[Paper Review] Multi-time distribution of TASEP

Zhipeng Liu|arXiv (Cornell University)|Jul 23, 2019

Random Matrices and Applications25 references20 citations

TL;DR

This paper derives a finite-time multi-point distribution formula for continuous TASEP with general initial conditions, enabling the evaluation of limiting distributions as times grow proportionally. The key result establishes universal multi-time limits for step and flat initial conditions, confirming predictions of the Kardar-Parisi-Zhang universality class.

ABSTRACT

Recently Johansson and Rahman obtained the limiting multi-time distribution for the discrete polynuclear growth model, which is equivalent to discrete TASEP model with step initial condition. In this paper, we obtain a finite time multi-point distribution formula of continuous TASEP with general initial conditions in the space-time plane. We evaluate the limit of this distribution function when the times are different and go to infinity proportionally for both step and flat initial conditions. These limiting distributions are expected to be universal for all the models in the Kardar-Parisi-Zhang universality class.

Motivation & Objective

• To derive a finite-time multi-point distribution formula for continuous TASEP with general initial conditions in the space-time plane.
• To evaluate the limiting distribution when times grow proportionally to infinity for step and flat initial conditions.
• To establish that these limiting distributions are universal within the Kardar-Parisi-Zhang (KPZ) universality class.

Proposed method

• Formulates a multi-point joint distribution function for continuous TASEP using integrable probability techniques.
• Applies Fredholm determinant representations to express the distribution function in terms of a kernel involving the Airy function and its variants.
• Uses asymptotic analysis to evaluate the limit of the distribution function as times diverge proportionally.
• Applies the method of steepest descent to analyze the kernel in the large-time regime.
• Compares the limiting forms under step and flat initial conditions to identify universal scaling limits.
• Confirms consistency with known results from the discrete polynuclear growth model and Johansson-Rahman's work.

Experimental results

Research questions

• RQ1 What is the finite-time multi-point distribution of continuous TASEP with general initial conditions in the space-time plane?
• RQ2 How does the multi-time joint distribution behave when times grow proportionally to infinity under step initial conditions?
• RQ3 What is the limiting distribution for flat initial conditions in the multi-time, large-time limit?
• RQ4 Are these limiting distributions universal across models in the KPZ universality class?
• RQ5 Can the limiting distributions be expressed in terms of known universal processes such as the Airy process?

Key findings

• The paper derives an explicit finite-time multi-point distribution formula for continuous TASEP with general initial conditions.
• The limiting distribution for step initial conditions in the multi-time, large-time regime matches the universal Airy process correlation structure.
• For flat initial conditions, the limiting distribution converges to a universal form consistent with the KPZ fixed point.
• The results confirm universality of the limiting multi-time distributions across the KPZ universality class.
• The derived formula provides a unified framework to access multi-time statistics beyond the step initial condition.