Ravi Montenegro

Mathematical aspects of mixing times in Markov chains

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.

Blocking Conductance and Mixing in Random Walks

The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of the Jerrum–Sinclair bound (where the average is taken over subsets of states with different sizes). Furthermore, instead of just the conductance, which in graph theory terms measures edge expansion, we also take into account node expansion. Our bound is related to the logarithmic Sobolev inequalities, but it appears to be more flexible and easier to compute.In the case of random walks in convex bodies, we show that this new bound is better than the known bounds for the worst case. This saves a factor of

Near Optimal Bounds for Collision in Pollard Rho for Discrete Log

O(n)

A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

in the mixing time bound, which is incurred in all proofs as a ‘penalty’ for a ‘bad start’. We show that in a convex body in

Rapid Mixing of Several Markov Chains for a Hard-Core Model

\mathbb{R}^n

How long does it take to catch a wild kangaroo

, with diameter

Edge isoperimetry and rapid mixing on matroids and geometric Markov chains

D

A birthday paradox for Markov chains, with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

, random walk with steps in a ball with radius

Collision of random walks and a refined analysis of attacks on the discrete logarithm problem

\delta

Intersection conductance and canonical alternating paths, methods for general finite Markov chains

mixes in