arXiv subjects

Peter J. Forrester

Publications and source records attributed to Peter J. Forrester.

At least 127 records · Page 7Linked to original sources

arXiv2011

One-component plasma on a spherical annulus and a random matrix ensemble

The two-dimensional one-component plasma at the special coupling β= 2 is known to be exactly solvable, for its free energy and all of its correlations, on a variety of surfaces and with various boundary conditions. Here we study this system confined to a spherical annulus with soft wall boundary conditions, paying special attention to the resulting asymptotic forms from the viewpoint of expected general properties of the two-dimensional plasma. Our study is motivated by the realization of the Boltzmann factor for the plasma system with β= 2, after stereographic projection from the sphere to the complex plane, by a certain random matrix ensemble constructed out of complex Gaussian and Haar distributed unitary matrices.

math-ph↗

arXiv2011

Fluctuation universality for a class of directed solid-on-solid models

Our interest is in a class of directed solid-on-solid models, which may be regarded as continuum versions of boxed plane partitions. In the case that the heights are chosen from a uniform distribution, the joint PDF of the heights is the same as that for the positions in a finitized bead process recently introduced by the authors and Nordenstam. We use knowledge of the correlation functions for the latter to show that upon a certain scaling the fluctuations of the heights along the back row of the solid-on-solid model are given by the Airy process from random matrix theory, as is the case for boxed plane partitions. Moreover, we show that this limiting distribution remains true if instead of the uniform distribution, the heights are sampled from a general absolutely continuous distribution.

math-ph↗

arXiv2011

Pfaffian point process for the Gaussian real generalised eigenvalue problem

The generalised eigenvalues for a pair of $N\times N$ matrices $(X_1,X_2)$ are defined as the solutions of the equation $\det (X_1-λX_2)=0$, or equivalently, for $X_2$ invertible, as the eigenvalues of $X_2^{-1}X_1$. We consider Gaussian real matrices $X_1,X_2$, for which the generalised eigenvalues have the rotational invariance of the half-sphere, or after a fractional linear transformation, the rotational invariance of the unit disk. In these latter variables we calculate the joint eigenvalue probability density function, the probability $p_{N,k}$ of finding $k$ real eigenvalues, the densities of real and complex eigenvalues (the latter being related to an average over characteristic polynomials), and give an explicit Pfaffian formula for the higher correlation functions $ρ_{(k_1,k_2)}$. A limit theorem for $p_{N,k}$ is proved, and the scaled form of $ρ_{(k_1,k_2)}$ is shown to be identical to the analogous limit for the correlations of the eigenvalues of real Gaussian matrices. We show that these correlations satisfy sum rules characteristic of the underlying two-component Coulomb gas.

math-ph↗

arXiv2011

Non-intersecting Brownian walkers and Yang-Mills theory on the sphere

We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the Tracy-Widom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the Tracy-Widom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the Tracy-Widom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N non-intersecting Brownian excursions ("watermelons" with a wall) whose distribution in the asymptotic scaling limit is then described by GOE Tracy-Widom law. In addition, large deviation formulas for the maximum height are also computed.

math-ph↗

arXiv2010

A generalized plasma and interpolation between classical random matrix ensembles

The eigenvalue probability density functions of the classical random matrix ensembles have a well known analogy with the one component log-gas at the special couplings β= 1,2 and 4. It has been known for some time that there is an exactly solvable two-component log-potential plasma which interpolates between the β=1 and 4 circular ensemble, and an exactly solvable two-component generalized plasma which interpolates between β= 2 and 4 circular ensemble. We extend known exact results relating to the latter --- for the free energy and one and two-point correlations --- by giving the general (k_1+k_2)-point correlation function in a Pfaffian form. Crucial to our working is an identity which expresses the Vandermonde determinant in terms of a Pfaffian. The exact evaluation of the general correlation is used to exhibit a perfect screening sum rule.

math-ph↗

arXiv2010

A Fuchsian matrix differential equation for Selberg correlation integrals

We characterize averages of $\prod_{l=1}^N|x - t_l|^{α- 1}$ with respect to the Selberg density, further contrained so that $t_l \in [0,x]$ $(l=1,...,q)$ and $t_l \in [x,1]$ $(l=q+1,...,N)$, in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated, thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able to compute power series for the marginal distributions of the $t_j$ $(j=1,...,N)$. In the case $q=0$ and $α< 1$ we compute the explicit leading order term in the $x \to 0$ asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations. In the case $q=0$ and $α\in \mathbb Z^+$, and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured.

math-ph↗

arXiv2010

Difference system for Selberg correlation integrals

The Selberg correlation integrals are averages of the products $\prod_{s=1}^m\prod_{l=1}^n (x_s - z_l)^{μ_s}$ with respect to the Selberg density. Our interest is in the case $m=1$, $μ_1 = μ$, when this corresponds to the $μ$-th moment of the corresponding characteristic polynomial. We give the explicit form of a $(n+1) \times (n+1)$ matrix linear difference system in the variable $μ$ which determines the average, and we give the Gauss decomposition of the corresponding $(n+1) \times (n+1)$ matrix. For $μ$ a positive integer the difference system can be used to efficiently compute the power series defined by this average.

math-ph↗

arXiv2010

A Finitization of the Bead Process

The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an $abc$-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after approriate identification of the anisotropy parameter therein.

math.PR↗

arXiv2010

The limiting Kac random polynomial and truncated random orthogonal matrices

An exact calculation of the eigenvalue statistics of truncated random Haar distributed real orthogonal matrices has recently been carried out by Khoruzhenko, Sommers and Zyczkowski. We further develop this calculation, and use it to deduce a Pfaffian form of the correlations for the zeros of the limiting Kac random polynomial. This contrasts with the forms known from previous studies of the real zeros (a multidimensional Gaussian integral with the integrand multiplied by the absolute values of the variables) and the complex zeros (a Hafnian).

math-ph↗

arXiv2010

Jack polynomial fractional quantum Hall states and their generalizations

In the the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with $α= - (r-1)/(k+1)$, $(r-1)$ and $(k+1)$ relatively prime, and with partition given in terms of its frequencies by $[n_00^{(r-1)s}k 0 ^{r-1}k 0 ^{r-1}k \cdots 0 ^{r-1} m]$ satisfy this clustering condition. Our proof makes essential use of the fact that these Jack polynomials are translationally invariant. We also consider nonsymmetric Jack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerre polynomials, and Macdonald polynomials from the viewpoint of the clustering.

math-ph↗

arXiv2010

Connection preserving deformations and $q$-semi-classical orthogonal polynomials

We present a framework for the study of $q$-difference equations satisfied by $q$-semi-classical orthogonal systems. As an example, we identify the $q$-difference equation satisfied by a deformed version of the little $q$-Jacobi polynomials as a gauge transformation of a special case of the associated linear problem for $q$-$\mathrm{P}_{\mathrm{VI}}$. We obtain a parameterization of the associated linear problem in terms of orthogonal polynomial variables and find the relation between this parameterization and that of Jimbo and Sakai.

nlin.SI↗

arXiv2010

Interlaced particle systems and tilings of the Aztec diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on $N$ lines, with line $j$ containing $j$ particles. The particles are restricted to lattice points from 0 to $N$, and particles on successive lines are subject to an interlacing constraint. It is shown that marginal distributions for this particle system can be computed exactly. This in turn is used to give unified derivations of a number of fundamental properties of the tiling problem, for example the evaluation of the number of distinct configurations and the relation to the GUE minor process. An interlaced particle system associated with the domino tiling of a certain half Aztec diamond is similarly defined and analyzed.

math-ph↗

arXiv2009

Tridiagonal realization of the anti-symmetric Gaussian $β$-ensemble

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $β$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $β$-ensemble to those realizing particular examples of the Laguerre $β$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Prüfer phases of the random matrices.

math-ph↗

arXiv2009

Derivation of an eigenvalue probability density function relating to the Poincare disk

A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives the eigenvalue probability density function for the top N x N sub-block of a Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition, and integrating over all variables except the eigenvalues. The integration is done by identifying a recursive structure which reduces the dimension. This approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices A^{-1} B, where A and B are random matrices with entries standard complex normals. We relate the eigenvalue distribution of the sub-blocks to a many body quantum state, and to the one-component plasma, on the pseudosphere.

math-ph↗

arXiv2009

Matrix averages relating to the Ginibre ensembles

The theory of zonal polynomials is used to compute the average of a Schur polynomial of argument $AX$, where $A$ is a fixed matrix and $X$ is from the real Ginibre ensemble. This generalizes a recent result of Sommers and Khorozhenko [J. Phys. A {\bf 42} (2009), 222002], and furthermore allows analogous results to be obtained for the complex and real quaternion Ginibre ensembles. As applications, the positive integer moments of the general variance Ginibre ensembles are computed in terms of generalized hypergeometric functions, these are written in terms of averages over matrices of the same size as the moment to give duality formulas, and the averages of the power sums of the eigenvalues are expressed as finite sums of zonal polynomials.

math-ph↗

arXiv2009

Differential equations for deformed Laguerre polynomials

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painleve transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painleve equation. The methodology used to derive these results rely on two theories regarding differential equations for orthogonal polynomial systems, one involving isomonodromic deformations and the other ladder operators. We compare the two theories by showing how either can be used to obtain a characterization of a more general Laguerre unitary ensemble average in terms of the Hamiltonian system for Painleve V.

math.CA↗

arXiv2008

A method to calculate correlation functions for $β=1$ random matrices of odd size

The calculation of correlation functions for $β=1$ random matrix ensembles, which can be carried out using Pfaffians, has the peculiar feature of requiring a separate calculation depending on the parity of the matrix size N. This same complication is present in the calculation of the correlations for the Ginibre Orthogonal Ensemble of real Gaussian matrices. In fact the methods used to compute the $β=1$, N odd, correlations break down in the case of N odd real Ginibre matrices, necessitating a new approach to both problems. The new approach taken in this work is to deduce the $β=1$, N odd correlations as limiting cases of their N even counterparts, when one of the particles is removed towards infinity. This method is shown to yield the correlations for N odd real Gaussian matrices.

math-ph↗

arXiv2008

Eigenvalue Separation in Some Random Matrix Models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c goes to infinity, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogues an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.

math-ph↗