On the distribution of the Wigner time delay in one-dimensional disordered systems

We consider the scattering by a one-dimensional random potential and derive the probability distribution of the corresponding Wigner time delay. It is shown that the limiting distribution is the same for two different models and coincides with the one predicted by random matrix theory. It is also shown that the corresponding stochastic process is given by an exponential functional of the potential.Comment: 11 pages, four references adde

Integer Partitions and Exclusion Statistics

We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.Comment: 16 pages, 4 .eps figures include

Distribution of the Oscillation Period in the Underdamped One Dimensional Sinai Model

We consider the Newtonian dynamics of a massive particle in a one dimemsional random potential which is a Brownian motion in space. This is the zero temperature nondamped Sinai model. As there is no dissipation the particle oscillates between two turning points where its kinetic energy becomes zero. The period of oscillation is a random variable fluctuating from sample to sample of the random potential. We compute the probability distribution of this period exactly and show that it has a power law tail for large period, P(T)\sim T^{-5/3} and an essential singluarity P(T)\sim \exp(-1/T) as T\to 0. Our exact results are confirmed by numerical simulations and also via a simple scaling argument.Comment: 9 pages LateX, 2 .eps figure

General flux to a trap in one and three dimensions

The problem of the flux to a spherical trap in one and three dimensions, for diffusing particles undergoing discrete-time jumps with a given radial probability distribution, is solved in general, verifying the Smoluchowski-like solution in which the effective trap radius is reduced by an amount proportional to the jump length. This reduction in the effective trap radius corresponds to the Milne extrapolation length.Comment: Accepted for publication, in pres

Universality of the Wigner time delay distribution for one-dimensional random potentials

We show that the distribution of the time delay for one-dimensional random potentials is universal in the high energy or weak disorder limit. Our analytical results are in excellent agreement with extensive numerical simulations carried out on samples whose sizes are large compared to the localisation length (localised regime). The case of small samples is also discussed (ballistic regime). We provide a physical argument which explains in a quantitative way the origin of the exponential divergence of the moments. The occurence of a log-normal tail for finite size systems is analysed. Finally, we present exact results in the low energy limit which clearly show a departure from the universal behaviour.Comment: 4 pages, 3 PostScript figure

Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models

We study the probability density function $P(h_m,L)$ of the maximum relative height $h_m$ in a wide class of one-dimensional solid-on-solid models of finite size $L$. For all these lattice models, in the large $L$ limit, a central limit argument shows that, for periodic boundary conditions, $P(h_m,L)$ takes a universal scaling form $P(h_m,L) \sim (\sqrt{12}w_L)^{-1}f(h_m/(\sqrt{12} w_L))$, with $w_L$ the width of the fluctuating interface and $f(x)$ the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using transfer matrix technique, valid for any $L>0$. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a non universal amplitude, and simply given by the derivative of the Airy distribution function $f'(x)$.Comment: 13 pages, 4 figure

Explicit formulae in probability and in statistical physics

We consider two aspects of Marc Yor's work that have had an impact in statistical physics: firstly, his results on the windings of planar Brownian motion and their implications for the study of polymers; secondly, his theory of exponential functionals of Levy processes and its connections with disordered systems. Particular emphasis is placed on techniques leading to explicit calculations.Comment: 14 pages, 2 figures. To appear in Seminaire de Probabilites, Special Issue Marc Yo

Sample-size dependence of the ground-state energy in a one-dimensional localization problem

We study the sample-size dependence of the ground-state energy in a one-dimensional localization problem, based on a supersymmetric quantum mechanical Hamiltonian with random Gaussian potential. We determine, in the form of bounds, the precise form of this dependence and show that the disorder-average ground-state energy decreases with an increase of the size $R$ of the sample as a stretched-exponential function, $\exp( - R^{z})$, where the characteristic exponent $z$ depends merely on the nature of correlations in the random potential. In the particular case where the potential is distributed as a Gaussian white noise we prove that $z = 1/3$. We also predict the value of $z$ in the general case of Gaussian random potentials with correlations.Comment: 30 pages and 4 figures (not included). The figures are available upon reques

Area Distribution of Elastic Brownian Motion

We calculate the excursion and meander area distributions of the elastic Brownian motion by using the self adjoint extension of the Hamiltonian of the free quantum particle on the half line. We also give some comments on the area of the Brownian motion bridge on the real line with the origin removed. We will stress on the power of self adjoint extension to investigate different possible boundary conditions for the stochastic processes.Comment: 18 pages, published versio

A PBW basis for Lusztig's form of untwisted affine quantum groups

Let $\mathfrak{g}$ be an untwisted affine Kac-Moody algebra over the field $K \,$, and let $U_q(\mathfrak{g})$ be the associated quantum enveloping algebra; let $\mathfrak{U}_q(g)$ be the Lusztig's integer form of $U_q(\mathfrak{g}) \,$, generated by $q$-divided powers of Chevalley generators over a suitable subring $R$ of $K(q) \,$. We prove a Poincar\'e-Birkhoff-Witt like theorem for $\mathfrak{U}_q(\mathfrak{g}) \,$, yielding a basis over $R$ made of ordered products of $q$-divided powers of suitable quantum root vectors.Comment: 22 pages, AMS-TeX C, Version 2.1c. This is the author's final version, corresponding to the printed journal versio