The Toda chain is the prime example of a classical integrable system with strictly local
conservation laws. Relying on the Dumitriu–Edelman matrix model, we obtain the …

The lecture notes cover the emergence of generalized hydrodynamics for the classical and
quantum Toda chain, the classical Calogero fluid, the Ablowitz-Ladik discretization of the …

We find Stieltjes-type and Jacobi-type continued fractions for some “master polynomials” that
enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) …

We consider the discrete defocusing nonlinear Schrödinger equation in its integrable
version, which is called defocusing Ablowitz–Ladik lattice. We consider periodic boundary …

1. Introduction, 2. Dynamics of the classical Toda lattice, 3. Static properties, 4. Mean-field
Dyson Brownian motion, 5. Hydrodynamics for hard rods, 6. Generalized hydrodynamic …

In the classical β-ensembles of random matrix theory, setting β= 2α/N and taking the N→∞
limit gives a statistical state depending on α. Using the loop equations for the classical β …

We study the limiting behavior of Gaussian beta ensembles in the regime where β n= const
β n= const as n → ∞ n→∞. The results are (1) Gaussian fluctuations for linear statistics of …

This paper establishes universal formulas describing the global asymptotics of two different
models of discrete $\beta $-ensembles in high, low and fixed temperature regimes. Our …

We consider N particles in the plane, influenced by a general external potential, that are
subject to the Coulomb interaction in two dimensions at inverse temperature β β. At large …

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where $\frac
{N\beta}{2}\to c $, called the high temperature regime and show that it can be used to …