Comparability of random permutations in the strong Bruhat order

TL;DR

This paper investigates the probability that two random permutations are comparable in the strong Bruhat order, showing it decreases faster than polynomially and is approximately exponential in the square of the logarithm of permutation size.

Contribution

It establishes that the probability of comparability in the strong Bruhat order diminishes faster than any polynomial, refining previous bounds and providing a precise asymptotic estimate.

Findings

Probability decreases faster than polynomially

Probability is on the order of exp(-Theta(log^2 n))

Improves understanding of permutation comparability in algebraic combinatorics

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 $\exp(-\Theta(\log^2 n))$.