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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?

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For Markov chains, this is close to the concept of "Geometric Ergodicity"; see Chapter 5 of "Markov Chains and Stochastic Stability", by Richard Tweedie and Sean Meyn.

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  • $\begingroup$ Is it? My property implies that the probability of being in any state decreases exponentially so we definitely don't have ergodicity. $\endgroup$ – Mr. G Mar 15 at 1:04

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