Solitons in the Calogero model for distinguishable particles

We consider a large $- N,$ two-family Calogero model in the Hamiltonian, collective-field approach. The Bogomol'nyi limit appears and the corresponding solutions are given by the static-soliton configurations. Solitons from different families are localized at the same place. They behave like a paired hole and lump on the top of the uniform vacuum condensates, depending on the values of the coupling strengths. When the number of particles in the first family is much larger than that of the second family, the hole solution goes to the vortex profile already found in the one-family Calogero model.Comment: 14 pages, no figures, late

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

{\bf τ\tau-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $\tau$-functions for certain Painlev\'e systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $\tau$-functions for certain Painlev\'e systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $\tau$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $\tau$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $\tau$-functions is the gap probability in the corresponding orthogonal symmetry case.Comment: AMS-Late

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

Eigenvalue distributions for some correlated complex sample covariance matrices

The distributions of the smallest and largest eigenvalues for the matrix product $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$ determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently for the same distributions.Comment: 11 page

A real quaternion spherical ensemble of random matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form \bY= \bA^{-1} \bB, where \bA and \bB are independent $N\times N$ matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue jpdf and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices $\beta=1,2,4$. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure \bY= \bA^{-1} \bB, where \bA and \bB are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.Comment: 25 pages, 3 figures. Added another citation in version

Data mining: a tool for detecting cyclical disturbances in supply networks.

Disturbances in supply chains may be either exogenous or endogenous. The ability automatically to detect, diagnose, and distinguish between the causes of disturbances is of prime importance to decision makers in order to avoid uncertainty. The spectral principal component analysis (SPCA) technique has been utilized to distinguish between real and rogue disturbances in a steel supply network. The data set used was collected from four different business units in the network and consists of 43 variables; each is described by 72 data points. The present paper will utilize the same data set to test an alternative approach to SPCA in detecting the disturbances. The new approach employs statistical data pre-processing, clustering, and classification learning techniques to analyse the supply network data. In particular, the incremental k-means clustering and the RULES-6 classification rule-learning algorithms, developed by the present authors’ team, have been applied to identify important patterns in the data set. Results show that the proposed approach has the capability automatically to detect and characterize network-wide cyclical disturbances and generate hypotheses about their root cause

On the evaluation formula for Jack polynomials with prescribed symmetry

The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of smallest degree, a symmetric polynomial results. Of interest in applications is the value of the latter polynomial when all the variables are set equal. Dunkl has obtained this evaluation, making use of a certain skew symmetric operator. We introduce a simpler operator for this purpose, thereby obtaining a new derivation of the evaluation formula. An expansion formula of a certain product in terms of Jack polynomials with prescribed symmetry implied by the evaluation formula is used to derive a generalization of a constant term identity due to Macdonald, Kadell and Kaneko. Although we don't give the details in this work, the operator introduced here can be defined for any reduced crystallographic root system, and used to provide an evaluation formula for the corresponding Heckman-Opdam polynomials with prescribed symmetry.Comment: 18 page

Solving 1d plasmas and 2d boundary problems using Jack polynomials and functional relations

The general one-dimensional ``log-sine'' gas is defined by restricting the positive and negative charges of a two-dimensional Coulomb gas to live on a circle. Depending on charge constraints, this problem is equivalent to different boundary field theories. We study the electrically neutral case, which is equivalent to a two-dimensional free boson with an impurity cosine potential. We use two different methods: a perturbative one based on Jack symmetric functions, and a non-perturbative one based on the thermodynamic Bethe ansatz and functional relations. The first method allows us to compute explicitly all coefficients in the virial expansion of the free energy and the experimentally-measurable conductance. Some results for correlation functions are also presented. The second method provides in particular a surprising fluctuation-dissipation relation between the free energy and the conductance.Comment: 19 page

Off-diagonal correlations of the Calogero-Sutherland model

We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B, 610, 578 (2001)]. The leading analytic and non-analytic terms in the short-distance expansion of the one-body density matrix are discussed. Exact numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasi-condensation, quasi-crystallization and quasi-supersolid regimes.Comment: 17 pages, 7 figure