Filmus–O'Donnell–Wu log-Sobolev conjecture for the multislice

Let $\kappa=(\kappa_1,\ldots,\kappa_L)$ be the parameter defining the multislice, let $\tau_{\textsc{ls}}(\kappa)$ denote the log-Sobolev constant of its transposition walk, and set

$$\kappa_{\textsc{\min}}:=\min\{\kappa_1,\ldots,\kappa_L\}.$$

Here $\asymp$ means equality up to universal pre-factors. Filmus–O'Donnell–Wu conjecture. For any choice of the parameter $\kappa$,

$$\tau_{\textsc{ls}}(\kappa)\asymp\log\left(\frac{n}{\kappa_{\textsc{\min}}}\right).$$

The conjecture predicts the correct order of the log-Sobolev constant throughout all ranges of $\kappa$, interpolating between the sharp extreme cases. It asserts that the transposition walk mixes essentially as well as independent refreshment of the coordinates, despite the multislice probability space being far from a product space.