Space-time fluctuations for currents in the thermal Toda lattice converge to an explicit Gaussian limit under diffusive scaling, implying Brownian motion for single-particle trajectories and explicit 1/time correlation decays.
Asymptotic Scattering Relation for the Toda Lattice
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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. We justify the notion from the physics literature that this model can be thought of as a dense collection of ``quasiparticles'' that act as solitons by, (i) precisely defining the locations of these quasiparticles; (ii) showing that local charges and currents for the Toda lattice are well-approximated by simple functions of the quasiparticle data; and (iii) proving an asymptotic scattering relation that governs the dynamics of the quasiparticle locations. Our arguments are based on analyzing properties about eigenvector entries of the Toda lattice's (random) Lax matrix, particularly, their rates of exponential decay and their evolutions under inverse scattering.
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Toda flows are extended to a class of unbounded initial conditions with sublinear growth, including η-ensembles from random matrix theory that yield invariant measures.
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Space-time fluctuations for currents in the thermal Toda lattice converge to an explicit Gaussian limit under diffusive scaling, implying Brownian motion for single-particle trajectories and explicit 1/time correlation decays.
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Establishes a large deviation principle for the spectral measure of the Lax matrix of the periodic Toda chain under generalised Gibbs ensemble statistics.