We consider the Generalized Gibbs ensembles of the Ablowitz-Ladik lattice and of the Schur flow. We derive large deviations principles for the distribution of the empirical measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limits in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble, respectively.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while G.M. participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 20-21 semester "Universality and Integrability in Random Matrix Theory and Interacting Particle Systems". G.M. has received funding from the European Union’s H2020 research and innovation programme under the Marie Skłodowska–Curie grant No. 778010 IPaDEGAN. R.M. has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 884584).
We thank the two anonymous referees for their comments and suggestions.
"Large deviations for Ablowitz-Ladik lattice, and the Schur flow." Electron. J. Probab. 28 1 - 29, 2023. https://doi.org/10.1214/23-EJP941