Comparability of random permutations in the strong Bruhat order
Nicholas Christo, Marcus Michelen
TL;DR
The paper determines the asymptotic decay of the probability that two random permutations are comparable in the strong Bruhat order, showing $\mathbb{P}(\pi \le \tau)=\exp(-\Theta(\log^2 n))$, which is faster than any polynomial. The authors recast the event as a two-dimensional persistence problem via $Z(a,b)=X(a,b)-Y(a,b)$ and prove matching upper and lower bounds by combining discrete Bernoulli-to-Gaussian reductions (via Rio’s strong approximation) with Gaussian-persistence techniques and a correlation-based lower bound using the Johnson–Leader–Long–FKG inequality and dyadic chaining. The results resolve prior conjectures suggesting polynomial decay and provide a robust probabilistic framework linking Bruhat-order comparability to Brownian-sheet-type persistence. The methods introduce a versatile toolkit for two-parameter persistence phenomena at the intersection of combinatorics and Gaussian process theory.
Abstract
The (strong) Bruhat order for permutations provides a partial ordering defined as follows: two permutations are comparable if one can be obtained from the other by a sequence of adjacent transpositions that each increase the number of inversions by $1$. Given two random permutations, what is the probability that they are comparable in the Bruhat order? This problem was first considered in a 2006 work of Hammett and Pittel, which showed an exponential lower bound and a polynomial upper bound. The lower bound was very recently improved to the subexponential bound of $\exp(-n^{1/2 + o(1)})$ by Boretsky, Cornejo, Hodges, Horn, Lesnevich, and McAllister. Hammett and Pittel predicted that the probability should decrease polynomially. We show that the probability decreases faster than any polynomial and is on the order of $\exp(-Θ(\log^2 n))$.