2,181 research outputs found
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
Random walks and random fixed-point free involutions
A bijection is given between fixed point free involutions of
with maximum decreasing subsequence size and two classes of vicious
(non-intersecting) random walker configurations confined to the half line
lattice points . In one class of walker configurations the maximum
displacement of the right most walker is . Because the scaled distribution
of the maximum decreasing subsequence size is known to be in the soft edge GOE
(random real symmetric matrices) universality class, the same holds true for
the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page
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
Painlev\'e 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 Painlev\'e II that arises
is a double shifted B\"acklund 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
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.Comment: 26 pages, 1 Figure, 2 Table
Finite N Fluctuation Formulas for Random Matrices
For the Gaussian and Laguerre random matrix ensembles, the probability
density function (p.d.f.) for the linear statistic
is computed exactly and shown to satisfy a central limit theorem as . For the circular random matrix ensemble the p.d.f.'s for the linear
statistics and are calculated exactly by using a constant term identity
from the theory of the Selberg integral, and are also shown to satisfy a
central limit theorem as .Comment: LaTeX 2.09, 11 pages + 3 eps figs (needs epsf.sty
Applications and generalizations of Fisher-Hartwig asymptotics
Fisher-Hartwig asymptotics refers to the large form of a class of
Toeplitz determinants with singular generating functions. This class of
Toeplitz determinants occurs in the study of the spin-spin correlations for the
two-dimensional Ising model, and the ground state density matrix of the
impenetrable Bose gas, amongst other problems in mathematical physics. We give
a new application of the original Fisher-Hartwig formula to the asymptotic
decay of the Ising correlations above , while the study of the Bose gas
density matrix leads us to generalize the Fisher-Hartwig formula to the
asymptotic form of random matrix averages over the classical groups and the
Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our
generalizations is that they extend to Hankel determinants the Fisher-Hartwig
asymptotic form known for Toeplitz determinants.Comment: 25 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 particles is studied and it is described by use of
the probability density function of eigenvalues of Gaussian random
matrices. The particle distribution depends on the ratio of the observation
time and the time interval in which the nonintersecting condition is
imposed. As 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
Critical behavior in Angelesco ensembles
We consider Angelesco ensembles with respect to two modified Jacobi weights
on touching intervals [a,0] and [0,1], for a < 0. As a \to -1 the particles
around 0 experience a phase transition. This transition is studied in a double
scaling limit, where we let the number of particles of the ensemble tend to
infinity while the parameter a tends to -1 at a rate of order n^{-1/2}. The
correlation kernel converges, in this regime, to a new kind of universal
kernel, the Angelesco kernel K^{Ang}. The result follows from the Deift/Zhou
steepest descent analysis, applied to the Riemann-Hilbert problem for multiple
orthogonal polynomials.Comment: 32 pages, 9 figure
Universal Behavior of Correlations between Eigenvalues of Random Matrices
The universal connected correlations proposed recently between eigenvalues of
unitary random matrices is examined numerically. We perform an ensemble average
by the Monte Carlo sampling. Although density of eigenvalues and a bare
correlation of the eigenvalues are not universal, the connected correlation
shows a universal behavior after smoothing.Comment: ISSP-September-199
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
Correlations between eigenvalues of large random matrices with independent entries
We derive the connected correlation functions for eigenvalues of large
Hermitian random matrices with independently distributed elements using both a
diagrammatic and a renormalization group (RG) inspired approach. With the
diagrammatic method we obtain a general form for the one, two and three-point
connected Green function for this class of ensembles when matrix elements are
identically distributed, and then discuss the derivation of higher order
functions by the same approach. Using the RG approach we re-derive the one and
two-point Green functions and show they are unchanged by choosing certain
ensembles with non-identically distributed elements. Throughout, we compare the
Green functions we obtain to those from the class of ensembles with unitary
invariant distributions and discuss universality in both ensemble classes.Comment: 23 pages, RevTex, hard figures available from [email protected]
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