arXiv subjects

Peter J. Forrester

Publications and source records attributed to Peter J. Forrester.

At least 109 records · Page 6Linked to original sources

arXiv2014

Eigenvalue statistics for product complex Wishart matrices

The eigenvalue statistics for complex $N \times N$ Wishart matrices $X_{r,s}^\dagger X_{r,s}$, where $ X_{r,s}$ is equal to the product of $r$ complex Gaussian matrices, and the inverse of $s$ complex Gaussian matrices, are considered. In the case $r=s$ the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function ${}_{s+1} F_r$. For finite $N$ the eigenvalue probability density function is computed, and is shown to be an example of a biorthogonal ensemble. A double contour integral form of the corresponding correlation kernel is derived, which allows the hard edge scaled limit to be computed. The limiting kernel is given in terms of certain Meijer G-functions, and is identical to that found in the recent work of Kuijlaars and Zhang in the case $s=0$. Properties of the kernel and corresponding correlation functions are discussed.

math-ph↗

arXiv2014

The Golden-Thompson inequality --- historical aspects and random matrix applications

The Golden-Thompson inequality, ${\rm Tr} \, (e^{A + B}) \le {\rm Tr} \, (e^A e^B)$ for $A,B$ Hermitian matrices, appeared in independent works by Golden and Thompson published in 1965. Both of these were motivated by considerations in statistical mechanics. In recent years the Golden-Thompson inequality has found applications to random matrix theory. In this survey article we detail some historical aspects relating to Thompson's work, giving in particular an hitherto unpublished proof due to Dyson, and correspondence with Pólya. We show too how the $2 \times 2$ case relates to hyperbolic geometry, and how the original inequality holds true with the trace operation replaced by any unitarily invariant norm. In relation to the random matrix applications, we review its use in the derivation of concentration type lemmas for sums of random matrices due to Ahlswede-Winter, and Oliveira, generalizing various classical results.

math-ph↗

arXiv2014

Probability densities and distributions for spiked and general variance Wishart $β$-ensembles

A Wishart matrix is said to be spiked when the underlying covariance matrix has a single eigenvalue $b$ different from unity. As $b$ increases through $b=2$, a gap forms from the largest eigenvalue to the rest of the spectrum, and with $b-2$ of order $N^{-1/3}$ the scaled largest eigenvalues form a well defined parameter dependent state. Recent works by Bloemendal and Virág [BV], and Mo, have quantified this parameter dependent state for real Wishart matrices from different viewpoints, and the former authors have done similarly for the spiked Wishart $β$-ensemble. The latter is defined in terms of certain random bidiagonal matrices. We use a recursive structure to give an alternative construction of the spiked and more generally the general variance Wishart $β$-ensemble, and we give the exact form of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real quaternion Wishart matrices ($β= 4$) the latter is recognised as having appeared in earlier studies on symmetrized last passage percolation, allowing the exact form of the scaled distribution of the largest eigenvalue to be given. This extends and simplifies earlier work of Wang, and is an alternative derivation to a result in [BV]. We also use the construction of the spiked Wishart $β$-ensemble from [BV] to give a simple derivation of the explicit form of the eigenvalue PDF.

math-ph↗

arXiv2014

Raney distributions and random matrix theory

Recent works have shown that the family of probability distributions with moments given by the Fuss-Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss-Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss-Catalan distribution, an equilibrium problem characterizing the density is identified. The Stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from $q$ inverse standard Gaussian matrices, and $s$ standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on $(0,\infty)$, we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterisation. As an application, the leading asymptotic form of the density as the endpoints of the support are approached is computed, and is shown to have some universal features.

math-ph↗

arXiv2014

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| \to \infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.

math-ph↗

arXiv2013

Probability of all eigenvalues real for products of standard Gaussian matrices

With $\{X_i\}$ independent $N \times N$ standard Gaussian random matrices, the probability $p_{N,N}^{P_m}$ that all eigenvalues are real for the matrix product $P_m = X_m X_{m-1} \cdots X_1$ is expressed in terms of an $N/2 \times N/2$ ($N$ even) and $(N+1)/2 \times (N+1)/2$ ($N$ odd) determinant. The entries of the determinant are certain Meijer $G$-functions. In the case $m=2$ high precision computation indicates that the entries are rational multiples of $π^2$, with the denominator a power of 2, and that to leading order in $N$ $p_{N,N}^{P_m}$ decays as $(π/4)^{N^2/2}$. We are able to show that for general $m$ and large $N$, $p_{N,N}^{P_m} \sim b_m^{N^2}$ with an explicit $b_m$. An analytic demonstration that $p_{N,N}^{P_m} \to 1$ as $m \to \infty$ is given.

math-ph↗

arXiv2013

Ramanujan's $_1ψ_1$ summation theorem --- perspective, announcement of bilateral $q$-Dixon--Anderson and $q$-Selberg integral extensions, and context

The Ramanujan $_1ψ_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Bailey's very-well-poised $_6ψ_6$ summation. Bilateral Jackson integral generalizations of the Dixon--Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_1ψ_1$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulas for Jackson integrals relating to other roots systems obtained from the same perspective.

math.CV↗

arXiv2013

Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations

There are some distinguished ensembles of non-Hermitian random matrices for which the joint PDF can be written down explicitly, is unchanged by rotations, and furthermore which have the property that the eigenvalues form a Pfaffian point process. For these ensembles, in which the elements of the matrices are either real, or real quaternion, the kernel of the Pfaffian is completely determined by certain skew orthogonal polynomials, which permit an expression in terms of averages over the characteristic polynomial, and the characteristic polynomial multiplied by the trace. We use Schur polynomial theory, knowledge of the value of a Schur polynomial averaged against real, and real quaternion Gaussian matrices, and the Selberg integral to evaluate these averages.

math-ph↗

arXiv2013

Asymptotics of spacing distributions 50 years later

In 1962 Dyson used a physically based, macroscopic argument to deduce the first two terms of the large spacing asymptotic expansion of the gap probability for the bulk state of random matrix ensembles with symmetry parameter β. In the ensuing years, the question of asymptotic expansions of spacing distributions in random matrix theory has shown itself to have a rich mathematical content. As well as presenting the main known formulas, we give an account of the mathematical methods used for their proofs, and provide some new formulas. We also provide a high precision numerical computation of one of the spacing probabilities to illustrate the accuracy of the corresponding asymptotics.

math-ph↗

arXiv2012

Reunion probability of N vicious walkers: typical and large fluctuations for large N

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyse the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for L\ll O(\sqrt{N}) and L\geq O(\sqrt{N}). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value L=L_c(N)\sim \sqrt{N}. This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L \sim L_c(N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L \gg L_c(N)] and the left [L \ll L_c(N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L \sim L_c(N) and atypical fluctuations in the right tail L \gg L_c(N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices.

cond-mat.stat-mech↗

arXiv2012

Painleve II in random matrix theory and related fields

We review some occurrences of Painlevé II transcendents in the study of two-dimensional Yang-Mills theory, fluctuation formulas for growth models, and as distribution functions within random matrix theory. We first discuss settings in which the parameter $α$ in the Painlevé equation is zero, and the boundary condition is that of the Hasting-MacLeod solution. As well as expressions involving the Painlevé transcendent itself, one encounters the sigma form of the Painlevé II equation, and Lax pair equations in which the Painlevé transcendent occurs as coefficients. We then consider settings which give rise to general $α$ Painlevé II transcendents. In a particular random matrix setting, new results for the corresponding boundary conditions in the cases $α= \pm 1/2$, 1 and 2 are presented.

math-ph↗

arXiv2012

Asymptotics of spacing distributions at the hard edge for $β$-ensembles

In a previous work [J. Math. Phys. {\bf 35} (1994), 2539--2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large $s$ asymptotic form of the general $β> 0$ gap probability $E_β^{\rm hard}(0;(0,s);βa/2)$, provided both $βa /2 \in \mathbb Z_\ge 0$ and $2/β\in \mathbb Z^+$. It shown how the details of this method can be extended to remove the requirement that $2/β\in \mathbb Z^+$. Furthermore, a large deviation formula for the gap probability $E_β(n;(0,x);{\rm ME}_{β,N}(λ^{a β/2} e^{βN λ/2}))$ is deduced by writing it in terms of the charateristic function of a certain linear statistic. By scaling $x = s/(4N)^2$ and taking $N \to \infty$, this is shown to reproduce a recent conjectured formula for $E_β^{\rm hard}(n;(0,s);βa/2)$, $βa /2 \in \mathbb Z_{\ge 0}$, and moreover to give a prediction without the latter restriction. This extended formula, which for the constant term involves the Barnes double gamma function, is shown to satisfy an asymptotic functional equation relating the gap probability with parameters $(β,n,a)$, to a gap probability with parameters $(4/β,n',a')$, where $n'=β(n+1)/2-1$, $a'=β(a-2)/2+2$.

math-ph↗

arXiv2012

Lyapunov exponents for products of complex Gaussian random matrices

The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = Σ^{1/2} G_i^{\rm c}$, where $Σ$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.

math.PR↗

arXiv2012

Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma

The two-dimensional one-component plasma (2dOCP) is a system of $N$ mobile particles of the same charge $q$ on a surface with a neutralising background. The Boltzmann factor of the 2dOCP at temperature $T$ can be expressed as a Vandermonde determinant to the power $Γ=q^{2}/(k_B T)$. Recent advances in the theory of symmetric and anti-symmetric Jack polymonials provide an efficient way to expand this power of the Vandermonde in their monomial basis, allowing the computation of several thermodynamic and structural properties of the 2dOCP for $N$ values up to 14 and $Γ$ equal to 4, 6 and 8. In this work, we explore two applications of this formalism to study the moments of the pair correlation function of the 2dOCP on a sphere, and the distribution of radial linear statistics of the 2dOCP in the plane.

math-ph↗

arXiv2012

The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(β= 1)$, complex ($β= 2)$ and real quaternion $(β= 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $β> 0$. In the Gaussian case a further construction valid for $β> 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. The resulting functional forms are shown to be a special cases of duality formulas due to Desrosiers. New derivations of the general case of Desrosiers' dualities are given. A soft edge scaling limit of the averaged characteristic polynomial is identified, and an explicit evaluation in terms of so-called incomplete Airy functions is obtained.

math.PR↗

arXiv2012

Asymptotic forms for hard and soft edge general $β$ conditional gap probabilities

An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix $β$-ensembles. The conditioning is that there are $n$ eigenvalues in the gap, with $n \ll |t|$, $t$ denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consistent with known asymptotic expansions in the case $n=0$. With this modification made for general $n$, the derived expansions - which are for the logarithm of the gap probabilities - are conjectured to be correct up to and including terms O$(\log|t|)$. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating $β$ to $4/β$.

math-ph↗

arXiv2012

Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $β$-ensembles

We analyze the eigenvalue density for the Laguerre and Jacobi $β$-ensembles in the cases that the corresponding exponents are extensive. In particular, we obtain the asymptotic expansion up to terms $o(1)$, in the large deviation regime outside the limiting interval of support. As found in recent studies of the large deviation density for the Gaussian $β$-ensemble, and Laguerre $β$-ensemble with fixed exponent, there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.

math-ph↗

arXiv2011

Spectral density asymptotics for Gaussian and Laguerre $β$-ensembles in the exponentially small region

The first two terms in the large $N$ asymptotic expansion of the $β$ moment of the characteristic polynomial for the Gaussian and Laguerre $β$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.

math-ph↗