[Paper Review] Transition between Airy_1 and Airy_2 processes and TASEP fluctuations

Alexei Borodin, Patrik L. Ferrari+1 more|arXiv (Cornell University)|Mar 7, 2007

Random Matrices and Applications14 citations

TL;DR

This paper introduces the Airy₂→₁ transition process, a new stochastic process that describes the crossover between the Airy₁ and Airy₂ processes in the totally asymmetric simple exclusion process (TASEP) with specific initial conditions. Using signed determinantal point processes and asymptotic analysis, the authors derive a one-point distribution that smoothly interpolates between the GUE and GOE Tracy-Widom distributions, with the transition region scaling as $ t^{2/3} $, marking a universal limit law in the KPZ universality class.

ABSTRACT

We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy_1 and Airy_2 processes, whose one-point distributions are the GOE and GUE Tracy-Widom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its one-point distribution is a new interpolation between GOE and GUE edge distributions.

Motivation & Objective

• To understand the universal crossover between curved (Airy₂) and flat (Airy₁) regimes in the KPZ universality class.
• To analyze particle fluctuations in TASEP with initial conditions that yield both constant and linearly decreasing density regions.
• To derive the limit process governing the transition region between Airy₁ and Airy₂ fluctuations.
• To establish a new one-point distribution that interpolates between the GUE and GOE Tracy-Widom laws.
• To extend the framework of signed determinantal point processes to analyze non-step initial conditions beyond the RSK construction.

Proposed method

• The analysis is conducted within the framework of signed determinantal point processes, enabling treatment of non-step initial conditions.
• The authors decompose the correlation kernel into three parts: $ K_0 $, $ K_1 $, and $ K_2 $, each analyzed separately for trace-class properties.
• For each kernel component, they prove Hilbert-Schmidt and trace-class bounds using decay estimates of the Airy function and exponential weights.
• Asymptotic analysis is performed in the large-time limit, focusing on the transition region where density shifts from constant to linearly decreasing.
• The one-point distribution of the limit process is derived via integral kernel asymptotics and interpolation between GUE and GOE edge laws.
• The transition process $ \mathcal{A}_{2\to 1} $ is shown to have a width scaling as $ t^{2/3} $, consistent with KPZ scaling.

Experimental results

Research questions

• RQ1 What is the universal limit process that governs the crossover between the Airy₁ and Airy₂ processes in TASEP?
• RQ2 How does the one-point distribution of particle positions evolve from the GUE Tracy-Widom to the GOE Tracy-Widom law in the transition region?
• RQ3 Can the transition region between flat and curved KPZ regimes be described by a new universal process distinct from previously known interpolations?
• RQ4 What is the scaling behavior of the transition region in terms of time $ t $, and how does it compare to the $ t^{2/3} $ scaling of the KPZ class?
• RQ5 How can signed determinantal point processes be used to analyze initial conditions beyond the step or periodic cases?

Key findings

• The transition region between the Airy₁ and Airy₂ processes is governed by a new universal process, denoted $ \mathcal{A}_{2\to 1} $, which interpolates between the two.
• The one-point distribution of $ \mathcal{A}_{2\to 1} $ smoothly interpolates between the GUE Tracy-Widom distribution $ F_2(s) $ and the GOE Tracy-Widom distribution $ F_1(2^{2/3}s) $.
• The width of the transition region scales as $ t^{2/3} $, consistent with the universal KPZ scaling exponent.
• The process $ \mathcal{A}_{2\to 1} $ is distinct from previously known interpolations, as it features a GOE-type distribution over an extended region rather than at a single point.
• The analysis is valid for initial conditions with both constant and linearly decreasing density, such as particles starting from $ 2\mathbb{Z}_{-} $, enabling the coexistence of Airy₁ and Airy₂ regimes.
• The use of signed determinantal point processes allows for a unified asymptotic analysis of all four regions: constant density, linearly decreasing density, finite distance from the rightmost particle, and the transition region.