Mathematical Aspects of Mixing Times in Markov Chains

Abstract

In the past few years we have seen a surge in the theory of finite Markov chains, by way of new techniques to bounding the convergence to stationarity. This includes functional techniques such as logarithmic Sobolev and Nash inequalities, refined spectral and entropy techniques, and isoperimetric techniques such as the average and blocking conductance and the evolving set methodology. We attempt to give a more or less self-contained treatment of some of these modern techniques, after reviewing several preliminaries. We also review classical and modern lower bounds on mixing times. There have been other important contributions to this theory such as variants on coupling techniques and decomposition methods, which are not included here; our choice was to keep the analytical methods as the theme of this presentation. We illustrate the strength of the main techniques by way of simple examples, a recent result on the Pollard Rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.

Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

“…The next theorem answers (up to constant factors) the 5th open question in [MT06], which asks how small the log-Sobolev and entropy constants can be for an n-vertex unweighted connected graph. Here, we write f (n) = Θ g(n) to mean that there are positive finite constants c 1 and c 2 such that for all n ≥ 1, we have c 1 g(n) ≤ f (n) ≤ c 2 g(n).…”

“…As is well known, the barbell graph has Ω(n The 5th open question in [MT06] asks how small the log-Sobolev and entropy constants can be for an n-vertex unweighted connected graph. We can now answer this (up to constant factors).…”

Fast averaging

“…The linear iterative algorithm and its variants that has been studied since [1] takes roughly T mix (ε) iterations to compute an estimate of x ave that is within 1±ε multiplicative error. Here by T mix (ε), we mean the ε-mixing time of the probability matrix P which is characterized by the spectral gap of P (see [22] for example, for details). The total number of operations performed by such algorithms scale proportional to the number of edges in the G. Even the randomized or Gossip variant of such algorithm [3] requires Ω(n) operations per iteration.…”

How to Encipher Messages on a Small Domain

“…Morris proved that the mixing time for the Thorp shuffle-roughly, the number of steps until all q = N cards are ordered nearly uniformly-is polylogarithmic: it is O(lg 44 N ) [25]. This was subsequently improved to O(lg 19 N ) [22] and then to O(lg 4 N ) [23]. Naor and Reingold analyzed unbalanced Feistel constructions, showing, in particular, that one pass over a maximally unbalanced Feistel network that operates on n bits remains secure to nearly 2 n/2 queries.…”