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For $G$ a finitely generated group and $g in G$, we say $g$ is detected by a normal subgroup $N lhd G$ if $g otin N$. The depth $D_G(g)$ of $g$ is the lowest index of a normal, finite index subgroup $N$ that detects $g$. In this paper we study the expected depth, $mathbb E[D_G(X_n)]$, where $X_n$ is a random walk on $G$. We give several criteria that imply that $$mathbb E[D_G(X_n)] xrightarrow[nto infty]{} 2 + sum_{k geq 2}frac{1}{[G:Lambda_k]}, ,$$ where $Lambda_k$ is the intersection of all normal subgroups of index at most $k$. In particular, the equality holds in the class of all nilpotent groups and in the class of all linear groups satisfying Kazhdan Property $(T)$. We explain how the right-hand side above appears as a natural limit and also give an example where the convergence does not hold.
In this short note we give various near optimal characterizations of random walks over finite Abelian groups with large maximum discrepancy from the uniform measure. We also provide several interesting connections to existing results in the literature.
We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution be
A $k$-free like group is a $k$-generated group $G$ with a sequence of $k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$ is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away from 0. By a recent re
The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $
We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions $f(eta)$, including the case of Levy flights. We study the expected maximum ${mathbb E}[M_n]$ of bridge RWs, i.e., RWs starting an