[Paper Review] Dynamical properties of a tagged particle in the totally asymmetric simple exclusion process with the step-type initial condition

Takashi Imamura, Tomohiro Sasamoto|arXiv (Cornell University)|Feb 2, 2007

Random Matrices and Applications14 citations

TL;DR

This paper studies the dynamical behavior of a tagged particle in the one-dimensional totally asymmetric simple exclusion process (TASEP) with step-type initial conditions. Using connections to the Schur process, it derives the multi-time joint distribution of the particle's position as a Fredholm determinant and identifies universal scaling limits: the Airy process in the homogeneous case and a multi-matrix model eigenvalue process when slow particles are present in front.

ABSTRACT

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in TASEP with the step-type initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.

Motivation & Objective

• To analyze the time evolution of a tagged particle in the TASEP with step-type initial conditions.
• To derive the multi-time joint distribution function of the tagged particle's position using integrable probability techniques.
• To investigate the scaling limit of the system and identify universal behavior in the large-time regime.
• To explore how the presence of slow particles in front of the tagged particle alters the limiting stochastic process.

Proposed method

• Leverages the connection between TASEP dynamics and the Schur process to express the joint distribution as a Fredholm determinant.
• Applies techniques from integrable probability and random matrix theory to analyze the scaling limit.
• Uses the formalism of determinantal point processes to characterize the particle's position distribution over time.
• Performs asymptotic analysis of the Fredholm determinant to identify universal limits.
• Compares the limiting process under homogeneous hopping rates to that with external sources (slow particles).

Experimental results

Research questions

• RQ1 How does the joint distribution of the tagged particle's position evolve over time in TASEP with step initial conditions?
• RQ2 What is the functional form of the multi-time joint distribution in terms of determinantal processes?
• RQ3 How does the scaling limit of the tagged particle's motion relate to random matrix theory?
• RQ4 What happens to the limiting process when slow particles are present ahead of the tagged particle?
• RQ5 Does the limiting process transition from the Airy process to a multi-matrix model eigenvalue process under certain conditions?

Key findings

• The multi-time joint distribution of the tagged particle's position is represented as a Fredholm determinant via the Schur process.
• In the homogeneous case with equal hopping rates, the limiting process converges to the Airy process after the particle begins moving.
• When slow particles are present in front, the limiting process transitions from the Airy process to the distribution of the largest eigenvalue in a Hermitian multi-matrix model with external sources.
• The universality of the largest eigenvalue in random matrix theory is realized in the scaling limit of the system.
• The transition time in the limiting process depends on the configuration and hopping rates of the particles ahead of the tagged particle.