[Paper Review] KPZ limit theorems

Jinho Baik|arXiv (Cornell University)|Jun 28, 2022

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper surveys KPZ limit theorems for one-dimensional interacting particle systems, 1+1 random growth models, and two-dimensional directed polymers, establishing that appropriately scaled height functions converge to a universal 2D random field known as the KPZ fixed point. The key contribution is new results on the periodic domain case, extending the universality framework to finite, periodic geometries and highlighting connections to integrable probability, random matrix theory, and integrable differential equations.

ABSTRACT

One-dimensional interacting particle systems, 1+1 random growth models, and two-dimensional directed polymers define 2d height fields. The KPZ universality conjecture posits that an appropriately scaled height function converges to a model-independent universal random field for a large class of models. We survey limit theorems for a few models and discuss changes that arise in different domains. In particular, we present recent results on periodic domains. We also comment on integrable probability models, integrable differential equations, and universality.

Motivation & Objective

• To provide a comprehensive survey of KPZ limit theorems across key models including TASEP, corner growth, and directed polymers.
• To extend the universality framework to finite, periodic domains, where prior results were limited to infinite or half-infinite spaces.
• To explore connections between KPZ universality, integrable probability, and random matrix theory, particularly through Tracy-Widom distributions.
• To analyze the structure of multi-point and multi-time distributions in both infinite and periodic settings.
• To investigate the role of integrable differential equations in describing the limiting distributions of KPZ models.

Proposed method

• Uses 1:2:3 scaling (time T, space T^{2/3}, height T^{1/3}) to define the scaled height function h_T(γ,τ) = [h(γT^{2/3}, τT) - c(T)] / T^{1/3} for convergence to the KPZ fixed point.
• Applies the Komlós–Major–Tusnády embedding and Skorohod embedding techniques to control convergence in the k=O(n) regime for directed last-passage percolation.
• Establishes connections between exponential directed last-passage percolation (DLPP) and Coulomb gases with logarithmic repulsion and confining potentials.
• Uses random matrix theory to show that the last-passage time L(m,n) has the same distribution as the largest eigenvalue of a complex Wishart matrix.
• Compares limiting distributions in infinite and periodic domains using explicit formulas derived from integrable systems and Fredholm determinants.
• Analyzes differential equations (e.g., Painlevé-type) that govern the distribution functions of the KPZ fixed point in different geometries.

Experimental results

Research questions

• RQ1 How does the KPZ fixed point emerge as a universal limit in 1+1 growth models and interacting particle systems?
• RQ2 What are the limiting distributions of the height function in periodic spatial domains, and how do they differ from those in infinite or half-infinite spaces?
• RQ3 To what extent do the connections between KPZ models, Coulomb gases, and random matrices persist in finite or periodic geometries?
• RQ4 Can the 1:2:3 scaling limit be rigorously established for models with k=O(n) in the directed percolation setting?
• RQ5 How do integrable differential equations such as Painlevé II or Painlevé PIII describe the universal fluctuations in KPZ models across different boundary conditions?

Key findings

• The 1:2:3 scaled height function converges to the KPZ fixed point, a universal 2D random field, for a broad class of models including TASEP and directed polymers.
• For the exponential DLPP model, the last-passage time L(m,n) has the same distribution as the largest eigenvalue of a complex Wishart matrix, linking it to random matrix theory.
• In the periodic domain, the limiting multi-point distribution differs from the infinite-space case, with explicit formulas derived via integrable techniques.
• The Tracy–Widom distribution arises as the one-point marginal of the KPZ fixed point, confirming universality in the step initial condition case.
• For k=O(n), limit theorems are established only in isolated cases, and the proof breaks down for α≥3/7 due to non-negligible upward path contributions.
• The KPZ fixed point is fully characterized as a Markov process on the space of lower semicontinuous functions, with explicit transition probabilities derived from Fredholm determinants.