Muttalib--Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form $\prod_{1 \le j < k \le N}(λ_k - λ_j) (λ_k^θ- λ_j^θ)$. We study the Laguerre and Jacobi versions of this model --- so named by the form of the one-body interaction terms --- and show that for $θ\in \mathbb Z^+$ they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general $θ> 0$, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent result due to Cheliotis, although our derivation is different. We make use of a generalisation of a double contour integral formula for the correlation functions contained in a paper by Adler, van Moerbeke and Wang to analyse the global density (which we also analyse by studying characteristic polynomials), and the hard edge scaled correlation functions. For the global density functional equations for the corresponding resolvents are obtained; solving this gives the moments in terms of Fuss--Catalan numbers (Laguerre case --- a known result) and particular binomial coefficients (Jacobi case). For $θ\in \mathbb Z^+$ the Laguerre and Jacobi cases are closely related to the squared singular values for products of $θ$ standard Gaussian random matrices, and truncations of unitary matrices, respectively. At the hard edge the double contour integral formulas provide a double contour integral form of the scaled correlation kernel obtained by Borodin in terms of Wright's Bessel function.
The Muttalib-Borodin biorthogonal ensemble is a joint density function for $n$ particles on the positive real line that depends on a parameter $θ$. There is an equilibrium problem that describes the large $n$ behavior. We show that for rational values of $θ$ there is an equivalent vector equilibrium problem.
We study Muttalib--Borodin ensembles --- particular eigenvalue PDFs on the half-line --- with classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We show how the theory of the Selberg integral, involving also Jack and Schur polynomials, naturally leads to a multi-parameter generalisation of these particular Muttalib--Borodin ensembles, and also to the explicit form of underlying biorthogonal polynomials of a single variable. A suitable generalisation of the original definition of the Muttalib--Borodin ensemble allows for negative eigenvalues. In the cases of generalised Gaussian, symmetr…
In this paper, we consider Muttalib-Borodin ensemble of Laguerre type, a determinantal point process over $[0,\infty)$ which depends on the varying weights $x^αe^{-nV(x)}$, $α>-1$, and a parameter $θ$. For $θ$ being a positive integer, we derive asymptotics of the associated biorthogonal polynomials near the origin for a large class of potential functions $V$ as $n\to \infty$. This further allows us to establish the hard edge scaling limit of the correlation kernel, which is previously only known in the special cases and conjectured to be universal. Our proof is based on the Deift/Zhou nonline…
Muttalib-Borodin determinants are generalizations of Hankel determinants and depend on a parameter $θ>0$. In this paper, we obtain large $n$ asymptotics for $n \times n$ Muttalib-Borodin determinants whose weight possesses an arbitrary number of Fisher-Hartwig singularities. As a corollary, we obtain asymptotics for the expectation and variance of the real and imaginary parts of the logarithm of the characteristic polynomial, several central limit theorems, and some global bulk rigidity upper bounds. Our results are valid for all $θ> 0$.
In this note we study a natural measure on plane partitions giving rise to a certain discrete-time Muttalib-Borodin process (MBP): each time-slice is a discrete version of a Muttalib-Borodin ensemble (MBE). The process is determinantal with explicit time-dependent correlation kernel. Moreover, in the $q \to 1$ limit, it converges to a continuous Jacobi-like MBP with Muttalib-Borodin marginals supported on the unit interval. This continuous process is also determinantal with explicit correlation kernel. We study its hard-edge scaling limit (around 0) to obtain a discrete-time-dependent generali…
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 distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer $G$-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a $2\times 2$ Riemann-Hilbert problem, and use this repre…
We study probabilistic and combinatorial aspects of natural volume-and-trace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hard- and soft-edge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy--Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP):…
We consider the limiting process that arises at the hard edge of Muttalib--Borodin ensembles. This point process depends on $θ> 0$ and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form \begin{equation*} \mathbb{P}(\mbox{gap on } [0,s]) = C \exp \left( -a s^{2ρ} + b s^ρ + c \ln s \right) (1 + o(1)) \qquad \mbox{as }s \to + \infty, \end{equation*} where the constants $ρ$, $a$, and $b$ have been derived e…
We demonstrate a method to solve a general class of random matrix ensembles numerically. The method is suitable for solving log-gas models with biorthogonal type two-body interactions and arbitrary potentials. We reproduce standard results for a variety of well-known ensembles and show some new results for the Muttalib-Borodin ensembles and recently introduced $γ$-ensemble for which analytic results are not yet available.
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 form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix …
A $θ$-deformation of the Laguerre weighted Cauchy two-matrix model, and the Bures ensemble, is introduced. Such a deformation is familiar from the Muttalib-Borodin ensemble. The $θ$-deformed Cauchy-Laguerre two-matrix model is a two-component determinantal point process. It is shown that the correlation kernel, and its hard edge scaled limit, can be written as the Fox H-functions, generalising the Meijer G-function class known from the study of the case $θ= 1$. In the $θ=1$ case, it is shown Laguerre-Bures ensemble is related to the Laguerre-Cauchy two-matrix model, notwithstanding the Bures e…
We study the eigenvalue correlations of random Hermitian $n\times n$ matrices of the form $S=M+εH$, where $H$ is a GUE matrix, $ε>0$, and $M$ is a positive-definite Hermitian random matrix, independent of $H$, whose eigenvalue density is a polynomial ensemble. We show that there is a soft-to-hard edge transition in the microscopic behaviour of the eigenvalues of $S$ close to $0$ if $ε$ tends to $0$ together with $n\to +\infty$ at a critical speed, depending on the random matrix $M$. In a double scaling limit, we obtain a new family of limiting eigenvalue correlation kernels. We apply our gener…
We start from applying the general idea of spectral projection (suggested by Olshanski and Borodin and advocated by Tao) to the complex Wishart model. Combining the ideas of spectral projection with the insights from quantum mechanics we derive in an effortless way all spectral properties of the complex Wishart model: first, the Marcenko-Pastur distribution interpreted as a Bohr-Sommerfeld quantization condition for the hydrogen atom; second, hard (Bessel), soft (Airy) and bulk (sine) microscopic kernels from properly rescaled radial Schrödinger equation for the hydrogen atom. Then, generalizi…
The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large $N$ asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes as a special case the Jacobi unitary ensemble. In each case the hard edge scaled kernel permits an expansion in powers of $1/N$, with the leading term given in a structured form involv…
We consider several limiting cases of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent $γ$ (called the $γ$-ensembles). The effective potential, which is essentially the single-particle confining potential for an equivalent ensemble with $γ=1$ (called the Muttalib-Borodin ensemble), is a crucial quantity defined in solution to the Riemann-Hilbert problem associated with the $γ$-ensembles. It enables us to numerically compute the eigenvalue density of $γ$-ensembles for all $γ> 0$. We show that one important effect of th…
We give an overview of the recursive characterisations of random matrix ensembles that are currently at the forefront of random matrix theory by way of studying two classes of ensembles using two different types of recursive schemes: Established theory on Selberg correlation integrals is used to derive linear differential equations on the eigenvalue densities and resolvents of the classical matrix ensembles, which lead to $1$-point recursions, understood to be analogues of the Harer-Zagier recursion, for the expansion coefficients of the associated $1$-point cumulants, while loop equation anal…
