Mixing time bounds via the spectral profile

Abstract

On complete, non-compact manifolds and infinite graphs, Faber- Krahn inequalities have been used to estimate the rate of decay of the heat kernel. We develop this technique in the setting of finite Markov chains, prov-ing upper and lower L∞ mixing time bounds via the spectral profile. This ap-proach lets us recover and refine previous conductance-based bounds of mixing time (including the Morris-Peres result), and in general leads to sharper esti- mates of convergence rates. We apply this method to several models including groups with moderate growth, the fractal-like Viscek graphs, and the product group ℤa × ℤb, to obtain tight bounds on the corresponding mixing times. © 2006 Applied Probability Trust.