Discrete Integrable Systems and Random Lax Matrices

Tamara Grava, Massimo Gisonni, Giorgio Gubbiotti, Guido Mazzuca*

*Corresponding author for this work

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

5 Citations (Scopus)
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Abstract

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 β-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.
Original languageEnglish
Article number10
Pages (from-to)1-35
JournalJournal of Statistical Physics
Volume190
Issue number10
Early online date1 Nov 2022
DOIs
Publication statusPublished - 1 Oct 2023

Bibliographical note

Funding Information:
This work was made in the framework of the Project “Meccanica dei Sistemi discreti” of the GNFM unit of INDAM. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the authors 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”. M.G., T.G. and G.M. acknowledge the Marie Sklodowska-Curie Grant No. 778010 IPaDEGAN and the support of GNFM-INDAM group. TG and GM acknowledge the hospitality and support of the Galileo Galilei Institute, programme "Randomness, Integrability, and Universality". For the numerical simulations that we presented, we made extensive use of the NumPy [], seaborn [], and matplotlib [] libraries.

Publisher Copyright:
© 2022, The Author(s).

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