The record statistic and forward stability of Schubert products

TL;DR

This paper explores the probabilistic behavior of forward stability in Schubert polynomial products by analyzing permutation record statistics across various classes, providing asymptotic results and new sampling methods.

Contribution

It introduces a detailed probabilistic analysis of forward stability for Schubert products, including explicit formulas, asymptotic behaviors, and a new linear-time uniform sampler for permutations.

Findings

Derived asymptotics for the mean record probabilities in permutation families.

Established limiting distribution results for uniform and Grassmannian permutations.

Proved linear growth of the mean record statistic for Boolean permutations.

Abstract

We initiate a probabilistic study of forward stability for products of Schubert polynomials through the record statistic (left-to-right maxima) of permutations. Building on the explicit record formula for forward stability obtained by Hardt and Wallach, we study random pairs of permutations drawn from three natural families: uniform permutations, Grassmannian permutations, and Boolean permutations. For each family, we determine record probabilities and use them to analyze the asymptotic behavior of forward stability. For uniform and Grassmannian permutations, we obtain asymptotics for the mean together with limiting distribution results. For Boolean permutations, we prove linear-order growth of the mean, and our analysis also produces an explicit time-inhomogeneous Markov chain that yields an exact linear-time uniform sampler. Beyond these cases, we prove that the record-set statistic is equidistributed on the avoidance classes of $132$ and $231$, and consequently the corresponding forward stability distributions coincide. We conclude with conjectures for numerous further permutation classes and a conjectural recursive criterion for when two avoidance classes have the same record-set distribution.