Mixing on k Columns of the Transvection Walk

In Diaconis and Saloff-Coste (1996), the authors introduced the simple ``transvection" walk on $\mathrm{GL}_n(\mathbb F_2)$: at each step, choose two distinct rows and add one to the other. In Ben-Hamou (2025), the author recently proved that this walk has mixing time $O(n^2\log n)$. Inspired by applications in cryptography (see Sotiraki (2016)), Ben-Hamou and Peres (2018) conjectured that the first $k$ columns of this walk mixed in $O(nk \log(n))$ steps. Our main result is a proof of this conjecture uniformly in $n$ and $k.$ Our proof is based on a local-to-global entropy estimate, in the spirit of block factorization results such as Caputo et al (2015), Caputo et al (2021). In our setting, the kernels that correspond roughly to the block kernels of Caputo et al (2021) do not have uniformly large log-Sobolev constants, and so naively applying these techniques does not improve over Ben-Hamou (2025). We avoid these bad blocks by combining our entropy estimates with a burn-in argument similar to classical drift-and-minorization arguments of Rosenthal (1995). This method may be of broader interest, and so we illustrate it by proving an analogous result for a family of product-replacement algorithms on the Heisenberg group.