# Biorthogonal ensembles with two-particle interactions

@article{Claeys2014BiorthogonalEW, title={Biorthogonal ensembles with two-particle interactions}, author={Tom Claeys and Stefano Romano}, journal={Nonlinearity}, year={2014}, volume={27}, pages={2419 - 2443} }

We investigate determinantal point processes on [0, +∞) of the form We prove that the biorthogonal polynomials associated with such models satisfy a recurrence relation and a Christoffel–Darboux formula if , and that they can be characterized in terms of 1 × 2 vector-valued Riemann–Hilbert problems, which exhibit some non-standard properties. In addition, we obtain expressions for the equilibrium measure associated with our model if w(λ) = e−nV (λ) in the one-cut case with and without a hard…

## 39 Citations

### On biorthogonal ensembles with two particle interaction

- Mathematics
- 2015

We prove asymptotics of one-point correlation functions and derive a large deviation principle for biorthogonal ensembles associated to probability density functions Probk of the form 1 Zk Y i<j |zi…

### The local universality of Muttalib–Borodin biorthogonal ensembles with parameter θ=12

- MathematicsNonlinearity
- 2019

The Muttalib–Borodin biorthogonal ensemble is a probability density function for n particles on the positive real line that depends on a parameter and an external field . For we find the large n…

### A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

- MathematicsJournal of Functional Analysis
- 2021

### Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral

- MathematicsAnnales Henri Poincaré
- 2018

We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues…

### Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral

- Mathematics
- 2017

We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues…

### Some biorthogonal polynomials arising in numerical analysis and approximation theory

- MathematicsJournal of Computational and Applied Mathematics
- 2022

### Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

- Mathematics
- 2016

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit…

### Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model.

- MathematicsPhysical review. E
- 2021

It is shown that appropriate models of γ ensembles can be used as a possible framework to study the effects of disorder on the distribution of conductances and reduces γ can lead to a large nonmonotonicity in the effective potential, which in turn leads to significant changes in the density of eigenvalues.

## References

SHOWING 1-10 OF 45 REFERENCES

### A Christoffel-Darboux formula for multiple orthogonal polynomials

- MathematicsJ. Approx. Theory
- 2004

### Random Matrices with Equispaced External Source

- Mathematics
- 2012

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on…

### Moment determinants as isomonodromic tau functions

- Mathematics
- 2008

We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which…

### Christoffel-Darboux-type formulae and a recurrence for biorthogonal polynomials

- Mathematics
- 1989

It is proved that biorthogonal polynomials obey two different kinds of Christoffel-Darboux-type formulae, one linking polynomials with a different parameter and one combining polynomials with…

### Energy correlations for a random matrix model of disordered bosons

- Physics
- 2006

Linearizing the Heisenberg equations of motion around the ground state of an interacting quantum many-body system, one gets a time-evolution generator in the positive cone of a real symplectic Lie…

### The Isomonodromy Approach to Matrix Models in 2 D Quantum Gravity

- Physics
- 2004

We consider the double-scaling limit in the hermitian matrix model for N 2D quantum gravity associated with the measure exp £ tjZ\ N^3. We show 7 = 1 that after the appropriate modification of the…

### The isomonodromy approach to matric models in 2D quantum gravity

- Physics
- 1992

AbstractWe consider the double-scaling limit in the hermitian matrix model for 2D quantum gravity associated with the measure exp
$$\sum\limits_{j = 1}^N {t_{j^{Z^{2j,} } } N \geqq 3} $$
. We show…

### On the theory of biorthogonal polynomials

- Mathematics
- 1988

Let (x,,u) be a distribution in x E R for every ,u in a real parameter set Q. Subject to additional technical conditions, we study mth degree monic polynomials Pm that satisfy the biorthogonality…