Cutoff for Cayley Graphs of Nilpotent Groups

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We study the mixing behavior of random walks on Cayley graphs of nilpotent groups, with a focus on their connection to the projected walks on the Abelianizations. For a generating set $S$ of a nilpotent group $G$, we establish that under a degree condition on $|S|$, the spectral gap and the $\varepsilon$-mixing time of the simple random walk $X = (X_t)_{t \geq 0}$ on the corresponding Cayley graph are asymptotically equivalent to those of the projected walk on the Abelianization, represented by $[G,G] X_t$. Consequently, $X$ exhibits cutoff if and only if its projection does. Furthermore, when the generating set $S$ consists of $k$ elements sampled uniformly at random with replacement from $G$, and $1 \ll \log k \ll \log |G|$, we show that the simple random walk on the resulting Cayley graph exhibits cutoff with high probability. The cutoff time, in this case, is determined by the cutoff time of the projected walk on the Abelianization. This work is based on joint research with Jonathan Hermon.