2012/08/26 by Abrams, Aaron, Babson, Eric, Landau, Henry +2
#60B15 #60G50 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1208.5235
Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with Peres' conjecture in [3] relating sharp cut off, spectral gap, and mixing time.