# Asymptotics of the largest eigenvalue distribution of the Laguerre unitary ensemble

@article{Lyu2015AsymptoticsOT, title={Asymptotics of the largest eigenvalue distribution of the Laguerre unitary ensemble}, author={Shulin Lyu and Chao Min and Yang Chen}, journal={Journal of Mathematical Physics}, year={2015} }

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are between 0 and t, i.e., the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely {\alpha}n(t) and \b{eta}n(t), as well as three auxiliary quantities, denoted by rn(t), Rn(t) and sigma n(t). We establish the second order differential equations for both beta n(t) and rn(t). By…

## 15 Citations

### The smallest eigenvalue distribution of the Jacobi unitary ensembles

- MathematicsMathematical Methods in the Applied Sciences
- 2020

In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight xα(1 − x)β, x ∈ [0, 1], α, β > −1, the probability that all eigenvalues of Hermitian matrices from this…

### Painlevé IV, σ-form, and the deformed Hermite unitary ensembles

- Mathematics
- 2021

We study the Hankel determinant generated by a deformed Hermite weight with one jump w(z,t,γ)=e−z2+tz|z−t|γ(A+Bθ(z−t)), where A ≥ 0, A + B ≥ 0, t ∈ R, γ > −1, and z ∈ R. By using the ladder operators…

### Painlev\'{e} IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants

- Mathematics
- 2022

This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche [3], Filipuk et al. [17] and Clarkson and Jordaan [9]. Filipuk, Van Assche and Zhang…

### Orthogonal polynomials, asymptotics, and Heun equations

- MathematicsJournal of Mathematical Physics
- 2019

The Painleve equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable…

### Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble

- MathematicsRandom Matrices: Theory and Applications
- 2019

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the…

### Painlev\'{e} IV, $\sigma-$Form and the Deformed Hermite Unitary Ensembles

- Mathematics
- 2021

We study the Hankel determinant generated by a deformed Hermite weight with one jump w(z, t, γ) = e 2+tz|z − t|γ(A + Bθ(z − t)), where A ≥ 0, A + B ≥ 0, t ∈ R, γ > −1 and z ∈ R. By using the ladder…

### On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles

- Mathematics, Computer Science
- 2019

Under the Kolmogorov-Smirnov metric, an upper bound for the rate of convergence to the Gaussian law is obtained for linear statistics of matrix ensembles corresponding to Gaussian, Laguerre, and…

### Gap probabilities in the Laguerre unitary ensemble and discrete Painlevé equations

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2020

In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre unitary ensemble, from the point of view of Sakai’s geometric theory of Painlevé…

## References

SHOWING 1-10 OF 68 REFERENCES

### The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble

- MathematicsJournal of Nonlinear Mathematical Physics
- 2018

We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function , are greater than s. This probability is expressed as the quotient of Dn(s,t) and its…

### The Distribution of the first Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble

- Mathematics
- 2007

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlev\'e V system, and the solution of its…

### Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices

- Mathematics
- 2015

We study the probability distribution function (PDF) of the smallest eigenvalue of Laguerre-Wishart matrices $W = X^\dagger X$ where $X$ is a random $M \times N$ ($M \geq N$) matrix, with complex…

### Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

- MathematicsJ. Approx. Theory
- 2010

### Asymptotic gap probability distributions of the Gaussian unitary ensembles and Jacobi unitary ensembles

- Mathematics
- 2018

### Random Matrix Ensembles with Singularities and a Hierarchy of Painlevé III Equations

- Mathematics
- 2015

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with…

### Critical edge behavior in the modified Jacobi ensemble and Painlevé equations

- Mathematics
- 2015

We study the Jacobi unitary ensemble perturbed by an algebraic singularity at t > 1. For fixed t, this is the modified Jacobi ensemble studied by Kuijlaars et al. The main focus here, however, is the…

### Painlevé III′ and the Hankel determinant generated by a singularly perturbed Gaussian weight

- MathematicsNuclear Physics B
- 2018

### Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

### Gap Probability of the Circular Unitary Ensemble with a Fisher–Hartwig Singularity and the Coupled Painlevé V System

- MathematicsCommunications in Mathematical Physics
- 2020

We consider the circular unitary ensemble with a Fisher–Hartwig singularity of both jump type and root type at $$z=1$$ z = 1 . A rescaling of the ensemble at the Fisher–Hartwig singularity leads to…