Mixing Time Bounds via the Spectral Profile

By Sharad Goel, Ravi Montenegro, Prasad Tetali. 2005

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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, proving upper and lower $L^{\infty}$ 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 $Z_a \times Z_b$, to obtain tight bounds on the corresponding mixing times.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/09e7e4504308d307ec146f392f4891c427320f53