## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 17, Number 3 (2007), 931-952.

### A general lower bound for mixing of single-site dynamics on graphs

Thomas P. Hayes and Alistair Sinclair

#### Abstract

We prove that any Markov chain that performs local, reversible updates on randomly chosen vertices of a bounded-degree graph necessarily has mixing time at least Ω(*n* log*n*), where *n* is the number of vertices. Our bound applies to the so-called “Glauber dynamics” that has been used extensively in algorithms for the Ising model, independent sets, graph colorings and other structures in computer science and statistical physics, and demonstrates that many of these algorithms are optimal up to constant factors within their class. Previously, no superlinear lower bound was known for this class of algorithms. Though widely conjectured, such a bound had been proved previously only in very restricted circumstances, such as for the empty graph and the path. We also show that the assumption of bounded degree is necessary by giving a family of dynamics on graphs of unbounded degree with mixing time *O*(*n*).

#### Article information

**Source**

Ann. Appl. Probab. Volume 17, Number 3 (2007), 931-952.

**Dates**

First available in Project Euclid: 22 May 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoap/1179839178

**Digital Object Identifier**

doi:10.1214/105051607000000104

**Mathematical Reviews number (MathSciNet)**

MR2326236

**Zentralblatt MATH identifier**

1125.60075

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 68W20: Randomized algorithms 68W25: Approximation algorithms 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

**Keywords**

Glauber dynamics mixing time spin systems Markov random fields

#### Citation

Hayes, Thomas P.; Sinclair, Alistair. A general lower bound for mixing of single-site dynamics on graphs. Ann. Appl. Probab. 17 (2007), no. 3, 931--952. doi:10.1214/105051607000000104. http://projecteuclid.org/euclid.aoap/1179839178.