For every psilon>0, there exists a constant C_psilon<∞ such that, for every n ngreater than or equal to 1 and every increasing set A not=emptyset in 0,1^n satisfying mu(A)≥ varepsilon, the lazy random walk on the Boolean cube censored to A has mixing time t_mix(P_A)≤ C_varepsilonnlog n. The cited preprint gives the more explicit bound t_mix(P_A)≤ Kmu(A)^-3nlog(en) for an absolute constant K.

Ding–Mossel monotone-censoring mixing question

Ding and Mossel asked in 2013 whether every monotone set of density at least $\varepsilon>0$ yields mixing time $O_\varepsilon(n\log n)$ . The question concerns the censored random walk on the Boolean cube.

Known results

Ding–Mossel (2013): a conductance argument gave $O_\varepsilon(n^3)$ mixing for constant-density sets.
Fei and Pinto Jr. (2025), with an alternate proof cited as Chen–Stein–Yau (2025): an optimal spectral-gap scale and an $O_\varepsilon(n^2)$ mixing bound.
A 2026 preprint proved $O(n\log n)$ mixing with high probability for a uniformly random monotone set, but explicitly not for every set.

September 15, 2026 claimed proof

Yiming Chen and Yuval Peres report a uniform mixing bound for every increasing censoring set, with explicit density dependence, establishing the conjectured $O_\varepsilon(n\log n)$ order. This is a complete-resolution claim in an unrefereed preprint and has not been independently verified in the retrieved record.

Current status (as of September 2026): A new preprint claims the universal $O_\varepsilon(n\log n)$ bound, while the claim remains unverified; before it, the universal bound was only $O_\varepsilon(n^2)$ and the conjecture remained open.

Ding–Mossel monotone-censoring mixing question

Primary source

Additional references

Progress summary

Solutions 0