A team of researchers from ICTS has achieved a significant breakthrough by studying a novel class of random matrices that arise from classical integrable systems. This work offers a promising ground for future analytical insights into several important quantities in the area of Random Matrix Theory such as level spacing statistics, gap ratios, and spectral form factors.
Jitendra Kethepalli, Manas Kulkarni and Anupam Kundu along with Herbert Spohn (Technische Universitat, Munchen, Germany) have recently published their work, titled “Lax Random Matrices from Calogero Systems,” in the Journal of Statistical Mechanics (J. Stat. Mech. 033101(2025). The publication has been selected for the Highlights collection by the JSTAT Scientific Directors.
This is a remarkable class of random matrices rooted in the elegant mathematical framework of classical integrable systems, for example, the Calogero family of models. These random matrices naturally arise from Lax matrices associated with integrable systems. The central agenda of this study was the fundamental question about the distribution of eigenvalues. The authors show that the rich interplay between integrability and randomness, allows one to reinterpret the problem as a generalized form of well-known random matrix models. These results not only deepen our understanding of new random matrix ensembles arising from many-body integrable systems, but also lay the groundwork for advancing generalized hydrodynamics which is a framework for describing large-scale behavior in integrable systems.
Beyond particle models, this work opens the door to exploring integrable lattice systems with Lax Pair structures, such as the Ablowitz-Ladik equation and discretized Landau-Lifshitz models, paving the way for deeper connections between randomness and integrability.