Fluctuations for the Toda lattice

TL;DR

This paper analyzes the space-time fluctuations of the Toda lattice at thermal equilibrium, showing they converge to a Gaussian limit and deriving explicit scaling behaviors for particle trajectories and correlations.

Contribution

It establishes the Gaussian nature of fluctuations in the Toda lattice under diffusive scaling and characterizes the joint limit of quasi-particle fluctuations as a dressed Lévy-Chentsov field.

Findings

Fluctuations converge to an explicit Gaussian limit.

Single particle trajectory scales as a Brownian motion.

Correlation functions decay inversely with time, with explicit scaling distributions.

Abstract

In this paper we consider the Toda lattice $(\mathbf{p}(t);\mathbf{q}(t))$ at thermal equilibrium, meaning that its variables $(p_j)$ and $(e^{q_j-q_{j+1}})$ are independent Gaussian and Gamma random variables, respectively. We show under diffusive scaling that the space-time fluctuations for the model's currents converge to an explicit Gaussian limit. As consequences, we deduce, (i) the scaling limit for the trajectory of a single particle $q_0$ is a Brownian motion; (ii) space-time two-point correlation functions for the model decay inversely with time, with explicit scaling distributions predicted by Doyon (SciPost Phys. 5 (2018), 054) and Spohn (J. Phys. A 53 (2020), 265004). Our starting point is the notion that the Toda lattice can be thought of as a dense collection of many ``quasi-particles'' that interact through scattering. The core of our work is to establish that the full joint scaling limit of the fluctuations for these quasi-particles is given by a Gaussian process, called a dressed L\'evy-Chentsov field.