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Isoperimetric inequalities and decay of iterated kernels for almost-transitive Markov chains. (English) Zbl 0842.60070

This paper gives precise isoperimetric inequalities for infinite graphs on which a group acts with finite quotient. It also gives estimates of the iterated kernel of the associated random walks. In the case of Cayley graphs of finitely generated groups one recovers known results that are mainly due to N. Varopoulos. The treatment of transitive graphs requires some further arguments.
This paper uses the transitive (or almost transitive) action in a rather naive way to transport geodesic paths from one place to another. The heart of the matter is to show that there are not to many geodesic paths of a given length that use a given edge. This is closely related to arguments used in a paper of L. Babai and M. Szegedy [ibid. 1, No. 1, 1-11 (1992; Zbl 0792.05064)]. There is another route that leads to the same results. It consists in lifting the problem to the locally compact group \(G\) that acts on the graph. This approach is developed in a recent paper of W. Woess [to appear in Rend. Semin. Mat. Fis. Milano].

MSC:

60G50 Sums of independent random variables; random walks
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Citations:

Zbl 0792.05064
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References:

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