35 research outputs found
Biased random walks on random graphs
These notes cover one of the topics programmed for the St Petersburg School
in Probability and Statistical Physics of June 2012.
The aim is to review recent mathematical developments in the field of random
walks in random environment. Our main focus will be on directionally transient
and reversible random walks on different types of underlying graph structures,
such as , trees and for .Comment: Survey based one of the topics programmed for the St Petersburg
School in Probability and Statistical Physics of June 2012. 64 pages, 16
figure
Bouchaud's model exhibits two different aging regimes in dimension one
Let E_i be a collection of i.i.d. exponential random variables. Bouchaud's
model on Z is a Markov chain X(t) whose transition rates are given by
w_{ij}=\nu \exp(-\beta ((1-a)E_i-aE_j)) if i, j are neighbors in Z. We study
the behavior of two correlation functions: P[X(t_w+t)=X(t_w)] and
P[X(t')=X(t_w) \forall t'\in[t_w,t_w+t]]. We prove the (sub)aging behavior of
these functions when \beta >1 and a\in[0,1].Comment: Published at http://dx.doi.org/10.1214/105051605000000124 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Free extreme values
Free probability analogs of the basics of extreme-value theory are obtained,
based on Ando's spectral order. This includes classification of freely
max-stable laws and their domains of attraction, using ``free extremal
convolutions'' on the distributions. These laws coincide with the limit laws in
the classical peaks-over-threshold approach. A free extremal projection-valued
process over a measure-space is constructed, which is related to the free
Poisson point process.Comment: Published at http://dx.doi.org/10.1214/009117906000000016 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Poisson convergence for the largest eigenvalues of Heavy Tailed Random Matrices
We study the statistics of the largest eigenvalues of real symmetric and
sample covariance matrices when the entries are heavy tailed. Extending the
result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of
the fourth moment, the top eigenvalues behave, in the limit, as the largest
entries of the matrix.Comment: 22 pages, to appear in Annales de l'Institut Henri Poincar
Cugliandolo-Kurchan equations for dynamics of Spin-Glasses
We study the Langevin dynamics for the family of spherical -spin
disordered mean-field models and prove that in the limit of system size
approaching infinity, the empirical state correlation and integrated response
functions converge almost surely and uniformly in time, to the non-random
unique strong solution of a pair of explicit non-linear integro-differential
equations introduced by Cugliandolo and Kurchan.Comment: 31 page