In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain

Markov chain

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...

. It can be seen as a special case of Cheeger inequalities in expander graphs.

Let be a finite set and let be the transition probability for a reversible Markov chain on . Assume this chain has stationary distribution

Stationary distribution

Stationary distribution may refer to:* The limiting distribution in a Markov chain* The marginal distribution of a stationary process or stationary time series* The set of joint probability distributions of a stationary process or stationary time series...

 .

Define


and for define

Define the constant as

The operator acting on the space of functions from to , defined by

has eigenvalues . It is known that . The Cheeger bound is a bound on the second largest eigenvalue .

Theorem (Cheeger bound):