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Mixing time for the Ising model : a uniform lower bound for all graphs

Jian DingYuval Peres — 2011

Annales de l'I.H.P. Probabilités et statistiques

Consider Glauber dynamics for the Ising model on a graph of vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least log /(), where is the maximum degree and () = (log2). Their result applies to more general spin systems, and in that generality, they showed that some dependence on is necessary. In this paper, we focus on the ferromagnetic Ising model and prove that the mixing time of Glauber dynamics on any -vertex graph is at least (1/4 + o(1))log .

The infinite valley for a recurrent random walk in random environment

Nina GantertYuval PeresZhan Shi — 2010

Annales de l'I.H.P. Probabilités et statistiques

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE...

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