Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition dependent on it's location in the grid. This comes from a random conductance model. The theorem that concerns me is a general result for Markov chains, but I leave this motivation to assist in its proof (see below).

Let $P^{2k}(0,0)$ be the probability of going from the origin and back in $2k$ steps. Moreover, suppose $P$ is reversible. The theorem that concerns me is:

$P^{2n}(0,0)$ is decreasing in $n$.

I am interested in a probabilistic proof of this. The proof that I know is of a spectral nature:

Define $\langle f,g\rangle:= \sum_{X\in\mathbb{Z}^d} \pi(x)f(x)g(x)$,

where $\pi(x)$ is the stationary measure. This gives an inner product on $L^2(\mathbb{Z}^d)$. In the case of a random conductance model, $\pi(x)$ would be the sum of random edge weights at $x$.

Then

$P^{2k}(0,0)=\langle \delta_0,P^{2k}\delta_0\rangle$,

and since $P$ is self adjoint with $\|P\|_2\leq 1$, the desired result follows.

I have tried various approaches such as conditioning on the hitting times of the origin and as well trying to prove the result by induction. I would like to see a proof that showcases a probabilistic argument. For example, is it possible to show the result from the machinery of evolving sets of Morris and Peres?