Mixing Time Bounds via the Spectral Profile Sharad Goel, Ravi Montenegro and Prasad Tetali

On complete, non-compact manifolds and infinite graphs, FaberKrahn 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, proving upper and lower L mixing time bounds via the spectral profile. This approach lets us recover and refine previous conductance-based bounds of mixing time (including the Morris-Peres result), and in general leads to sharper estimates 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.