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Seminario di Probabilità e Finanza Matematica: Time scales separation and dynamical heterogeneities in the East model

Martedì 17 Dicembre 2013, ore 16:00 - Aula 1BC50 - Fabio Martinelli

ARGOMENTI: Seminari

Seminario di Probabilità e Finanza Matematica

Martedì 17 dicembre 2013, alle ore 16:00 in aula 1BC50 il Prof. Fabio Martinelli del Dipartimento di Matematica dell'Università di Roma Tre terrà un seminario come intitolato "Time scales separation and dynamical heterogeneities in the East model".

Abstract
The East model is a finite linear Markov chain of 0-1 spins, evolving according to a very simple rule:
i) with rate 1 and independently for each vertex, a new value 1/0 is proposed with probability 1-q and q respectively;
ii) the proposed value is accepted iff the spin immediately to the left is 0.
The model and its generalizations play an important role as models of the dynamics of real glasses. The parameter q, which turns out to be the density of the zeros (the facilitating spins) in the stationary measure, is assumed to be very small. In the physical literature this setting corresponds to a low temperature case. We will examine the problem of dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium. A key result will be a very precise computation of the relaxation time of the chain as a function of q and of its length, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Our findings reject non rigorous approaches based on numerical simulations. We will conclude with a conjecture due to D. Aldous and P. Diaconis about the scaling limit of the East chain as q-->0.

Joint work with A. Faggionato and P. Chleboun

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