Differential Equations for Dyson Processes

We call "Dyson process" any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n by n Hermitian matrices, and the eigenvalues describe n curves. Given sets X_1,...,X_m the probability that for each k no curve passes through X_k at time \tau_k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process. Generalizing and strengthening earlier work, we assume that each X_k is a finite union of intervals and find for the Airy process a system of partial differential equations, with the end-points of the intervals of the X_k as independent variables, whose solution determines the probability that for each k no curve passes through X_k at time \tau_k. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.Comment: 36 pages, LaTeX. Version 3 corrects an error in the earlier version

On Exact Solutions to the Cylindrical Poisson-Boltzmann Equation with Applications to Polyelectrolytes

Using exact results from the theory of completely integrable systems of the Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann approximation.Comment: 12 pages, 4 figures, LaTeX fil

Universality of the Pearcey process

Consider non-intersecting Brownian motions on the line leaving from the origin and forced to two arbitrary points. Letting the number of Brownian particles tend to infinity, and upon rescaling, there is a point of bifurcation, where the support of the density of particles goes from one interval to two intervals. In this paper, we show that at that very point of bifurcation a cusp appears, near which the Brownian paths fluctuate like the Pearcey process. This is a universality result within this class of problems. Tracy and Widom obtained such a result in the symmetric case, when the two target points are symmetric with regard to the origin. This asymmetry enabled us to improve considerably a result concerning the non-linear partial differential equations governing the transition probabilities for the Pearcey process, obtained by Adler and van Moerbeke

Characteristic polynomials of random matrices at edge singularities

We have discussed earlier the correlation functions of the random variables \det(\la-X) in which $X$ is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the \la's, instead of belonging to the bulk of the spectrum, approach the edge, a cross-over takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of $X$, various new phenomenons may occur and one can tune the spectrum of this source matrix to new critical points. Again there are remarkably simple formulae for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.Comment: 22 pages, late

On the partial connection between random matrices and interacting particle systems

In the last decade there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects appearing in the limit of large matrices arise also in the long time limit for interacting particles and growth models. Examples of these are the famous Tracy-Widom distribution functions and the Airy_2 process. The link is however sometimes fragile. For example, the connection between the eigenvalues in the Gaussian Orthogonal Ensembles (GOE) and growth on a flat substrate is restricted to one-point distribution, and the connection breaks down if we consider the joint distributions. In this paper we first discuss known relations between random matrices and the asymmetric exclusion process (and a 2+1 dimensional extension). Then, we show that the correlation functions of the eigenvalues of the matrix minors for beta=2 Dyson's Brownian motion have, when restricted to increasing times and decreasing matrix dimensions, the same correlation kernel as in the 2+1 dimensional interacting particle system under diffusion scaling limit. Finally, we analyze the analogous question for a diffusion on (complex) sample covariance matrices.Comment: 31 pages, LaTeX; Added a section concerning the Markov property on space-like path

On the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions

The problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions is addressed by considering a particular nonlocal correlation function, called row configuration probability. This correlation function can be used as building block for computing various (both local and nonlocal) correlation functions in the model. The row configuration probability is calculated using the quantum inverse scattering method; the final result is given in terms of a multiple integral. The connection with the emptiness formation probability, another nonlocal correlation function which was computed elsewhere using similar methods, is also discussed.Comment: 15 pages, 2 figure

Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\mathcal O(n^\gamma)$, for $0< \gamma<1$. For a Gaussian potential, P\'ech\'e \cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\times r$ Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.Comment: 41 pages, 4 figure

Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit

We consider the double scaling limit in the random matrix ensemble with an external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.Comment: 36 pages, 9 figure

Smallest Dirac Eigenvalue Distribution from Random Matrix Theory

We derive the hole probability and the distribution of the smallest eigenvalue of chiral hermitian random matrices corresponding to Dirac operators coupled to massive quarks in QCD. They are expressed in terms of the QCD partition function in the mesoscopic regime. Their universality is explicitly related to that of the microscopic massive Bessel kernel.Comment: 4 pages, 1 figure, REVTeX. Minor typos in subscripts corrected. Version to appear in Phys. Rev.

Universality of Correlation Functions in Random Matrix Models of QCD

We demonstrate the universality of the spectral correlation functions of a QCD inspired random matrix model that consists of a random part having the chiral structure of the QCD Dirac operator and a deterministic part which describes a schematic temperature dependence. We calculate the correlation functions analytically using the technique of Itzykson-Zuber integrals for arbitrary complex super-matrices. An alternative exact calculation for arbitrary matrix size is given for the special case of zero temperature, and we reproduce the well-known Laguerre kernel. At finite temperature, the microscopic limit of the correlation functions are calculated in the saddle point approximation. The main result of this paper is that the microscopic universality of correlation functions is maintained even though unitary invariance is broken by the addition of a deterministic matrix to the ensemble.Comment: 25 pages, 1 figure, Late