Expanding and Linking Random Matrix Theory

Correlation functions between singular values and eigenvalues

M Allard, M Kieburg

2026-01-01

Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant..

Moments of characteristic polynomials for classical β ensembles

BJ Shen, PJ Forrester

2025-11-01

For random matrix ensembles with unitary symmetry, there is interest in the large N form of the moments of the absolute value of t..

Winding number statistics for chiral random matrices: Universal correlations and statistical moments in the unitary case

N Hahn, M Kieburg, O Gat, T Guhr

2025-10-01

The winding number is the topological invariant that classifies chiral symmetric Hamiltonians with one-dimensional parametric depe..

Orthogonal polynomials in the normal matrix model with two insertions

M Kieburg, ABJ Kuijlaars, S Lahiry

2025-06-30

We consider orthogonal polynomials with respect to the weight | z 2 + a 2 | c N e − N | z | 2 in the whole complex plane. We obtai..

Complex symmetric, self-dual, and Ginibre random matrices: analytical results for three classes of bulk and edge statistics

G Akemann, N Aygün, M Kieburg, P Päßler

2025-03-24

Recently, a conjecture about the local bulk statistics of complex eigenvalues has been made based on numerics. It claims that ther..

Discrete orthogonal ensemble on the exponential lattices

SH Li, BJ Shen, GF Yu, PJ Forrester

2025-03-01

Inspired by Aomoto's q-Selberg integral, a study is made of an orthogonal ensemble on an exponential lattice. By introducing a ske..

Power spectra of Dyson's circular ensembles

PJ Forrester, NS Witte

2025-01-01

The power spectrum is a Fourier series statistic associated with the covariances of the displacement from average positions of the..

Yang-Lee zeros for real-space condensation

Z Burda, DA Johnston, M Kieburg

2025-01-01

Using the electrostatic analogy, we derive an exact formula for the limiting Yang-Lee zero distribution in the random allocation m..

Random Pure Gaussian States and Hawking Radiation

E Aurell, L Hackl, P Horodecki, RH Jonsson, M Kieburg

2024-08-09

A black hole evaporates by Hawking radiation. Each mode of that radiation is thermal. If the total state is nevertheless to be pur..

On the gamma difference distribution

PJ Forrester

2024-08-01

The gamma difference distribution is defined as the difference of two independent gamma distributions, with in general different s..

Spectral moments of the real Ginibre ensemble

SS Byun, PJ Forrester

2024-08-01

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differentia..

DIP-RAMP-PLATEAU FOR DYSON BROWNIAN MOTION FROM THE IDENTITY ON U(N)

PJ Forrester, M Kieburg, SH Li, J Zhang

2024-01-01

In recent work, the authors have shown that the eigenvalue probability density function for Dyson Brownian motion from the identit..

Local central limit theorem for real eigenvalue fluctuations of elliptic GinOE matrices

PJ Forrester

2024-01-01

Random matrices from the elliptic Ginibre orthogonal ensemble (GinOE) are a certain linear combination of a real symmetric, and re..

Winding number statistics for chiral random matrices: Averaging ratios of parametric determinants in the orthogonal case

N Hahn, M Kieburg, O Gat, T Guhr

2023-11-01

We extend our recent study of winding number density statistics in Gaussian random matrix ensembles of the chiral unitary (AIII) a..

Expanding the Fourier Transform of the Scaled Circular Jacobi β Ensemble Density

PJ Forrester, BJ Shen

2023-10-01

The family of circular Jacobi β ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz..

Rank 1 perturbations in random matrix theory — A review of exact results

PJ Forrester

2023-10-01

A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this pr..

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

M Kieburg, SH Li, J Zhang, PJ Forrester

2023-06-01

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance proper..

Computable structural formulas for the distribution of the beta-Jacobi edge eigenvalues

Peter J Forrester, Santosh Kumar

2023-05-01

The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eig..

Finite size corrections relating to distributions of the length of longest increasing subsequences

PJ Forrester, A Mays

2023-04-01

Considered are the large N, or large intensity, forms of the distribution of the length of the longest increasing subsequences for..

Winding number statistics for chiral random matrices: Averaging ratios of determinants with parametric dependence

N Hahn, M Kieburg, O Gat, T Guhr

2023-02-01

Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations...

Derivative principles for invariant ensembles

M Kieburg, J Zhang

2023-01-15

In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential..

q-Pearson pair and moments in q-deformed ensembles

PJ Forrester, SH Li, BJ Shen, GF Yu

2023-01-01

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the q-lattice setting is considered...

High-low temperature dualities for the classical β-ensembles

PJ Forrester

2022-10-01

The loop equations for the β-ensembles are conventionally solved in terms of a 1/N expansion. We observe that it is also possible ..

Volume-Law Entanglement Entropy of Typical Pure Quantum States

E Bianchi, L Hackl, M Kieburg, M Rigol, L Vidmar

2022-07-01

The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has recently been conjectured to b..

Differential recurrences for the distribution of the trace of the β-Jacobi ensemble

PJ Forrester, S Kumar

2022-06-01

Examples of the β-Jacobi ensemble in random matrix theory specify the joint distribution of the transmission eigenvalues in scatte..

Consecutive level spacings in the chiral Gaussian unitary ensemble: From the hard and soft edge to the bulk

G Akemann, V Gorski, M Kieburg

2022-05-13

The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the sp..

Applications in random matrix theory of a PIII ′ τ-function sequence from Okamoto's Hamiltonian formulation

D Dai, PJ Forrester, SX Xu

2022-04-01

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the ei..

Global and local scaling limits for the β = 2 Stieltjes-Wigert random matrix ensemble

PJ Forrester

2022-04-01

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann fa..

JOINT MOMENTS OF A CHARACTERISTIC POLYNOMIAL AND ITS DERIVATIVE FOR THE CIRCULAR β-ENSEMBLE

PJ Forrester

2022-01-01

The problem of calculating the scaled limit of the joint moments of the characteristic polynomial, and the derivative of the chara..

Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices

PJ Forrester, SH Li

2022-01-01

The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal..

Circulant L-ensembles in the thermodynamic limit

PJ Forrester

2021-11-05

L-ensembles are a class of determinantal point processes which can be viewed as a statistical mechanical systems in the grand cano..

Dyson’s disordered linear chain from a random matrix theory viewpoint

PJ Forrester

2021-10-01

The first work of Dyson relating to random matrix theory, “The dynamics of a disordered linear chain,” is reviewed. Contained in t..

Quantifying Dip–Ramp–Plateau for the Laguerre Unitary Ensemble Structure Function

PJ Forrester

2021-10-01

The ensemble average of |∑j=1Neikλj|2 is of interest as a probe of quantum chaos, as is its connected part, the structure function..

Relations between moments for the Jacobi and Cauchy random matrix ensembles

PJ Forrester, AA Rahman

2021-07-01

We outline a relation between the densities for the β-ensembles with respect to the Jacobi weight (1 − x)^a(1 + x)<sup>b..

The classical β-ensembles with β proportional to 1/N: From loop equations to Dyson’s disordered chain

PJ Forrester, G Mazzuca

2021-07-01

In the classical β-ensembles of random matrix theory, setting β = 2α/N and taking the N → ∞ limit gives a statistical state depend..

Cascade of phase transitions in a planar Dirac material

T Kanazawa, M Kieburg, JJM Verbaarschot

2021-06-01

We investigate a model of interacting Dirac fermions in 2 + 1 dimensions with M flavors and N colors having the U(M)×SU(N) symmetr..

Differential Identities for the Structure Function of Some Random Matrix Ensembles

PJ Forrester

2021-05-01

The structure function of a random matrix ensemble can be specified in terms of the covariance of the linear statistics ∑j=1Neik1λ..