Year: 2026
Type: preprint
Abstract: We prove the conjecture of Ding and Mossel~\cite{DM}: the lazy random walk on the discrete cube, censored to any increasing set of fixed positive density, mixes in time $O(n\log n)$. More precisely, for every nonempty increasing set $A\subseteq\{0,1\}^n$, with $a=μ(A)$ and $μ$ uniform on the cube, we prove $$ t_{\mathrm{mix}}(P)\le C_a n\log(en), $$ where $C_a$ depends only on $a$. The proof uses hypercontractivity on the ambient cube to strengthen the Poincaré inequality on coordinate sections. An optimal stopping time occupation inequality for increasing sets converts the resulting local bound into a uniform bound on hitting times of large sets. An appendix by Y.~Peres improves the density dependence to $C_a=13a^{-1}\log(e/a)$, within a logarithmic factor of the conjectured $a^{-1}$ dependence.
Source: arXiv (Cornell University)
Authors: Chen, Yiming, Peres, Yuval
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8700735110
OpenAlex ID: W7213399702