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Cut times for random walks on the discrete Heisenberg group. (English) Zbl 1022.60004
Let \((S(n))_{n\geq 0}\) be a random walk on a discrete group \(\Gamma\). Then a time \(n\) is called a cut time if for all \(k> n\), \(S(k)\) is different from \(S(0),\dots, S(n)\). It was recently proved by N. James and Y. Perez that transient random walks on \(\mathbb{Z}^d\) with finite range have infinitely many cut times with probability one. Based on sharp estimates on the Green function due to G. Alexopoulos, it is shown in this interesting paper that the same is true for the discrete Heisenberg group \(H_3\). This group was the last unsolved case in the study of cut times for finite range random walks on finitely generated discrete groups.
MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F15 Strong limit theorems
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