Monotonicity of average return probabilities for random walks in random environments

We extend a result of Lyons (2016) from fractional tiling of finite graphs to a version for infinite random graphs. The most general result is as follows. Let $\mathbf{P}$ be a unimodular probability measure on rooted networks $(G,o)$ with positive weights $w_G$ on its edges and with a percolation subgraph $H$ of $G$ with positive weights $w_H$ on its edges. Let ${\mathbf{P}}_{(G,o)}$ denote the conditional law of $H$ given $(G,o)$. Assume that $\alpha := {\bf P}_{(G, o)}\bigl[{o \in V(H)}\bigr] > 0$isaconstant$\bf P$-a\mathrm{.}s\mathrm{.}Weshowthatif$\bf P$-a\mathrm{.}s\mathrm{.}whenever$e \in E(G)$isadjacentto$o$, $${\mathbf{E}}_{(G,o)}\left[w_H(e)\mid e\in E(H)\right]{\mathbf{P}}_{(G,o)}\left[e\in E(H)\mid o\in V(H)\right]\le w_G(e),$$ then $$\mathrm{\forall }t>0\mathbf{E}\left[p_t(o;G)\right]\le \mathbf{E}\left[p_t(o;H)\mid o\in V(H)\right]\mathrm{.}$$