# First Colonization of a Hard-Edge in Random Matrix Theory

@article{Bertola2008FirstCO, title={First Colonization of a Hard-Edge in Random Matrix Theory}, author={Marco Bertola and S. Y. Lee}, journal={Constructive Approximation}, year={2008}, volume={31}, pages={231-257} }

We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model, a phenomenon also known as the “birth of a cut” near a hard-edge. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials.

## 10 Citations

### Mesoscopic colonization of a spectral band

- Mathematics
- 2009

We consider the unitary matrix model in the limit where the size of the matrices becomes infinite and in the critical situation when a new spectral band is about to emerge. In previous works, the…

### Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

- Mathematics
- 2010

Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues,…

### Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Supercritical and Subcritical Regimes

- Mathematics
- 2012

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers and sample covariance matrices. We consider the case when the n×n external source…

### Eigenvalue separation in some random matrix models

- Mathematics
- 2009

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s/(2N)1/2, where N…

### Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

- Mathematics
- 2018

We study n × n Hankel determinants constructed with moments of a Hermite
weight with a Fisher–Hartwig singularity on the real line. We consider the case when
the singularity is in the bulk and is…

### Universality in the Profile of the Semiclassical Limit Solutions to the Focusing Nonlinear Schrödinger Equation at the First Breaking Curve

- Mathematics
- 2009

We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schrodinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating…

### A numerical study of the small dispersion limit of the Korteweg–de Vries equation and asymptotic solutions

- Mathematics
- 2012

### Orthogonal Polynomials for a Class of Measures with Discrete Rotational Symmetries in the Complex Plane

- Mathematics
- 2015

We obtain the strong asymptotics of polynomials $$p_n(\lambda )$$pn(λ), $$\lambda \in {\mathbb {C}}$$λ∈C, orthogonal with respect to measures in the complex plane of the form $$\begin{aligned} \hbox…

### Orthogonal Polynomials for a Class of Measures with Discrete Rotational Symmetries in the Complex Plane

- Materials ScienceConstructive Approximation
- 2016

We obtain the strong asymptotics of polynomials pn(λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

### Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

- Mathematics
- 2008

The eigenvalue statistics of a pair (M1, M2) of n × n Hermitian matrices taken randomly with respect to the measure $${1 \over Z_{n}} \exp \left({-n} {\rm Tr}\,(V(M_{1})+ W(M_{2})- \tau…

## References

SHOWING 1-10 OF 22 REFERENCES

### First Colonization of a Spectral Outpost in Random Matrix Theory

- Mathematics
- 2007

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the…

### Universal distribution of random matrix eigenvalues near the ‘birth of a cut’ transition

- Mathematics
- 2006

We study the eigenvalue distribution of a random matrix, at a transition where a new connected component of the eigenvalue density support appears away from other connected components. Unlike…

### Birth of a Cut in Unitary Random Matrix Ensembles

- Mathematics
- 2007

We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to…

### The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the

- Mathematics
- 2007

In this paper we studied the double scaling limit of a random unitary matrix ensemble near a singular point where a new cut is emerging from the support of the equilibrium measure. We obtained the…

### Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration

- Mathematics
- 2002

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed…

### Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

- Mathematics
- 2005

Abstract We consider polynomials orthogonal on [0,∞) with
respect to Laguerre-type weights w(x) = xα e-Q(x),
where α > -1 and where Q denotes a polynomial with
positive leading coefficient. The…

### Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model

- Mathematics
- 1999

We derive semiclassical asymptotics for the orthogonal polynomials Pn(z) on the line with respect to the exponential weight exp(iNV(z)), where V (z) is a double-well quartic polynomial, in the limit…

### Random Matrices

- Mathematics
- 2005

The elementary properties of random matrices are reviewed and widely used mathematical methods for both hermitian and nonhermitian random matrix ensembles are discussed.

### Logarithmic Potentials with External Fields

- Mathematics
- 1997

This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to…

### New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field

- Mathematics
- 1998

In this paper we use techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for…