# The property that there exists an $r \in (0, 1)$ such that $P(Z_n = x) \leq r^n$ for a random walk $(Z_n)_{n \geq 0}$.

If we have a random walk $$(Z_n)_{n \geq 0}$$, one can ask whether there exists an $$r \in (0, 1)$$ such that $$P(Z_n = x) \leq r^n$$ for all $$n, x$$. Based on my own (possibly wrong) observations, this property isn't true for a simple 1D random walk since the probability of the most likely state goes like $$1/\sqrt{n}$$. However, this would be true for something like a random walk on a free group with more than two generators. And who's to say about a general Markov chain on the free group?

My question is: What is the name of the property I described (I'm having a hard time looking it up), and are there any standard techniques for showing a given random walk has it?