Triangular random matrices and biorthogonal ensembles

Large deviations for biorthogonal ensembles and variational formulation for the Dykema-Haagerup distribution

“…The aim of this note is to extend the work of Eichelbascher, Sommerauer and Stolz [ESS11] in order to prove a large deviations principle for a wide class of biorthgonal ensembles which include the matrix models introduced by Cheliotis in [Che14]. The authors of [ESS11] proved a large deviations principle for wide variety of models, such as the biorthogonal Laguerre ensembles or the matrix model of Lueck, Sommers and Zirnbauer [LSZ06] for disordered bosons.…”

A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

“…The model has attracted considerable attention in recent years. Random matrix models whose eigenvalues (or singular values) have the distribution (1.1) were recently given in [6,13]. The model is also related to products of random matrices [13,16].…”

Local Universality in Biorthogonal Laguerre Ensembles

“…These ensembles are further studied by Borodin [10] under a more general framework, namely, biorthogonal ensembles. It is also worthwhile to mention the work of Cheliotis [12], where the author constructed certain triangular random matrices in terms of a Wishart matrix whose squared singular values are distributed according to (1.1); see also [22]. Note that when θ = 1, (1.1) reduces to the well-known Wishart-Laguerre unitary ensemble and plays a fundamental role in random matrix theory; cf.…”

“…From (1.9) and (1.10), one can check directly that ρ blows up with a rate x −θ/(1+θ) near the origin, and vanishes as a square root near (1 + θ) 1+θ /θ θ , which is compatible with the changes of variables. We finally note that the other description of macroscopic behavior with the notion of a DT-element [16] can be found in [12].…”