Large deviations of the periodic Toda chain

This work establishes a large deviation principle for the spectral measure of the Lax matrix associated to the periodic Toda chain of $N$ particles, subject to a generalised Gibbs measure. This large deviation principle is governed by a rate function which can be regarded as a generalisation of the free energy of the system. Such a large deviation principle is proven both for the model when the momentum is constrained to be zero and when it is allowed to fluctuate. Moreover, the large deviation principle is proven directly at the level of the representation of the generalised Gibbs partition function given in terms of the variables realising the classical separation of variables, \textit{i.e.} rectifying the equations of motion. As such, this work paves the way towards the computation of the thermodynamic limit of dynamical correlation functions in the Toda chain subject to generalised Gibbs ensemble statistics.