About the $L^2$ analyticity of Markov operators on graphs

TL;DR

This paper investigates the $L^2$ analyticity of Markov operators on graphs, showing that certain odd iterates are 'lazy' with uniformly bounded return probabilities, relying only on polynomial volume control.

Contribution

It establishes a new connection between $L^2$ analyticity and laziness of odd iterates of Markov operators without requiring doubling conditions.

Findings

Odd iterates of Markov operators are 'lazy' with bounded return probabilities.

The results hold under polynomial volume growth, not doubling conditions.

Provides a new perspective on the behavior of reversible random walks on graphs.

Abstract

Let $\Gamma$ be a graph and $P$ be a reversible random walk on $\Gamma$. From the $L^2$ analyticity of the Markov operator $P$, we deduce that an iterate of odd exponent of $P$ is `lazy', that is there exists an integer $k$ such that the transition probability (for the random walk $P^{2k+1}$) from a vertex $x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $\Gamma$ but only a polynomial control of the volume.