338 research outputs found
Zero distribution of polynomials satisfying a differential-difference equation
In this paper we investigate the asymptotic distribution of the zeros of
polynomials satisfying a first order differential-difference
equation. We give several examples of orthogonal and non-orthogonal families.Comment: 26 pages, 2 figure
A product formula and combinatorial field theory
We treat the problem of normally ordering expressions involving the standard boson operators a, ay where [a; ay] = 1. We show that a simple product formula for formal power series | essentially an extension of the Taylor expansion | leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions | in essence, a combinatorial eld theory. We apply these techniques to some examples related to specic physical Hamiltonians
An algebraic approach to Polya processes
P\'olya processes are natural generalization of P\'olya-Eggenberger urn
models. This article presents a new approach of their asymptotic behaviour {\it
via} moments, based on the spectral decomposition of a suitable finite
difference operator on polynomial functions. Especially, it provides new
results for {\it large} processes (a P\'olya process is called {\it small} when
1 is simple eigenvalue of its replacement matrix and when any other eigenvalue
has a real part ; otherwise, it is called large)
Random incidence matrices: moments of the spectral density
We study numerically and analytically the spectrum of incidence matrices of
random labeled graphs on N vertices : any pair of vertices is connected by an
edge with probability p. We give two algorithms to compute the moments of the
eigenvalue distribution as explicit polynomials in N and p. For large N and
fixed p the spectrum contains a large eigenvalue at Np and a semi-circle of
"small" eigenvalues. For large N and fixed average connectivity pN (dilute or
sparse random matrices limit), we show that the spectrum always contains a
discrete component. An anomaly in the spectrum near eigenvalue 0 for
connectivity close to e=2.72... is observed. We develop recursion relations to
compute the moments as explicit polynomials in pN. Their growth is slow enough
so that they determine the spectrum. The extension of our methods to the
Laplacian matrix is given in Appendix.
Keywords: random graphs, random matrices, sparse matrices, incidence matrices
spectrum, momentsComment: 39 pages, 9 figures, Latex2e, [v2: ref. added, Sect. 4 modified
Gaussian fluctuations of Young diagrams and structure constants of Jack characters
In this paper, we consider a deformation of Plancherel measure linked to Jack
polynomials. Our main result is the description of the first and second-order
asymptotics of the bulk of a random Young diagram under this distribution,
which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the
first order asymptotics) and Kerov (for the second order asymptotics). This
gives more evidence of the connection with Gaussian -ensemble, already
suggested by some work of Matsumoto.
Our main tool is a polynomiality result for the structure constant of some
quantities that we call Jack characters, recently introduced by Lassalle. We
believe that this result is also interested in itself and we give several other
applications of it.Comment: 71 pages. Minor modifications from version 1. An extended abstract of
this work, with significantly fewer results and a different title, is
available as arXiv:1201.180
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