[Paper Review] The scaling limit of the longest increasing subsequence

Duncan Dauvergne|arXiv (Cornell University)|Jan 1, 2020

Random Matrices and Applications45 references18 citations

TL;DR

This paper establishes a general framework for proving convergence of last passage percolation models to the directed landscape, a universal limit in the Kardar-Parisi-Zhang (KPZ) universality class. By introducing directed metrics and showing that compact convergence to the Airy line ensemble implies convergence to the Airy sheet, the authors prove that the scaled longest increasing subsequence in a uniform random permutation converges to the directed geodesic in the directed landscape.

ABSTRACT

I will describe a framework for proving convergence to the directed landscape, the central limit object in the KPZ universality class. The directed landscape is a random scale-invariant `directed' metric on the plane. One highlight of this work is that the scaling limit of the longest increasing subsequence in a uniformly random permutation is a geodesic in the directed landscape. Joint work with Balint Virag.

Motivation & Objective

• To establish a general convergence framework for last passage percolation models to the directed landscape, the universal limit in the KPZ universality class.
• To resolve the long-standing open problem of the scaling limit of the longest increasing subsequence in a uniform random permutation.
• To introduce and formalize the concept of directed metrics, a generalization of metrics better suited for limits in random geometry.
• To show that convergence of last passage models to the Airy line ensemble implies convergence to the Airy sheet, a key building block of the directed landscape.
• To extend convergence results to geodesics and particle systems, including the totally asymmetric simple exclusion process (TASEP).

Proposed method

• Introduce the notion of a directed metric, a generalization of classical metrics that behaves well under limits in random geometry.
• Prove that compact convergence of last passage models to the Airy line ensemble implies convergence to the Airy sheet, a fundamental object in the directed landscape.
• Use the RSK correspondence to link last passage percolation in i.i.d. environments to the geometry of the Airy line ensemble.
• Establish hypograph and graph convergence topologies for planar directed metrics to control convergence of distances and geodesics.
• Apply exponential tightness and maximal inequalities to control fluctuations in last passage percolation models.
• Leverage integrable probability techniques for specific models (e.g., geometric, exponential, Poisson, Brownian) to prove convergence to the directed landscape.

Experimental results

Research questions

• RQ1 Does the scaled longest increasing subsequence in a uniform random permutation converge to the directed geodesic in the directed landscape?
• RQ2 Can convergence of last passage models to the Airy line ensemble be upgraded to convergence to the Airy sheet and the directed landscape?
• RQ3 How can directed metrics be used to control convergence of distances and geodesics in random planar geometry?
• RQ4 What conditions ensure that convergence of last passage models implies convergence of their geodesics to those in the directed landscape?
• RQ5 Can the convergence framework be applied to integrable models like TASEP and last passage percolation with various weight distributions?

Key findings

• The scaled longest increasing subsequence in a uniform random permutation converges to the directed geodesic in the directed landscape, solving a long-standing open problem.
• Compact convergence of last passage models to the Airy line ensemble implies convergence to the Airy sheet, establishing a key link in the universality framework.
• Convergence of distances and geodesics in last passage models to their counterparts in the directed landscape is implied by convergence to the Airy sheet.
• The framework applies to classical models including i.i.d. last passage percolation, TASEP, and integrable models with geometric, exponential, Poisson, and Brownian weights.
• The convergence of TASEP to the KPZ fixed point is established via coupling with exponential last passage percolation, with explicit scaling limits provided.
• The paper provides a complete coupling framework for discrete- and continuous-time TASEP, showing convergence to the KPZ fixed point under general initial conditions, including non-sloped and sloped profiles.