Isoperimetric inequalities and Markov chains

Isoperimetric Invariants For Product Markov Chains and Graph Products

Bounds on some isoperimetric constants of the Cartesian product of Markov chains are obtained in terms of related isoperimetric quantities of the individual chains.

Comparison Inequalities and Fastest-mixing Markov Chains

We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution π on a given finite partially ordered state space X . When K L in this partial order we say that K and L satisfy a comparison inequality. We establish that if K1, . . . ,Kt and L1, . . . , Lt are reversible and Ks Ls for s = 1, . . . , t, then K1 · · ·Kt L1 · · ·Lt. In ...

Isoperimetric Inequalities and Eigenvalues

Cheeger inequalities for transient Markov chains

We construct upper and lower bounds for Cheeger-type constants for transient Markov chains. We first treat the situation where there is a detailed balance condition and obtain bounds that rely exclusively on the first and second eigenvalues of the substochastic transition matrix. Secondly, we consider general substochastic transition matrices and develop bounds in this broader setting. The effi...

Some Inequalities for Reversible Markov Chains