Effective Velocities in the Toda Lattice
Amol Aggarwal
TL;DR
This work establishes a rigorous law of large numbers for quasiparticle trajectories in the Toda lattice at thermal equilibrium. By encoding the quasiparticle data via the Lax matrix and its spectral density $\varrho$, the authors define a dressing-based effective velocity $v_{\mathrm{eff}}$ and prove that each bulk quasiparticle follows the linear law $Q_j(t) = Q_j(0) + t v_{\mathrm{eff}}(\lambda_j)$ up to a subdiffusive error, for small inverse temperature $\theta$. The proof combines a regularization of the asymptotic scattering relation with sharp concentration bounds for random Lax matrices, and introduces proxy dynamics $\mathfrak{Q}_j(t)$ to linearize the dynamics and exploit diagonal-dominance of an auxiliary matrix $\mathbf{S}$. The results rigorously justify the soliton-like transport picture and the flea-gas-style collision rate ansatz in a classical integrable system with random initial data, providing explicit formulas for $v_{\mathrm{eff}}$ and quantifying fluctuations via concentration and localization theory. Overall, the paper advances the mathematical understanding of hydrodynamic-like transport in integrable systems with random initial data and connects spectral data of the Lax matrix to macroscopic quasiparticle motion.
Abstract
In this paper we consider the Toda lattice $(\boldsymbol{p}(t); \boldsymbol{q}(t))$ at thermal equilibrium, meaning that its variables $(p_i)$ and $(e^{q_i-q_{i+1}})$ are independent Gaussian and Gamma random variables, respectively. This model can be thought of a dense collection of many ``quasiparticles'' that act as solitons. We establish a law of large numbers for the trajectory of these quasiparticles, showing that they travel with approximately constant velocities, which are explicit. Our proof is based on a direct analysis of the asymptotic scattering relation, an equation (proven in previous work of the author) that approximately governs the dynamics of quasiparticles locations. This makes use of a regularization argument that essentially linearizes this relation, together with concentration estimates for the Toda lattice's (random) Lax matrix.