Given an infinite connected graph $G$ of bounded degree with vertex set $X$, let $P\_x^n$ the time $n$ distribution of the simple random walk started at the vertex $x$ (so $P^n\_x(y)$ is the probability that a simple random walk started at $x$ ends at $y$ after $n$ steps).
Let further
$$
H\_n :=\displaystyle \sup\_{x \in X} h(P\_x^n),
\qquad
\eta\_n := \displaystyle \inf\_{x \in X} h(P\_x^n)
\quad \text{ and } \quad
r\_n := \displaystyle \sup\_{x,y \in X} P\_x^n(y).
$$
Here $h(\mu) = \sum\_y -\mu(y) \ln \mu(y)$ is the entropy of $\mu$.
Given that for any measure $\mu$ one has $\displaystyle \sup\_{y \in X} \mu(y) \geq e^{-h(\mu)}$, one gets the bound $r\_n \geq e^{-H\_n}$. One question is:


**Question 1:** what are *upper* bounds on $r\_n$ in terms of $H\_n$ and $h\_n$?


Note that in the case of [Cayley graphs of] groups, there is such a bound (the sharpest version known to me is by [Peres & Zheng](https://arxiv.org/abs/1609.05174)). My question is for "generic" [infinite connected] graphs [of bounded degree].


There is a natural counterpart to the question (which is hopefully easier to get):


**Question 2:** what are [family of] examples where upper bounds of $r\_n$ in terms of $H\_n$ and $\eta\_n$ are "bad"?


Of course "bad" is not well-defined, but in Cayley graphs it may happen that $r\_n$ behaves roughly like $C\_1e^{-C\_2\sqrt{H\_n}}$. I would expect much worse for a graph.


PS: if "generic" needed to be made explicit, then I would say something like: "no finitely generated subgroup of the subgroup of self quasi-isometries of the graph acts co-compactly".


PPS: any result with "lazy random walk" in place of "simple random walk" is also OK.



Does entropy of the random walk control the return probability
This is a very partial answer.
For simple random walks on Cayley graphs, linear growth of $H\_n$ implies that
for any $\epsilon>0$, the inequality $r\_n \le \exp((\epsilon-1/2)n)$
must hold for infinitely many $n$, see Theorem 1.1 in [1].


For infinite connected graphs of bounded degree, it is possible for $H\_n$ to grow linearly while $r\_n \asymp cn^{-3/2}$ for even $n$. A simple example is a graph $G$ obtained by adjoining an infinite path to the root of an infinite binary tree. I suspect that slower decay of $r\_n$ is not compatible with linear growth of entropy.


(Edit: @tmh Indicated in a comment below that one could get close to decay rate of $1/n$ for $r\_n$. So perhaps $1/n$ is a barrier? recall that without any entropy assumptions, the slowest possible decay rate for $r\_n$ on an infinite graph of bounded degree is $\;$ const.$/\sqrt{n}$.)


[1] Peres, Yuval, and Tianyi Zheng. "On groups, slow heat kernel decay yields liouville property and sharp entropy bounds." International Mathematics Research Notices 2020, no. 3 (2020): 722-750. <https://arxiv.org/abs/1609.05174>


