The $k$-ball cutoff conjecture for balanced generalised Bernoulli--Laplace chains

Fix $d$ and a distribution $\mu$ on $S_d$, and let $\gamma$ denote the relevant single-ball spectral-gap parameter. In the balanced generalised Bernoulli--Laplace chain with $n$ slots in each urn, at each step $k$ balls are moved from each urn according to a permutation sampled from $\mu$. Write $t_{\operatorname{mix}}^{(n),k}$ for the mixing time with $k$ balls moved at each step.

$k$-ball cutoff conjecture. For $k=o(n)$, the balanced chain exhibits cutoff around the time

$$\frac{t_{\operatorname{mix}}^{(n),1}}{k}=\frac{n\log n}{2k\gamma}(1+o(1)),$$

where $t_{\operatorname{mix}}^{(n),1}$ denotes the cutoff time for the corresponding $k=1$ chain. Furthermore, if $k$ satisfies $\Omega(n)=k=n-\Omega(n)$, the chain exhibits cutoff around its mixing time $t_{\operatorname{mix}}^{(n),k}$, which is of the same order as the expressions in the displayed formula.

The conjecture has already been proved for the classical Bernoulli--Laplace model, but remains open for the stated generalised model.