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The record statistic and forward stability of Schubert products

Andrew Hardt, Reuven Hodges, Hanzhang Yin

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.

The record statistic and forward stability of Schubert products

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 and , 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.

Paper Structure

This paper contains 36 sections, 48 theorems, 255 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Record Statistic
  3. Backward and Forward Stability
  4. Expectation and distribution of forward stability
  5. Conjectures and record-equidistribution phenomena
  6. Outline of the paper
  7. Preliminaries and Notation
  8. Random walk increments
  9. Some probabilistic preliminaries
  10. Uniform Permutations
  11. An upper bound
  12. Asymptotic Distribution
  13. Grassmannian Permutations
  14. Probabilities of records of Unif(Grass(n))
  15. The Modified Distribution G_n
  16. ...and 21 more sections

Key Result

Theorem 1.1

Fix $n\ge 1$ and $j\in[n]$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1.1: Empirical density histograms of $\mathop{\mathrm{\mathrm{FS}}}\nolimits(u,v)$ at $n=500$ for the three sampling models considered in the paper: uniform permutations, Boolean permutations, and Grassmannian permutations.

Theorems & Definitions (109)

  • Theorem 1.1: Record probabilities across permutation families
  • Theorem 1.2
  • Theorem 1.2
  • Theorem 1.2
  • Definition 2.3: Total variation distance
  • Definition 2.5: Distributional equivalence
  • Definition 2.6
  • Definition 2.7
  • Lemma 3.1: Rényi's record theorem
  • proof
  • ...and 99 more