The Eigenvalue Gap and Mixing Time
نویسنده
چکیده
The eigenvalue or spectral gap of a Markov chain is the difference between the two largest eigenvalues of the transition matrix of its underlying (state space) graph. In this paper we explore the intimate relationship between the spectral gap of a Markov chain and its mixing time, as well as another closely related structural property of a Markov chain known as conductance. The relationships among these properties can be used to put bounds on a chain’s mixing time, and can be used to prove both rapid and slow mixing. As the spectral gap and conductance of a Markov chain are often difficult to calculate, an additional tool, canonical paths, is introduced which can be used to put a lower bound on the spectral gap. Several theorems relating these properties to mixing time as well as an example of using these techniques to prove rapid mixing are given.
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