Mixing time of the tranvection walk on $\mathrm{SL}_n(\mathbb{Z}_p)$

Source abstract

We consider a Markov chain on $\mathrm{SL}_n(\mathbb{Z}_p)$, the space of $n \times n$ matrices with determinant $1$ and entries in $\mathbb{Z}_p$, for $p$ a prime number. At every time step, the chain updates by adding one row (or its negation) to another. By way of a log-Sobolev inequality, we show the mixing time of this chain is at most of order $O(n^2p^2(\log n +\log \log p))$.