arXiv subjects

P. J. Forrester

Publications and source records attributed to P. J. Forrester.

At least 19 recordsLinked to original sources

arXiv2018

Parametrising correlation matrices

Correlation matrices are the sub-class of positive definite real matrices with all entries on the diagonal equal to unity. Earlier work has exhibited a parametrisation of the corresponding Cholesky factorisation in terms of partial correlations, and also in terms of hyperspherical co-ordinates. We show how the two are relating, starting from the definition of the partial correlations in terms of the Schur complement. We extend this to the generalisation of correlation matrices to the cases of complex and quaternion entries. As in the real case, we show how the hyperspherical parametrisation leads naturally to a distribution on the space of correlation matrices $\{R\}$ with probability density function proportional to $( \det R)^a$. For certain $a$, a construction of random correlation matrices realising this distribution is given in terms of rectangular standard Gaussian matrices.

math.ST↗

arXiv2018

Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model

In the neighbourhood of the critical point, the correlation length of the spin-spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation $N$ between spins goes to infinity, but the scaling variable $s = N(1-t)/2$ remains fixed, where $t$ is the coupling, and $t=1$ the critical point. Previous work has specified these scaling functions (there is one for the critical point being approached from above, and another if approached from below) in terms of transcendents defined by a particular $σ$-form of the degenerate Painlevé V equation. For the diagonal-diagonal correlation, we characterise the first two leading large $N$ correction terms to the scaling functions --- these occur at orders $N^{-1}$ and $N^{-2}$ --- in terms of solutions of a second order linear differential equation with coefficients given in terms of these transcendents, and show how they can be computed. We show that the order $N^{-1}$ is trivial and can be eliminated through appropriate variables so that the leading non-trivial correction is of order $N^{-2}$. In this respect our result gives precise and full characterisation of claims made in the earlier literature.

math-ph↗

arXiv2018

Kac-Rice fixed point analysis for single- and multi-layered complex systems

We present a null model for single- and multi-layered complex systems constructed using homogeneous and isotropic random Gaussian maps. By means of a Kac-Rice formalism, we show that the mean number of fixed points can be calculated as the expectation of the absolute value of the characteristic polynomial for a product of independent Gaussian (Ginibre) matrices. Furthermore, using techniques from Random Matrix Theory, we show that the high-dimensional limit of our system has a third-order phase transition between a phase with a single fixed point and a phase with exponentially many fixed points. This is result is universal in the sense that it does not depend on finer details of the correlations for the random maps.

math-ph↗

arXiv2018

Lyapunov exponents for some isotropic random matrix ensembles

A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the explicit form of the distribution of the determinant. We use these result to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix product, and we further give explicit expressions for the largest Lyapunov exponent. Generalisations to the case of complex or quaternion entries are also given. For standard Gaussian matrices $X$, the full Lyapunov spectrum for products of random matrices $I_N + {1 \over c} X$ is computed in terms of a generalised hypergeometric function in general, and in terms of a single single integral involving a modified Bessel function for the largest Lyapunov exponent.

math.PR↗

arXiv2018

Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral

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 transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra-Itzykson-Zuber integral.

math-ph↗

arXiv2017

How many eigenvalues of a product of truncated orthogonal matrices are real?

A truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product $P_m$ of $m$ independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of $P_m$ form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulae, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity.

math-ph↗

arXiv2016

Selberg integral theory and Muttalib--Borodin ensembles

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, symmetric Jacobi and Cauchy weights, we show that the problem of computing the normalisations and the biorthogonal polynomials can be reduced down to Muttalib--Borodin ensembles with classical weights on the positive half-line.

math-ph↗

arXiv2016

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly $k$ real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these - the eigenvalue density of the real eigenvalues - gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.

math-ph↗

arXiv2016

Singular Values of Products of Ginibre Random Matrices

The squared singular values of the product of $M$ complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G-functions, or equivalently hypergeometric functions ${}_0 F_M$, also referred to as hyper-Bessel functions. In the case $M=1$ it is well known that the corresponding gap probability for no squared singular values in $(0,s)$ can be evaluated in terms of a solution of a particular sigma form of the Painlevé III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalised this formalism to general $M \ge 1$, but has not exhibited its reduction. After detailing the necessary working in the case $M=1$, we consider the problem of reducing the 12 coupled differential equations in the case $M=2$ to a single differential equation for the resolvent. An explicit 4-th order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third order nonlinear equation. The small and large $s$ asymptotics of the 4-th order equation are discussed, as is a possible relationship of the $M=2$ systems to so-called 4-dimensional Painlevé-type equations.

math.CA↗

arXiv2015

Hydrodynamical spectral evolution for random matrices

The eigenvalues of the matrix structure $X + X^{(0)}$, where $X$ is a random Gaussian Hermitian matrix and $X^{(0)}$ is non-random or random independent of $X$, are closely related to Dyson Brownian motion. Previous works have shown how an infinite hierarchy of equations satisfied by the dynamical correlations become triangular in the infinite density limit, and give rise to the complex Burgers equation for the Green's function of the corresponding one-point density function. We show how this and analogous partial differential equations, for chiral, circular and Jacobi versions of Dyson Brownian motion follow from a macroscopic hydrodynamical description involving the current density and continuity equation. The method of characteristics gives a systematic approach to solving the PDEs, and in the chiral case we show how this efficiently reclaims the characterisation of the global eigenvalue density for non-central Wishart matrices due to Dozier and Silverstein. Collective variables provide another approach to deriving the complex Burgers equation in the Gaussian case, and we show that this approach applies equally as well to chiral matrices. We relate both the Gaussian and chiral cases to the asymptotics of matrix integrals.

math-ph↗

arXiv2014

Loop Equation Analysis of the Circular $ β$ Ensembles

We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,β>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(ζ)=(ζ+z)/(ζ-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment $ m_{k} $ to an equivalent length. The leading large $ N $, large $ k $, $ k/N $ fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low $ k $ moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of $ β= 1,2,4 $, general $ N $ and even, positive $ β$, $ N=2,3 $.

math-ph↗

arXiv2014

Moments of the Gaussian $β$ Ensembles and the large-$N$ expansion of the densities

The loop equation formalism is used to compute the $1/N$ expansion of the resolvent for the Gaussian $β$ ensemble up to and including the term at $O(N^{-6})$. This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at $O(N^{-6})$. The latter contain non-integrable singularities at the endpoints of the support --- we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings $β= 1$, $2$ and $4$ there are characterisations of both the resolvent and the moments which allows for the corresponding expansions to be extended, in some recursive form at least, to arbitrary order. In this regard we give fifth order linear differential equations for the density and resolvent at $β= 1$ and $4$, which complements the known third order linear differential equations for these quantities at $β= 2$.

math.CA↗

arXiv2014

Singular Behavior At The Edge of Laughlin States

A distinguishing feature of fractional quantum Hall (FQH) states is a singular behavior of equilibrium densities at boundaries. In contrast to states at integer filling fraction, such quantum liquids posses an additional dipole moment localized near edges. It enters observable quantities such as universal dispersion of edge states and Lorentz shear stress. For a Laughlin state, this behavior is seen as a peak, or overshoot, in the single particle density near the edge, reflecting a general tendency of electrons in FQH states to cluster near edges. We compute the singular edge behavior of the one particle density by a perturbative expansion carried out around a completely filled Landau level. This correction is shown to fully capture the dipole moment and the major features of the overshoot observed numerically. Furthermore, it exhibits the Stokes phenomenon with the Stokes line at the boundary of the droplet, decaying like a Gaussian inside and outside the liquid with different decay lengths. In the limit of vanishing magnetic length the shape the overshoot is a singular double layer with a capacity that is a universal function of the filling fraction. Finally, we derive the edge dipole moment of Pfaffian FQH states. The result suggests an explicit connection between the magnitude of the dipole moment and the bulk odd viscosity.

cond-mat.str-el↗

arXiv2013

Exact and asymptotic features of the edge density profile for the one component plasma in two dimensions

There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires analytic continuation beyond the finite geometry used in its derivation. We consider both disk and cylinder geometry, and focus attention on the exact and asymptotic features of the edge density. At the special coupling Γ:= q^2/k_BT=2 the system is exactly solvable. In particular the k-point correlation can be written as a k \times k determinant, allowing the edge density to be computed to first order in Γ- 2. A double layer structure is found, which in turn implies an overshoot of the density as the edge of the leading support is approached from inside the plasma. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for into the plasma than for outside. For general Γ, a Gaussian fluctuation formula is used to study the large deviation form of the density for N large but finite. This asymptotic form involves thermodynamic quantities which we independently study, and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma.

cond-mat.stat-mech↗

arXiv2012

Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlevé II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlevé II that arises is a double shifted Bäcklund transformation of the Hasting-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge transitions to existing results for the joint distribution of the first and second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under $a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm determinant type characterisation is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.

math.CA↗

arXiv2011

On the Variance of the Index for the Gaussian Unitary Ensemble

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a $τ$-function of the fourth Painlevé equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance.

math.CA↗

arXiv2011

Rank probabilities for real random $N\times N\times 2$ tensors

We prove that the probability $P_N$ for a real random Gaussian $N\times N\times 2$ tensor to be of real rank $N$ is $P_N=(Γ((N+1)/2))^N/G(N+1)$, where $Γ(x)$, $G(x)$ denote the gamma and Barnes $G$-functions respectively. This is a rational number for $N$ odd and a rational number multiplied by $π^{N/2}$ for $N$ even. The probability to be of rank $N+1$ is $1-P_N$. The proof makes use of recent results on the probability of having $k$ real generalized eigenvalues for real random Gaussian $N\times N$ matrices. We also prove that $\log P_N= (N^2/4)\log (e/4)+(\log N-1)/12-ζ'(-1)+{\rm O}(1/N)$ for large $N$, where $ζ$ is the Riemann zeta function.

math.PR↗

arXiv2011

Fredholm Determinant evaluations of the Ising Model diagonal correlations and their λ- generalisation

The diagonal spin-spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants - one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural λ- parameter generalisation and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case, applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi-orthogonal situation.

math.CA↗