# Asymptotics of the return probabilities of a random walk on a transitive graph

Consider a random walk on an infinite connected vertex-transitive graph. Let $$f(t)=P_{o,o}^{2t}$$ be the probability that the random walk is at its origin at time $$2t$$. What can be said about the asymptotics of $$f(t)$$? Specifically, I would like to know whether $$\lim_{t\to\infty}\frac {\log f(t)}{\log t}$$ always exists (possibly $$-\infty$$). If not, how large can the ratio between the limit-superior and limit-inferior be?

• My guess would be that if your graph $G$ "looks" like a $2 + (-1)^n$-dimensional lattice at scale, say, $n^n$, then the limit inerior/superior will be something like $-(2 \pm 1)/2$. Commented Jan 13, 2022 at 12:31
• But how can you make the dimension change in different scales if the graph is transitive?
– Dor
Commented Jan 13, 2022 at 16:52
• @Dor: Good point, of course. I got the question wrong. Commented Jan 14, 2022 at 17:29

the limit $$L=\lim_{t\to\infty}\frac {\log f(t)}{\log t}$$ always exists, in the wide sense. If a transitive graph does not have polynomial growth, then the limit is $$-\infty$$, while if it has polynomial growth, the exponent is necessarily an integer $$d$$ by Trofimov's [1] extension of Gromov's Theorem, and in that case the limit $$L$$ equals $$-d/2$$. For a discussion with references see page 226 in [2], or Theorems 14.12 and 14.19 in [3] or Corr. 6.2.5 in [4].