Large deviations for generalized Gibbs ensembles of the classical Toda chain
Alice Guionnet, Ronan Memin
TL;DR
The paper develops a comprehensive large-deviation framework for the empirical eigenvalue distribution of Toda Lax matrices under generalized Gibbs ensembles. By bridging Toda matrices with Dumitriu–Edelman tri-diagonal $\beta$-ensembles, it derives LDPs with rate functions $T_P^V$ and $I_P^V$ that encode Coulomb-gas interactions and entropic terms, and shows how the minimizing measures $\mu_P^V$ and $\nu_P^V$ are linked via $\nu_P^V=\partial_P(P\mu_P^V)$. The results extend Spohn’s polynomial-potential analysis to general smooth potentials, proving exponential tightness, weak and full LDPs for tri-diagonal representations, and establishing convergence of the free energy to its Coulomb-gas counterpart. This work unifies generalized Toda ensembles with Coulomb gas theory, providing explicit large-deviation characterizations and illuminating the density of states for Toda Lax matrices in broad potential settings. The techniques combine subadditivity, couplings between Toda and $\beta$-ensembles, and Varadhan-type arguments to handle unbounded potentials and general growth, yielding robust asymptotic descriptions with potential relevance to integrable systems and statistical mechanics.
Abstract
We prove large deviation principles for the distribution of the empirical measure of the eigenvalues of Lax matrices following the Generalized Gibbs ensembles of the classical Toda chain introduced in [10]. We deduce the almost sure convergence of this empirical measure towards a limit which we describe in terms of the limiting empirical measure of beta-ensembles. Our results apply to general smooth potentials.