On diagonal lower bound of Markov kernel from $L^2$ analyticity

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 an on-diagonal lower bound of an iterate of odd number of $P$. The proof does not require the doubling property on $\Gamma$ but only a polynomial control of the volume.