Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation

Abstract

We consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, which is called defocusing Ablowitz-Ladik lattice. We consider periodic boundary conditions with period $N$ and initial data sample according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz-Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular $\beta$-ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz-Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz-Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation.

33 pages, 1 figures. We corrected some typos, add some references and simplified some proofs

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Cited by in corpus (7)

• Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale
• Large Deviations for Ablowitz-Ladik lattice, and the Schur flow
• CLT for real beta-ensembles at high temperature
• Particle Scattering and Fusion for the Ablowitz-Ladik Chain
• Generalized Hydrodynamics for the Volterra lattice: Ballistic and nonballistic behavior of correlation functions
• A matrix model of a non-Hermitian $\beta$-ensemble
• Lax random matrices from Calogero systems