comment: 4pp, P
Spectral measure of heavy tailed band and covariance random matrices
We study the asymptotic behavior of the appropriately scaled and possibly
perturbed spectral measure of large random real symmetric matrices with
heavy tailed entries. Specifically, consider the N by N symmetric matrix
whose (i,j) entry is where is an infinite array of i.i.d real variables with common
distribution in the domain of attraction of an -stable law,
, and is a deterministic function. For a random diagonal
independent of and with appropriate rescaling , we
prove that the distribution of 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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