[Paper Review] Limit distributions for KPZ growth models with spatially homogeneous random initial conditions

Sunil Chhita, Patrik L. Ferrari+1 more|arXiv (Cornell University)|Nov 21, 2016

Random Matrices and Applications41 references4 citations

TL;DR

This paper establishes a one-parameter family of limit distributions for height fluctuations in the totally asymmetric simple exclusion process (TASEP) under spatially homogeneous random initial conditions, parameterized by the diffusion coefficient σ of the initial particle density. Using last-passage percolation and variational formulas involving the Airy process and two-sided Brownian motion, it proves that the limiting distribution is given by a variational formula involving suprema of a sum of Brownian and Airy processes, generalizing the Baik-Rains and Tracy-Widom distributions to a continuous class of initial conditions.

ABSTRACT

For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the Baik-Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the diffusion coefficient of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six-vertex model at its conical point.

Motivation & Objective

• To understand how height fluctuations in the KPZ universality class depend on the statistical properties of spatially homogeneous random initial conditions.
• To generalize known limit distributions—such as GOE Tracy-Widom and Baik-Rains—beyond the flat and stationary cases.
• To establish a continuous family of limit distributions parameterized by the diffusion coefficient σ of the initial particle density fluctuations.
• To prove convergence to these distributions in the large-time limit using last-passage percolation and functional limit theorems.

Proposed method

• The authors use the last-passage percolation (LPP) representation of TASEP to analyze height fluctuations under general spatially homogeneous initial conditions.
• They derive a variational formula for the limit distribution: $ F^{(\sigma)}(s) = \mathbb{P}\left(\sup_{x\in\mathbb{R}}\left\{\sqrt{2}\sigma B(x) + \mathcal{A}_2(x) - x^2\right\} \leq s \right) $, where $ \mathcal{A}_2 $ is the Airy process and $ B $ is a two-sided Brownian motion.
• The proof relies on recent results on tightness and slow-decorrelation in LPP, particularly functional slow-decorrelation and convergence of finite-dimensional distributions.
• Monte Carlo simulations of TASEP are used to numerically compute and validate the predicted distributions for various σ values.
• An analytical approximation $ F^{(\sigma)}_{\text{appr}}(s) $ is derived by replacing the Airy process with its marginal at zero, enabling convolution-based computation with known distributions.
• The connection to the six-vertex model at its conical point is discussed, suggesting broader relevance to integrable stochastic systems.

Experimental results

Research questions

• RQ1 How do height fluctuations in the KPZ equation or TASEP depend on the diffusion coefficient σ of spatially homogeneous initial particle configurations?
• RQ2 Can the Baik-Rains and Tracy-Widom distributions be unified into a continuous one-parameter family of limit distributions for general initial conditions?
• RQ3 What is the functional form of the limit distribution for TASEP with non-stationary, spatially homogeneous initial data?
• RQ4 How can the convergence to the limit distribution be rigorously established using last-passage percolation and functional limit theorems?
• RQ5 To what extent does the variational formula involving Brownian motion and the Airy process accurately describe the full finite-size behavior of height fluctuations?

Key findings

• The paper establishes a one-parameter family of limit distributions $ F^{(\sigma)}(s) $, indexed by the diffusion coefficient σ of the initial particle density, generalizing the Baik-Rains (σ=1) and GOE Tracy-Widom (σ=0) distributions.
• For σ=0, the limit distribution is the GOE Tracy-Widom distribution, confirming a conjecture by Quastel and Remenik in the context of the KPZ equation.
• Numerical simulations of TASEP show excellent agreement with the theoretical prediction of the variational formula for σ²=3/7 and the stationary case (σ=1).
• An analytical approximation $ F^{(\sigma)}_{\text{appr}}(s) $, based on replacing the Airy process with its value at zero, yields a convolution of GUE and Groeneboom distributions, and matches simulation results surprisingly well even for small σ.
• The approximation improves for larger s, where fluctuations are dominated by the Brownian motion, and the cumulants of the distribution can be computed as sums of cumulants of the component distributions.
• The results are connected to the six-vertex model at its conical point, suggesting a broader universality class for such growth models with spatially homogeneous initial data.