Large deviations of the periodic Toda chain
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Amphi Schwartz
The Toda chain is a celebrated classical integrable system. It was introduced as a discretisation of the Korteweg-De Vries equation and can be regarded as a one-dimensional lattice where the atoms are connected by nonlinear springs. The system's conserved quantities are encoded in the spectrum of its associated Lax matrix. Because the system exhibits strong interactions, its statistical mechanics are very interesting. The natural probability measure to place on this system is the so-called Generalised Gibbs Measure. In joint work with Alice Guionnet, Karol Kozlowski and Tamara Grava, we prove a Large Deviation Principle for the empirical spectral measure of the Lax matrix subject to a generalised Gibbs distribution. This result confirms a conjecture from the physics literature and provides an explicit rate function with a physical interpretation in terms of soliton scattering.