author={Persi Diaconis and Laurent Saloff-Coste},
  journal={Annals of Applied Probability},
This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a self-contained development. Examples of applications include the study of a Metropolis chain… 

Figures from this paper

Sum-of-Squares proofs of logarithmic Sobolev inequalities on finite Markov chains

Logarithmic Sobolev inequalities are a fundamental class of inequalities that play an important role in information theory. They play a key role in establishing concentration inequalities and in

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the

Nash inequalities for finite Markov chains

This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to

Logarithmic Harnack Inequalities

Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9,

Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains

We consider finite-state Markov chains that can be naturally decomposed into smaller ``projection'' and ``restriction'' chains. Possibly this decomposition will be inductive, in that the restriction

Quantum logarithmic Sobolev inequalities and rapid mixing

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative Lp-spaces is reviewed and the relationship between quantum

Logarithmic Sobolev inequalities for finite spin systems and applications

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various

Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities

We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean



Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow

  • A. Sinclair
  • Computer Science
    Combinatorics, Probability and Computing
  • 1992
A new upper bound on the mixing rate is presented, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph, and improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system.

The logarithmic sobolev inequality for discrete spin systems on a lattice

For finite range lattice gases with a finite spin space, it is shown that the Dobrushin-Shlosman mixing condition is equivalent to the existence of a logarithmic Sobolev inequality for the associated

Logarithmic Sobolev inequalities for gibbs states

(1.1) e-U ( 2: ) ,U(dx) = -zA(dx), where the constant Z is determined by the requirement that ,U(M) = 1. The naive physical principle underlying this terminology is that a dynamical system which when

Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics

We prove that the spectral gap of the Kawasaki dynamics shrink at the rate of 1/L2 for cubes of sizeL provided that some mixing conditions are satisfied. We also prove that the logarithmic Sobolev

Geometric Bounds for Eigenvalues of Markov Chains

We develop bounds for the second largest eigenvalue and spectral gap of a reversible Markov chain. The bounds depend on geometric quantities such as the maximum degree, diameter and covering number

Moderate growth and random walk on finite groups

We study the rate of convergence of symmetric random walks on finite groups to the uniform distribution. A notion of moderate growth is introduced that combines with eigenvalue techniques to give


By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state