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 Pr\"ahofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlev\'e II transcendent plays a prominent role.Comment: 27 pages, 5 figure

Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) \le l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page

Scaling limit of vicious walks and two-matrix model

We consider the diffusion scaling limit of the one-dimensional vicious walker model of Fisher and derive a system of nonintersecting Brownian motions. The spatial distribution of $N$ particles is studied and it is described by use of the probability density function of eigenvalues of $N \times N$ Gaussian random matrices. The particle distribution depends on the ratio of the observation time $t$ and the time interval $T$ in which the nonintersecting condition is imposed. As $t/T$ is going on from 0 to 1, there occurs a transition of distribution, which is identified with the transition observed in the two-matrix model of Pandey and Mehta. Despite of the absence of matrix structure in the original vicious walker model, in the diffusion scaling limit, accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution.Comment: REVTeX4, 12 pages, no figure, minor corrections made for publicatio

The plasma picture of the fractional quantum Hall effect with internal SU(K) symmetries

We consider trial wavefunctions exhibiting SU(K) symmetry which may be well-suited to grasp the physics of the fractional quantum Hall effect with internal degrees of freedom. Systems of relevance may be either spin-unpolarized states (K=2), semiconductors bilayers (K=2,4) or graphene (K=4). We find that some introduced states are unstable, undergoing phase separation or phase transition. This allows us to strongly reduce the set of candidate wavefunctions eligible for a particular filling factor. The stability criteria are obtained with the help of Laughlin's plasma analogy, which we systematically generalize to the multicomponent SU(K) case. The validity of these criteria are corroborated by exact-diagonalization studies, for SU(2) and SU(4). Furthermore, we study the pair-correlation functions of the ground state and elementary charged excitations within the multicomponent plasma picture.Comment: 13 pages, 7 figures; reference added, accepted for publication in PR

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.Comment: 25 pages, 9 figures include

Variance Calculations and the Bessel Kernel

In the Laguerre ensemble of N x N (positive) hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, as N->infinity. Furthermore, this statistic often contains an additional parameter alpha for which the limit alpha->infinity is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_{N->infinity} var_N f and lim_{alpha->infinity}lim_{N->infinity} var_N f.Comment: 7 pages; resubmitted to make postscript compatibl

The Partition Function of Multicomponent Log-Gases

We give an expression for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charge at inverse temperature {\beta} = 1 (restricted to the line in the presence of a neutralizing field) in terms of the Berezin integral of an associated non- homogeneous alternating tensor. This is the analog of the de Bruijn integral identities [3] (for {\beta} = 1 and {\beta} = 4) ensembles extended to multicomponent ensembles.Comment: 14 page

Designing Effective Habitat Studies: Quantifying Multiple Sources of Variability in Bat Activity

Common aims of habitat studies are to differentiate between (i) suitable and unsuitable sites for a given species, and (ii) sites used by different communities of species. To quantify differences between sites, field data of site use must be precise enough that true underlying between-site variability is not masked by within-site measurement error. We designed a pilot study to guide the development of a survey protocol for a habitat study on bats in an agricultural landscape in southeastern Australia. Three woodland sites and two scattered tree sites of 2 ha each were surveyed for nine consecutive nights. At three locations within each site (spaced > 50 m apart) one or two Anabat detectors were mounted 1 m above ground or in a tree (2 m above ground). We used mixed regression models to quantify multiple sources of variability in bat calling activity, and graphical data analysis to visualise how increases in survey effort were likely to affect inference. For the five most active species, we found that typically over 40% of variability in nightly detections occurred at the between-site level; approximately 10% occurred between locations within sites; approximately 20% was explained by night-to-night differences; and approximately 30% of variability was not attributable to systematic variation within experimental units. Differences in community composition between sites were clearly evident when two or more detectors per site were used for four or more nights. We conclude with six general considerations for the design of effective habitat studies. These are to (i) consider key contrasts of interest; (ii) use data from mild, calm, dry nights only; (iii) calibrate detectors; (iv) use multiple detectors where possible, or move a single detector within a site; (v) survey for multiple nights; and (vi) where vertical differentiation in habitat use is likely, mount detectors at different heights. These considerations need to be balanced within the context of financial and logistical constraints

Exact Dynamical Correlation Functions of Calogero-Sutherland Model and One-Dimensional Fractional Statistics

One-dimensional model of non-relativistic particles with inverse-square interaction potential known as Calogero-Sutherland Model (CSM) is shown to possess fractional statistics. Using the theory of Jack symmetric polynomial the exact dynamical density-density correlation function and the one-particle Green's function (hole propagator) at any rational interaction coupling constant $\lambda = p/q$ are obtained and used to show clear evidences of the fractional statistics. Motifs representing the eigenstates of the model are also constructed and used to reveal the fractional {\it exclusion} statistics (in the sense of Haldane's ``Generalized Pauli Exclusion Principle''). This model is also endowed with a natural {\it exchange } statistics (1D analog of 2D braiding statistics) compatible with the {\it exclusion} statistics. (Submitted to PRL on April 18, 1994)Comment: Revtex 11 pages, IASSNS-HEP-94/27 (April 18, 1994

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics. Changes in this revision: recomputed Monte Carlo simulations, added reference [19], fit into margins, performed minor editin