We study properties of Hamiltonian integrable systems with random initial
data by considering their Lax representation. Specifically, we investigate the
spectral behaviour of the corresponding Lax matrices when the number $N$ of
degrees of freedom of the system goes to infinity and the initial data is
sampled according to a properly chosen Gibbs measure. We give an exact
description of the limit density of states for the exponential Toda lattice and
the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian
$\beta$-ensemble in the high temperature regime. For generalizations of the
Volterra lattice to short range interactions, called INB additive and
multiplicative lattices, the focusing Ablowitz–Ladik lattice and the focusing
Schur flow, we derive numerically the density of states. For all these systems,
we obtain explicitly the density of states in the ground states.

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