81 research outputs found
Excursions of diffusion processes and continued fractions
It is well-known that the excursions of a one-dimensional diffusion process
can be studied by considering a certain Riccati equation associated with the
process. We show that, in many cases of interest, the Riccati equation can be
solved in terms of an infinite continued fraction. We examine the probabilistic
significance of the expansion. To illustrate our results, we discuss some
examples of diffusions in deterministic and in random environments.Comment: 28 pages. Minor changes to Section
Exact Results on Sinai's Diffusion
We study the continuum version of Sinai's problem of a random walker in a
random force field in one dimension. A method of stochastic representations is
used to represent various probability distributions in this problem (mean
probability density function and first passage time distributions). This method
reproduces already known rigorous results and also confirms directly some
recent results derived using approximation schemes. We demonstrate clearly, in
the Sinai scaling regime, that the disorder dominates the problem and that the
thermal distributions tend to zero-one laws.Comment: 14 pages Latex. To appear J. Phys.
Products of random matrices and generalised quantum point scatterers
To every product of matrices, there corresponds a one-dimensional
Schr\"{o}dinger equation whose potential consists of generalised point
scatterers. Products of {\em random} matrices are obtained by making these
interactions and their positions random. We exhibit a simple one-dimensional
quantum model corresponding to the most general product of matrices in
. We use this correspondence to find new examples of
products of random matrices for which the invariant measure can be expressed in
simple analytical terms.Comment: 38 pages, 13 pdf figures. V2 : conclusion added ; Definition of
function change
On the flux distribution in a one dimensional disordered system
We study some transport properties of a one dimensional disordered system of finite length N. In this system particles are subject to random forces resulting both from a thermal noise and from a quenched random force F(x) which models the inhomogeneous medium. The latter is distributed as a white noise with a non zero average bias. Imposing some fixed concentration of particles at the end points of the chain yields a steady current J(N) which depends on the environnent {F(x)}. The problem of computing the probabilility distribution P(J) over the environments is addressed. Our approchh is based on a path integral method and on a moment calculation. In the case of a non zero bias our results generalize those obtained recently by Oshanin et al
Integer partitions and exclusion statistics: Limit shapes and the largest part of Young diagrams
We compute the limit shapes of the Young diagrams of the minimal difference
partitions and provide a simple physical interpretation for the limit
shapes. We also calculate the asymptotic distribution of the largest part of
the Young diagram and show that the scaled distribution has a Gumbel form for
all . This Gumbel statistics for the largest part remains unchanged even for
general partitions of the form with where
is the number of times the part appears.Comment: 12 pages, 4 figures (minor corrections, a note and some references
added
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