## Mixing time bounds via the spectral profile.(English)Zbl 1109.60061

Summary: On complete, noncompact 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 $$\mathbb Z_a\times \mathbb Z_b$$, to obtain tight bounds on the corresponding mixing times.

### MSC:

 60J27 Continuous-time Markov processes on discrete state spaces 68Q99 Theory of computing
Full Text: