Abstract
A random walk on a graph is a Markov chain whose state space consists of the vertices of the graph and where transitions are only allowed along the edges. We study (strongly) reversible random walks and characterize the class of graphs where then-step transition probabilities tend to zero exponentially fast (δgeometric ergodicity”). These characterizations deal with an isoperimetric property, norm inequalities for certain associated operators, and eigenvalues of the Laplace operator. There is some (strong) similarity with the theory of (non)amenable groups.
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Gerl, P. Random walks on graphs with a strong isoperimetric property. J Theor Probab 1, 171–187 (1988). https://doi.org/10.1007/BF01046933
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DOI: https://doi.org/10.1007/BF01046933