37,963 research outputs found
Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model
We derive bulk asymptotics of skew-orthogonal polynomials (sop)
\pi^{\bt}_{m}, , 4, defined w.r.t. the weight , , and . We assume that as there
exists an , such that , where is the critical value which separates
sop with two cuts from those with one cut. Simultaneously we derive asymptotics
for the recursive coefficients of skew-orthogonal polynomials. The proof is
based on obtaining a finite term recursion relation between sop and orthogonal
polynomials (op) and using asymptotic results of op derived in \cite{bleher}.
Finally, we apply these asymptotic results of sop and their recursion
coefficients in the generalized Christoffel-Darboux formula (GCD) \cite{ghosh3}
to obtain level densities and sine-kernels in the bulk of the spectrum for
orthogonal and symplectic ensembles of random matrices.Comment: 6 page
Matrices coupled in a chain. I. Eigenvalue correlations
The general correlation function for the eigenvalues of complex hermitian
matrices coupled in a chain is given as a single determinant. For this we use a
slight generalization of a theorem of Dyson.Comment: ftex eynmeh.tex, 2 files, 8 pages Submitted to: J. Phys.
Zeros of some bi-orthogonal polynomials
Ercolani and McLaughlin have recently shown that the zeros of the
bi-orthogonal polynomials with the weight
, relevant to a model of two coupled
hermitian matrices, are real and simple. We show that their argument applies to
the more general case of the weight , a convolution of
several weights of the same form. This general case is relevant to a model of
several hermitian matrices coupled in a chain. Their argument also works for
the weight , , and for a convolution of
several such weights.Comment: tex mehta.tex, 1 file, 9 pages [SPhT-T01/086], submitted to J. Phys.
On the Decadal Modes of Oscillation of an Idealized Ocean-atmosphere System
Axially-symmetric, linear, free modes of global, primitive equation, ocean-atmosphere models are examined to see if they contain decadal (10 to 30 years) oscillation time scale modes. A two-layer ocean model and a two-level atmospheric model are linearized around axially-symmetric basic states containing mean meridional circulations in the ocean and the atmosphere. Uncoupled and coupled, axially-symmetric modes of oscillation of the ocean-atmosphere system are calculated. The main conclusion is that linearized, uncoupled and coupled, ocean-atmosphere systems can contain axially-symmetric, free modes of variability on decadal time scales. These results have important implications for externally-forced decadal climate variability
Moments of the characteristic polynomial in the three ensembles of random matrices
Moments of the characteristic polynomial of a random matrix taken from any of
the three ensembles, orthogonal, unitary or symplectic, are given either as a
determinant or a pfaffian or as a sum of determinants. For gaussian ensembles
comparing the two expressions of the same moment one gets two remarkable
identities, one between an determinant and an
determinant and another between the pfaffian of a anti-symmetric
matrix and a sum of determinants.Comment: tex, 1 file, 15 pages [SPhT-T01/016], published J. Phys. A: Math.
Gen. 34 (2001) 1-1
A column of grains in the jamming limit: glassy dynamics in the compaction process
We investigate a stochastic model describing a column of grains in the
jamming limit, in the presence of a low vibrational intensity. The key control
parameter of the model, , is a representation of granular shape,
related to the reduced void space. Regularity and irregularity in grain shapes,
respectively corresponding to rational and irrational values of , are
shown to be centrally important in determining the statics and dynamics of the
compaction process.Comment: 29 pages, 14 figures, 1 table. Various minor changes and updates. To
appear in EPJ
Calculation of some determinants using the s-shifted factorial
Several determinants with gamma functions as elements are evaluated. This
kind of determinants are encountered in the computation of the probability
density of the determinant of random matrices. The s-shifted factorial is
defined as a generalization for non-negative integers of the power function,
the rising factorial (or Pochammer's symbol) and the falling factorial. It is a
special case of polynomial sequence of the binomial type studied in
combinatorics theory. In terms of the gamma function, an extension is defined
for negative integers and even complex values. Properties, mainly composition
laws and binomial formulae, are given. They are used to evaluate families of
generalized Vandermonde determinants with s-shifted factorials as elements,
instead of power functions.Comment: 25 pages; added section 5 for some examples of application
Universality in survivor distributions: Characterising the winners of competitive dynamics
We investigate the survivor distributions of a spatially extended model of
competitive dynamics in different geometries. The model consists of a
deterministic dynamical system of individual agents at specified nodes, which
might or might not survive the predatory dynamics: all stochasticity is brought
in by the initial state. Every such initial state leads to a unique and
extended pattern of survivors and non-survivors, which is known as an attractor
of the dynamics. We show that the number of such attractors grows exponentially
with system size, so that their exact characterisation is limited to only very
small systems. Given this, we construct an analytical approach based on
inhomogeneous mean-field theory to calculate survival probabilities for
arbitrary networks. This powerful (albeit approximate) approach shows how
universality arises in survivor distributions via a key concept -- the {\it
dynamical fugacity}. Remarkably, in the large-mass limit, the survival
probability of a node becomes independent of network geometry, and assumes a
simple form which depends only on its mass and degree.Comment: 12 pages, 6 figures, 2 table
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