Convergence of the two-point function of the stationary TASEP

TL;DR

This paper proves the convergence of the two-point function of the stationary TASEP, strengthening previous distributional results to include moments, and confirms a conjecture about its asymptotic behavior.

Contribution

It improves the understanding of the two-point function by establishing moment convergence, confirming a conjecture by Ferrari and Spohn.

Findings

Convergence of the two-point function in a weak sense as time tends to infinity.

Moment convergence of the height function distribution.

Confirmation of Ferrari and Spohn's conjecture.

Abstract

We consider the two-point function of the totally asymmetric simple exclusion process with stationary initial conditions. The two-point function can be expressed as the discrete Laplacian of the variance of the associated height function. The limit of the distribution function of the appropriately scaled height function was obtained previously by Ferrari and Spohn. In this paper we show that the convergence can be improved to the convergence of moments. This implies the convergence of the two-point function in a weak sense along the near-characteristic direction as time tends to infinity, thereby confirming the conjecture in the paper of Ferrari and Spohn.