Problems

The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have…

For the switch-walk-switch lamplighter walk on Z_2 wr T_d, prove the sharp asymptotic p_2n(e,e) = ρ_d^2n exp[-(π^2 (log(d-1))^2 + o(1)) n/log^2 n] with ρ_d = frac2√d-1d.

Litvak conjectured in 2018 that for every p > 0 the quantity E[min_i ≤ n |g_i|^p], for g ~ N(0,Σ), is minimized over n × n correlation matrices by the Gram matrix of the regular simplex in R^n-1. False: the matrix Σ^cos_ij = cos(π(i-j)/n)…

For the switch-walk-switch walk on Z_2 wr Z started at (0,0) and (0,2), prove |P_t^x - P_t^y|_TV asymp t^-1/2.

The Gaussian Moments Conjecture asks whether, for complex polynomials P,Q in n independent standard real Gaussian variables, E(P^m)=0 for all m≥ 1 forces E(QP^m)=0 for all large m. Explicit counterexamples with E(P^m)=0 and E(QP^m)=m!≠ 0…

In the all-heads coin game a player starts with n coins, each showing heads with probability p; each round all remaining coins are flipped, the player must set aside at least one head (losing if none shows), and wins once all coins are set…

Let X = (X_1,…,X_n) be a centered Gaussian vector, not necessarily nondegenerate. Then, for every α_1,…,α_n > 0, E[∏_i=1^n |X_i|^α_i] ≥ ∏_i=1^n E[|X_i|^α_i]. Moreover, if Var(X_i) > 0 for every i, then equality holds if and only if…

Let X_1,…,X_n be independent nonnegative random variables with EX_i ≤ 1, and let S be their sum. Is P(S < ES + 1) ≥ 1/e? Feige proved the constant 1/13 and conjectured the sharp 1/e. Three independent July 2026 proofs settle it, both…