## Rapidly Mixing Markov Chains: A Comparison of Techniques (2000)

Venue: | A Survey |

Citations: | 15 - 0 self |

### BibTeX

@ARTICLE{Guruswami00rapidlymixing,

author = {Venkatesan Guruswami},

title = {Rapidly Mixing Markov Chains: A Comparison of Techniques},

journal = {A Survey},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

For many fundamental sampling problems, the best, and often the only known, approach to solving them is to take a long enough random walk on a certain Markov chain and then return the current state of the chain. Techniques to prove how long "long enough" is, i.e., the number of steps in the chain one needs to take in order to be sufficiently close to the stationary distribution of the chain, are of great importance in obtaining estimates of running times of such sampling algorithms. In this report, we survey existing techniques to bound the mixing time of Markov chains. The mixing time of a Markov chain is exactly captured by the "spectral gap" of its underlying transition matrix. The spectral gap is closely related to a geometric parameter called "conductance" which is a measure of the "edge-expansion" of the Markov chain. Conductance also captures the mixing time up to square factors. Lower bounds on conductance, which give upper bounds on the mixing time, are typically obta...