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

Super-diffusivity in a shear flow model from perpetual homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations $dy_t=d\omega_t -\nabla \Gamma(y_t) dt$, $y_0=0$ and $d=2$. $\Gamma$ is a $2\times 2$ skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales $\Gamma_{12}=-\Gamma_{21}=h(x_1)$, with $h(x_1)=\sum_{n=0}^\infty \gamma_n h^n(x_1/R_n)$ where $h^n$ are smooth functions of period 1, $h^n(0)=0$, $\gamma_n$ and $R_n$ grow exponentially fast with $n$. We can show that $y_t$ has an anomalous fast behavior (\E[|y_t|^2]\sim t^{1+\nu} with $\nu>0$) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization

Multi-scale homogenization with bounded ratios and Anomalous Slow Diffusion

We show that the effective diffusivity matrix $D(V^n)$ for the heat operator $\partial_t-(\Delta/2-\nabla V^n \nabla)$ in a periodic potential $V^n=\sum_{k=0}^n U_k(x/R_k)$ obtained as a superposition of Holder-continuous periodic potentials $U_k$ (of period \T^d:=\R^d/\Z^d, $d\in \N^*$, $U_k(0)=0$) decays exponentially fast with the number of scales when the scale-ratios $R_{k+1}/R_k$ are bounded above and below. From this we deduce the anomalous slow behavior for a Brownian Motion in a potential obtained as a superposition of an infinite number of scales: $dy_t=d\omega_t -\nabla V^\infty(y_t) dt$Comment: 29 pages, 1 figure, submitted versio

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 $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.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