Cheeger bound

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

Let ${\displaystyle X}$ be a finite set and let ${\displaystyle K(x,y)}$ be the transition probability for a reversible Markov chain on ${\displaystyle X}$. Assume this chain has stationary distribution ${\displaystyle \pi }$.

Define

${\displaystyle Q(x,y)=\pi (x)K(x,y)}$

and for ${\displaystyle A,B\subset X}$ define

${\displaystyle Q(A\times B)=\sum _{x\in A,y\in B}Q(x,y).}$

Define the constant ${\displaystyle \Phi }$ as

${\displaystyle \Phi =\min _{S\subset X,\pi (S)\leq {\frac {1}{2}}}{\frac {Q(S\times S^{c})}{\pi (S)}}.}$

The operator ${\displaystyle K,}$ acting on the space of functions from ${\displaystyle |X|}$ to ${\displaystyle |X|}$, defined by

${\displaystyle (K\phi )(x)=\sum _{y}K(x,y)\phi (y)}$

has eigenvalues ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}}$. It is known that ${\displaystyle \lambda _{1}=1}$. The Cheeger bound is a bound on the second largest eigenvalue ${\displaystyle \lambda _{2}}$.

Theorem (Cheeger bound):

${\displaystyle 1-2\Phi \leq \lambda _{2}\leq 1-{\frac {\Phi ^{2}}{2}}.}$