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Glauber dynamics on trees and hyperbolic graphs. (English) Zbl 1075.60003
The main goal of the paper is to determine which geometric properties of a graph are most relevant to the mixing rate of the Glauber dynamics on particale systems. The paper contains six sections. In Section 2.1 a connection between the geometry of a graph and mixing time of Glauber dynamics on it is described. In Sections 2.3–4 Glauber dynamics for the Ising model on regular trees is studied. For these trees it is shown that the mixing time is polynomial at all temperatures, the range of temperatures for which the spectral gap is bounded away from zero is characterized. It is proven that on infinite regular trees, there is a range of temperatures in which the inverse spectral gap is bounded, even though there are many different Gibbs measures. In Section 5 Glauber dynamics for families of finite graphs of bounded degree are studied. Section 6 contains relevant problems that are open.

60C05 Combinatorial probability
05C80 Random graphs (graph-theoretic aspects)
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