Related papers: Lax random matrices from Calogero systems
In this paper, we study a structured family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. For this family, we obtain explicit and unified formulas for several classical matrix invariants, including…
Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a…
Motivated by the importance ascribed to correlations in random matrices used to model phenomena in various scientific disciplines, we report how algebraic correlations between matrix elements affect the eigenvalue statistics and spectral…
The particular case of the integrable two component (2+1)-dimensional hydrodynamical type systems, which generalises the so-called Hamiltonian subcase, is considered. The associated system in involution is integrated in a parametric form. A…
Various physical models can be expressed in terms of matrices. A valuable tool for analysing matrix models is numerical simulations, often the Metropolis algorithm with various improvements. The downside of this approach is that the…
We investigate the algebraic structure of integrable hierarchies that, we propose, underlie models of $W$-gravity coupled to matter. More precisely, we concentrate on the dispersionless limit of the topological subclass of such theories, by…
A density matrix describes the statistical state of a quantum system. It is a powerful formalism to represent both the quantum and classical uncertainty of quantum systems and to express different statistical operations such as measurement,…
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 non-linear Schroedinger equation, and the Lieb-Liniger…
Using a character expansion method, we calculate exactly the eigenvalue density of random matrices of the form M^\dagger M where M is a complex matrix drawn from a normalized distribution P(M) ~ exp(-\Tr(A M B M^\dagger) with A and B…
We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding transfer matrices give rise to time evolution equations for the initial Lax…
The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The use of the exotic cohomology of the symmetry algebras opens a way to formulate such…
The paper reports the recent results on application and extension of the matrix formulation of lagrangian hydrodynamic equations. The matrix approach is based on the notion of continuous deformation of infinitesimal material elements and…
A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on e(3) parametrized by polynomial a with above Lax matrices are constructed. Five cases from the family are selected by the condition of…
We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of…
Applying the replica method of statistical mechanics, we evaluate the eigenvalue density of the large random matrix (sample covariance matrix) of the form $J = A^{\rm T} A$, where $A$ is an $M \times N$ real sparse random matrix. The…
We discuss integrable many-body systems in one dimension of Calogero-Moser-Sutherland type, both classical and quantum as well as nonrelativistic and relativistic. In particular, we consider fundamental properties such as integrability, the…
We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…
The observed pattern of neutrino mass splittings and mixing angles indicates that their family structure is significantly different from that of the charged fermions. We investigate the implications of these data for the fermion mass…
We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero-Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is…
We construct integrable Hamiltonian systems on $G/K$, where $G$ is a quasitriangular Poisson Lie group and $K$ is a Lie subgroup arising as the fixed point set of a group automorphism $\sigma$ of $G$ satisfying the classical reflection…