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Description

The Toda lattice is a system of classical mechanics discovered by Toda in 1967, which describes interacting particles on a line. Due to its integrability, the Toda lattice with N particles possesses N independent conserved quantities. In this work, we show that under a certain class of random initial data with a constant particle density, after a large time T infinitely many integrated currents of the Toda lattice converge at the T^{1/2} scale to an explicit Gaussian scaling limit. This scaling limit is expected to be universal in the context of generalized hydrodynamics, which is the study of hydrodynamic limits of integrable systems. In particular, this result places the Toda lattice in a different universality class from that of a large class of “chaotic” interacting particle systems in 1+1 dimensions, whose fluctuations (along characteristic directions) are at the T^{1/3} scale and exhibit non-Gaussian statistics, a phenomenon known as KPZ universality. As part of the proof, we also obtain joint convergence of certain “quasi-particles” to an explicit scaling limit. Joint work with Amol Aggarwal.