Mixing Time Bounds via the Spectral Profile
Sharad Goel, Standford University, USA
Ravi Montenegro, University of Massachusetts Lowell, USA
Prasad Tetali, Georgia Institute of Technology, USA
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, 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 Za x Zb, to obtain
tight bounds on the corresponding mixing times.
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