Conductance and Rapidly Mixing Markov Chains

Conductance is a measure of a Markov chain that quantifies its tendency to circulate around its states. A Markov chain with low conductance will tend to get ‘stuck’ in a subset of its states whereas one with high conductance will jump around its state space more freely. The mixing time of a Markov chain is the number of steps required for the chain to approach its stationary distribution. There is an inverse correlation between conductance and mixing time. Rapidly mixing Markov chains have very powerful applications, most notably in approximation schemes. It is therefore desirable to prove certain Markov chains to be rapidly mixing, and bounds involving conductance can help us do that. This survey covers many useful bounds involving conductance and gives several specific examples of applications of rapidly mixing Markov chains.