[Paper Review] Comments on David Aldous and Persi Diaconis' "Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem"

Ivan Corwin|arXiv (Cornell University)|Jun 27, 2018

Random Matrices and Applications42 references3 citations

TL;DR

This commentary extends Aldous and Diaconis' 1999 work on the longest increasing subsequence (LIS) in random permutations by linking it to three major developments: Kardar-Parisi-Zhang (KPZ) universality, integrable probability, and stochastic PDEs. It demonstrates how the LIS problem maps to the polynuclear growth model and converges to the KPZ equation via weakly asymmetric exclusion processes (ASEP), with convergence proven via microscopic Hopf-Cole transforms and exact solvability techniques.

ABSTRACT

This is a commentary on the article: David Aldous and Persi Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), no. 4, 413-432.

Motivation & Objective

• To extend the understanding of the longest increasing subsequence (LIS) in random permutations beyond Aldous and Diaconis' 1999 analysis by connecting it to broader mathematical and physical universality classes.
• To explain how the LIS problem maps to the polynuclear growth model and the KPZ equation via Poisson point processes and space-time scaling.
• To highlight the role of integrable probability in proving convergence of ASEP to the KPZ equation using exact solvability tools like Markov duality and the microscopic Hopf-Cole transform.
• To demonstrate that the KPZ equation arises as a universal scaling limit across diverse stochastic systems, including ASEP and random matrix theory.
• To illustrate the robustness of the KPZ fixed point and the universality of the Tracy-Widom distribution in the context of LIS and related stochastic growth models.

Proposed method

• Uses Poissonization to model the permutation length as a Poisson random variable with mean N, enabling a continuous space-time representation.
• Constructs a graphical representation of permutations via Poisson point processes on [0,1]×[0,1], where the x- and y-orderings define the permutation.
• Rotates the coordinate system by π/4 to interpret the vertical direction as time and the horizontal as space, transforming the LIS into a path-counting problem on a growing interface.
• Defines the height function h(t,x) as the length of the longest increasing path from (0,0) to (t,x), which evolves via nucleation and coalescence of steps at Poisson-distributed points.
• Applies the microscopic Hopf-Cole transform to the ASEP height function, converting the nonlinear KPZ dynamics into a linear stochastic heat equation (SHE), enabling convergence proofs.
• Uses integrable probability techniques, including Markov duality and quantum group symmetries, to establish exact solvability and convergence to the KPZ equation under weak asymmetry scaling.

Experimental results

Research questions

• RQ1 How does the distribution of the longest increasing subsequence in a uniformly random permutation relate to the Tracy-Widom distribution in random matrix theory?
• RQ2 What is the connection between the LIS problem and the Kardar-Parisi-Zhang (KPZ) universality class in stochastic growth models?
• RQ3 How does the weakly asymmetric exclusion process (ASEP) converge to the KPZ equation, and what role does the microscopic Hopf-Cole transform play in this convergence?
• RQ4 In what ways does integrable probability enable exact solvability and convergence results for stochastic PDEs like the KPZ equation?
• RQ5 What are the limitations and strengths of different convergence methods (e.g., energy solutions, regularity structures, paracontrolled distributions) in approximating the KPZ equation?

Key findings

• The length of the longest increasing subsequence in a random permutation converges in distribution to the Tracy-Widom distribution as N → ∞, linking combinatorics to random matrix theory.
• The LIS problem maps to the polynuclear growth model via a Poisson point process on the unit square, with the height function h(t,x) evolving as a Markov process with nucleation and coalescence dynamics.
• Under weak asymmetry scaling, the ASEP height function h^ASEP converges in distribution to the Hopf-Cole solution of the KPZ equation as ε → 0.
• The microscopic Hopf-Cole transform converts the nonlinear ASEP dynamics into a linear stochastic heat equation (SHE), enabling rigorous convergence proofs via exact solvability.
• The convergence of ASEP to the KPZ equation holds under weak asymmetry and is robust across different approximation schemes, including energy solutions and regularity structures.
• The KPZ equation arises as a universal scaling limit for a wide class of stochastic growth models, with the LIS problem serving as a foundational example of this universality.