In recent years, unimodularity has increasingly been recognised as a useful property in transitive graphs in terms of the asymptotics of random walks on percolation graphs [1]. While for percolations in the subcritical and citical case it has been shown [2] that this symmetry is useful to estimate the expected return probability, we will show a similar relation for the supercritical cluster of amenable graphs [3]. The results are useful for studying the asymptotics of random walks in terms of the spectra of finite graphs induced by the elements of increasing exhaustions.
[ 1 ] I. Benjamini, N. Curien: Ergodic theory on stationary random graphs [ 2 ] F. S.: Bounds for the annealed return probability on large finite percolation clusters [ 3 ] P.Soardi, W.Woess: Amenability, unimodularity, and the spectral radius of random walks on infinite graphs