Let Γ be a graph and P be a reversible random walk on Γ. From the L2 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 P2k+1) from a vertex x to itself is uniformly bounded from below. The proof does not require the doubling property on G but only a polynomial control of the volume.
Bibliographical noteFunding Information:
The author was supported by the ANR project “Harmonic Analysis at its Boundaries”, ANR-12-BS01-0013-03.
- Discrete analyticity
- Lazy random walks
- Markov chains