The KPZ Fixed Point and the Directed Landscape

TL;DR

This paper reviews the KPZ universality class, focusing on the KPZ fixed point, its origins, models, recent developments, and key proofs, providing a comprehensive and accessible overview for probabilists.

Contribution

It offers a detailed synthesis of the KPZ fixed point's theory, including recent proofs and future directions, making complex developments accessible to a broad audience.

Findings

Construction of KPZ fixed point from TASEP and Brownian last passage percolation

Overview of the 2018 DOV paper on KPZ universality

Proof of absolute continuity of the KPZ fixed point by Sarkar and Virag

Abstract

The term 'KPZ' stands for the initials of three physicists, namely Kardar, Parisi and Zhang, which, in 1986 conjectured the existence of universal scaling behaviours for many random growth processes in the plane. A process is said to belong to the KPZ universality class if one can associate to it an appropriate 'height function' and show that its 3:2:1 (time : space: fluctuation) scaling limit, see 1.2, converges to a universal random process, the KPZ fixed point. Alternatively, membership is loosely characterised by having: 1. Local dynamics; 2. A smoothing mechanism; 3. Slope-dependent growth rate (lateral growth); 4. Space-time random forcing with the rapid decay of correlations. The central object that we will study is the so-called KPZ fixed point, which belongs to the KPZ universality class. Many strides have been made in the last couple of decades in this field, with constructions of the KPZ fixed point from certain processes such as the totally asymmetric simple exclusion process (with arbitrary initial condition) and Brownian last passage percolation. In this article, we: 1. delineate the origins of KPZ universality; 2. describe and motivate canonical models; 3. give an overview of recent developments, especially those in the 2018 Dauvergne, Ortmann and Virag (DOV) paper; 4. present the strategy of and key points in the proof of the absolute continuity result of the KPZ fixed point by Sarkar and Virag; 5. conclude with remarks for future directions. The presentation is such that the content is displayed in a way that is as self-contained as possible and aimed at a motivated audience that has mastered the fundamentals of the theory of probability.