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

Tamara Grava*, Guido Mazzuca

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)


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 sampled 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 β-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.
Original languageEnglish
Pages (from-to)1689-1729
Number of pages41
JournalCommunications in Mathematical Physics
Issue number3
Early online date3 Feb 2023
Publication statusE-pub ahead of print - 3 Feb 2023

Bibliographical note

Funding Information:
Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement.

Funding Information:
We thank Thomas Kriecherbauer, Ken McLaughlin, Gaultier Lambert and Herbert Spohn for the many discussions and suggestions during our time at MSRI. We thank Rostyslav Kozhan for useful comments on the manuscript. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the author participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester “Universality and Integrability in Random Matrix Theory and Interacting Particle Systems”. This project has received funding from the European Union’s H2020 research and innovation programme under the Marie Skłodowska–Curie grant No. 778010 IPaDEGAN. TG acknowledges the support of GNFM-INDAM group and the research project Mathematical Methods in NonLinear Physics (MMNLP), Gruppo 4-Fisica Teorica of INFN. G.M. is financed by the KAM grant number 2018.0344.

Publisher Copyright:
© 2023, The Author(s).


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