Abstract
The critical constant for recurrence, $c_{rt}$, is an invariant of the quotient space $H/G$ of a finitely generated group. The constant is determined by the largest moment a probability measure on $G$ can have without the induced random walk on $H/G$ being recurrent. We present a description of which subgroups of groups of polynomial volume growth are recurrent. Using this we show that for such recurrent subgroups $c_{rt}$ corresponds to the relative growth rate of $H$ in $G$, and in particular $c_{rt}$ is either $0$, $1$ or $2$.
Highlights
Whether or not a Markov chain is recurrent is one of the most basic probabilistic questions we can ask
On a countably infinite state space the problem was first treated by Polya, who showed that the integer lattice carries a recurrent random walk only in dimensions one and two [Pól21]
This result was proven by Varopoulos [Var85], utilizing Gromov’s theorem that polynomial volume growth is equivalent to containing a nilpotent subgroup of finite index [Gro81]
Summary
Whether or not a Markov chain is recurrent is one of the most basic probabilistic questions we can ask. For example there are classical results that a finite first moment guarantees recurrence on and a finite second moment guarantees recurrence on 2 (see Chapter 2, section 8 in [Spi76]) To generalize this to the quotient case we begin with a measure μ on G and use it to drive a random walk on H\G. In this case the critical constant lies in the interval (1/2, 1) Such stabilizer subgroups are important in the study of self-similar groups [Nek05], and it would be interesting to calculate crt for more examples of this kind. Another interesting question is whether or not the critical constant can take on values larger than two
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