arXiv subjects

Peter J. Forrester

Publications and source records attributed to Peter J. Forrester.

At least 145 records · Page 8Linked to original sources

arXiv2008

Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble

The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly $k$ eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly non-symmetric.

math-ph↗

arXiv2008

The Anti-Symmetric GUE Minor Process

Our study is initiated by a multi-component particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species $j$ consists of $\lfloor j/2 \rfloor$ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of anti-symmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge.

math.PR↗

arXiv2008

Determinantal Correlations for Classical Projection Processes

Recent applications in queuing theory and statistical mechanics have isolated the process formed by the eigenvalues of successive minors of the GUE. Analogous eigenvalue processes, formed in general from the eigenvalues of nested sequences of matrices resulting from random corank 1 projections of classical random matrix ensembles, are identified for the LUE and JUE. The correlations for all these processes can be computed in a unified way. The resulting expressions can then be analyzed in various scaling limits. At the soft edge, with the rank of the minors differing by an amount proportional to $N^{2/3}$, the scaled correlations coincide with those known from the soft edge scaling of the Dyson Brownian motion model.

math-ph↗

arXiv2007

A random matrix decimation procedure relating $β= 2/(r+1)$ to $β= 2(r+1)$

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $β$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $β$-ensembles with $β= 2/(r+1)$ is equal to that of another $β$-ensemble with $β= 2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.

math-ph↗

arXiv2007

The importance of the Selberg integral

It has been remarked that a fair measure of the impact of Atle Selberg's work is the number of mathematical terms which bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory, and cases of the Macdonald conjectures. It further initiated the study of q-analogues, which in turn enriched the Macdonald conjectures. We review these developments and proceed to exhibit the sustained prominence of the Selberg integral, evidenced by its central role in random matrix theory, Calogero-Sutherland quantum many body systems, Knizhnik-Zamolodchikov equations, and multivariable orthogonal polynomial theory.

math.CA↗

arXiv2007

On the gap probability generating function at the spectrum edge in the case of orthogonal symmetry

The gap probability generating function has as its coefficients the probability of an interval containing exactly $k$ eigenvalues. For scaled random matrices with orthogonal symmetry, and the interval at the hard or soft spectrum edge, the gap probability generating functions have the special property that they can be evaluated in terms of Painlevé transcendents. The derivation of these results makes use of formulas for the same generating function in certain scaled, superimposed ensembles expressed in terms of its correlation functions. It is shown that by a judicious choice of the superimposed ensembles, the scaled limit necessary to derive these formulas can be rigorously justified by a straight forward analysis.

math-ph↗

arXiv2007

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

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé \IIId system and its associated linear isomonodromic system characterise the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter $ a $. In particular choosing $ a=\pm 1/2 $ allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed.

math.CA↗

arXiv2007

Eigenvalue statistics of the real Ginibre ensemble

The real Ginibre ensemble consists of random $N \times N$ matrices formed from i.i.d. standard Gaussian entries. By using the method of skew orthogonal polynomials, the general $n$-point correlations for the real eigenvalues, and for the complex eigenvalues, are given as $n \times n$ Pfaffians with explicit entries. A computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented. This is relevant to May's stability analysis of biological webs.

cond-mat.stat-mech↗

arXiv2007

Symmetrized models of last passage percolation and non-intersecting lattice paths

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups $U(l)$, $Sp(2l)$ and $O(l)$. We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.

math-ph↗

arXiv2006

Random discrete Schrödinger operators from Random Matrix Theory

We investigate random, discrete Schrödiner operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson's Coulomb gas inverse temperature $β$. They belong to the class of "critical" random Schrödiner operators with random potentials which diminish as $|x|^{-{1/2}}$. We show that as a function of $β$ their eigenstates undergo a transition from extended ($β\ge 2 $) to power-law localized ($0 < β< 2$).

math-ph↗

arXiv2006

A note on biorthogonal ensembles

We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term.

math-ph↗

arXiv2006

Asymptotic correlations for Gaussian and Wishart matrices with external source

We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre) $N\times N$ hermitian matrices. We study the effect of finite rank perturbations of these ensembles by a source term. The rank $r$ of the perturbation corresponds to the number of non-null eigenvalues of the source matrix. In the perturbed ensembles, the correlation functions can be written in terms of kernels. We show that for all $N$, the difference between the perturbed and the unperturbed kernels is a degenerate kernel of size $r$ which depends on multiple Hermite or Laguerre functions. We also compute asymptotic formulas for the multiple Laguerre functions kernels in terms multiple Bessel (resp. Airy) functions. This leads to the large $N$ limiting kernels at the hard (resp. soft) edge of the spectrum of the perturbed Laguerre ensemble. Similar results are obtained in the Hermite case.

math-ph↗

arXiv2006

Relationships between $τ$-function and Fredholm determinant expressions for gap probabilities in random matrix theory

The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of $τ$-functions. Extending recent work relating to the soft edge, it is shown that these $τ$-functions, and their generalizations to contain a generating function parameter, can be expressed as Fredholm determinants. These same Fredholm determinants also occur in exact expressions for gap probabilities in scaled random matrix ensembles with unitary and symplectic symmetry.

math-ph↗

arXiv2006

Hermite and Laguerre $β$-ensembles: asymptotic corrections to the eigenvalue density

We consider Hermite and Laguerre $β$-ensembles of large $N\times N$ random matrices. For all $β$ even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized (multivariate) classical orthogonal polynomials. The corrections to the bulk density are oscillatory terms that depends on $β$. At the edges, the density can be expressed as a multiple integral of the Konstevich type which constitutes a $β$-deformation of the Airy function. This allows us to obtain the main contribution to the soft edge density when the spectral parameter tends to $\pm\infty$.

math-ph↗

arXiv2006

Correlation functions for random involutions

Our interest is in the scaled joint distribution associated with $k$-increasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution for a Poissonized model in which both the number of symbols in the involution, and the number of fixed points, are random variables. From this, a de-Poissonization argument yields the scaled correlations and distribution function for the random involutions. These are found to coincide with the same quantities known in random matrix theory from the study of ensembles interpolating between the orthogonal and symplectic universality classes at the soft edge, the interpolation being due to a rank 1 perturbation.

math.CO↗

arXiv2005

Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices

In a recent work Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are beta-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f generalizing the circular beta-ensemble, and we show how this joint density is related to known inter-relations between circular ensembles. Projecting the joint density onto the real line leads to the derivation of a random three-term recurrence for polynomials with zeros distributed according to the circular Jacobi beta-ensemble.

math.PR↗

arXiv2005

Counting formulas associated with some random matrix averages

Moments of secular and inverse secular coefficients, averaged over random matrices from classical groups, are related to the enumeration of non-negative matrices with prescribed row and column sums. Similar random matrix averages are related to certain configurations of vicious random walkers and to the enumeration of plane partitions. The combinatorial meaning of the average of the characteristic polynomial of random Hermitian and Wishart matrices is also investigated, and consequently several simple universality results are derived.

math.CO↗

arXiv2003

Growth models, random matrices and Painleve transcendents

The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulas. The derivation of these formulas is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the KPZ universality class. Following works of Baik and Rains, and Prähofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlevé II transcendent plays a prominent role.

math-ph↗