arXiv subjects

Yves Tourigny

Publications and source records attributed to Yves Tourigny.

13 recordsLinked to original sources

arXiv2025

The continuum limit of some products of random matrices associated with renewing flows

We consider the continuum limit of some products of random matrices in $\text{SL}(d,{\mathbb R})$ that arise as discretisations of incompressible renewing flows -- that is, of flows corresponding to a divergence-free velocity field that takes independent, identically-distributed values in successive time intervals of duration proportional to $\tau$. The statistical properties of the product are encoded in its generalised Lyapunov exponent whose computation reduces to finding the leading eigenvalue of a certain transfer operator. In the continuum limit obtained by neglecting the terms of order $o(\tau^2)$, the transfer operator becomes a partial differential operator and, for a certain type of disorder which we call ``symmetric'', some calculations are feasible. For $d=2$, we compute the growth rate of the product in terms of complete elliptic integrals. By letting the elliptic modulus vary, we obtain a spectral problem, corresponding to a modulus-dependent random renewing flow, which may be viewed as a perturbation of the spectral problem for the angular Laplacian. In this way, we deduce expansions for the generalised Lyapunov exponent in ascending powers of the modulus. These expansions generalise to the case $d \ge 2$, and we compute the first few terms explicitly for $d \in \{2,3\}$.

math-ph↗

arXiv2022

The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results

We consider the one-dimensional Schrödinger equation with a random potential and study the cumulant generating function of the logarithm of the wave function $ψ(x)$, known in the literature as the "generalized Lyapunov exponent"; this is tantamount to studying the statistics of the so-called "finite size Lyapunov exponent". The problem reduces to that of finding the leading eigenvalue of a certain non-random non-self-adjoint linear operator defined on a somewhat unusual space of functions. We focus on the case of Cauchy disorder, for which we derive a secular equation for the generalized Lyapunov exponent. Analytical expressions for the first four cumulants of $\ln|ψ(x)|$ for arbitrary energy and disorder are deduced. In the universal (weak-disorder/high-energy) regime, we obtain simple asymptotic expressions for the generalized Lyapunov exponent and for all the cumulants. The large deviation function controlling the distribution of $\ln|ψ(x)|$ is also obtained in several limits. As an application, we show that, for a disordered region of size $L$, the distribution $\mathcal{W}_L$ of the conductance $g$ exhibits the power law behaviour $\mathcal{W}_L(g)\sim g^{-1/2}$ as $g\to0$.

cond-mat.dis-nn↗

arXiv2019

Representation theory and products of random matrices in $\text{SL}(2,{\mathbb R})$

The statistical behaviour of a product of independent, identically distributed random matrices in $\text{SL}(2,{\mathbb R})$ is encoded in the generalised Lyapunov exponent $Λ$; this is a function whose value at the complex number $2 \ell$ is the logarithm of the largest eigenvalue of the transfer operator obtained when one averages, over $g \in \text{SL}(2,{\mathbb R})$, a certain representation $T_\ell (g)$ associated with the product. We study some products that arise from models of one-dimensional disordered systems. These models have the property that the inverse of the transfer operator takes the form of a second-order difference or differential operator. We show how the ideas expounded by N. Ja. Vilenkin in his book [Special Functions and the Theory of Group Representations, American Mathematical Society, 1968.] can be used to study the generalised Lyapunov exponent. In particular, we derive explicit formulae for the almost-sure growth and for the variance of the corresponding products.

math-ph↗

arXiv2016

Impurity models and products of random matrices

This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2 by 2 case. The notion of impurity model--- that is, a system in which the interactions are highly localised--- links the two themes and enables their study by elementary mathematical tools. After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.

math-ph↗

arXiv2014

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.

math-ph↗

arXiv2013

One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers

We study the one-dimensional Schrödinger equation with a disordered potential of the form $V (x) = ϕ(x)^2+ϕ'(x) + κ(x) $ where $ϕ(x)$ is a Gaussian white noise with mean $μg$ and variance $g$, and $κ(x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $ρ$ and mean strength $v$. Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers~: $ϕ(x)$ models the force field acting on the diffusing particle and $κ(x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $ Ω(E) = γ(E) - \mathrm{i} πN(E) $, where $N$ is the integrated density of states per unit length and $γ$ the reciprocal of the localisation length. By using the continuous version of the Dyson-Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean $v=2g$. Building on this result, we then solve the general case -- in the low-energy limit -- in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour $$ N(E) \underset{E\to0+}{\sim} E^ν\hspace{0.5cm} \mbox{where } ν=\sqrt{μ^2+2ρ/g}\:. $$ This confirms and extends several results obtained previously by approximate methods.

cond-mat.dis-nn↗

arXiv2012

Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.

cond-mat.dis-nn↗

arXiv2011

Supersymmetric quantum mechanics with Levy disorder in one dimension

We consider the Schroedinger equation with a supersymmetric random potential, where the superpotential is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent, whose real and imaginary parts are, respectively, the Lyapunov exponent and the integrated density of states of the system. In the case where the Levy process is non-decreasing, we show that the calculation of the complex Lyapunov exponent reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Levy process. We review the known solvable cases, where the complex Lyapunov exponent can be expressed in terms of special functions, and discover a new one.

math-ph↗

arXiv2011

Continued fraction solution of Krein's inverse problem

The spectral data of a vibrating string are encoded in its so-called characteristic function. We consider the problem of recovering the distribution of mass along the string from its characteristic function. It is well-known that Stieltjes' continued fraction provides a solution of this inverse problem in the particular case where the distribution of mass is purely discrete. We show how to adapt Stieltjes' method to solve the inverse problem for a related class of strings. An application to the excursion theory of diffusion processes is presented.

math.SP↗

arXiv2010

Products of random matrices and generalised quantum point scatterers

To every product of $2\times2$ matrices, there corresponds a one-dimensional Schrödinger 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 $\text{SL}(2, {\mathbb R})$. 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.

cond-mat.dis-nn↗

arXiv2010

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.

math.PR↗

arXiv2007

Pade approximants of random Stieltjes series

We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Pade approximants of the function S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional disordered system, and the results of Marklof et al. (2005), we obtain explicit formulae (in terms of modified Bessel functions) for the almost-sure rate of convergence of these approximants, and for the almost-sure distribution of their poles.

math-ph↗

arXiv2006

Explicit invariant measures for products of random matrices

We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL(2,C). The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac and Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schroedinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

math-ph↗