Alain Comtet

On the Landau Levels on the Hyperbolic Plane

Abstract The classical and quantum mechanics of a charged particle moving on the hyperbolic plane in a constant magnetic field is discussed. The underlying SL(2,R) symmetry leads to a general description of various possible trajectories. In contrast with the flat case, it is shown that closed orbits only arise for sufficiently strong fields. At the quantum level a group-theoretical approach including both bound and continuum states is presented. It is shown that the semiclassical approximation leads to the exact bound state spectrum. The resolvent and its flat space limit are constructed in closed form.

Classical diffusion of a particle in a one-dimensional random force field

Abstract We present a comprehensive study of the motion of a damped Brownian particle evolving in a static, one dimensional gaussian random force field. We provide both a clear physical picture of the process and a variety of analytical techniques. As the average bias μ is increased, a succession of different “diffusion laws” is observed: Sinais diffusion ( μ = 0, x 2 ⋍ ln 4 t ), anomalous drift ( x ⋍ t μ , μ ), anomalous dispersion ( x − Vt ⋍ ±t 1 μ , 1 ), and finally normal diffusion (μ > 2), apart from algebraic tails outside the scaling region. We show that all those results can be understood in simple terms through a large scale description of the problem as a directed walk among traps characterized by a broad distribution of release time. From this analysis, the full asymptotic probability distributions (averaged over disorder) are precisely determined, in terms of Levy stable laws. Sample to sample fluctuations are discussed. The probability of presence at the initial point is more specifically adressed. It amounts to computing the density of states of a Schrodinger equation with a special type of random potential. We obtain exactly the average over disorder of this quantity, using the following two different approaches: the “Dyson-Schmidt” technique, and the replica method. Both reveal interesting technical features, and the latter can be used to obtain information on the full probability distribution (and Green function). Some physical applications of this model are discussed.

Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces

The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution function also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(hm,L) of the maximal height hm (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(hm,L)=L−1/2f(hmL−1/2) for all L>0 where the function f(x) is the Airy distribution function. This result is valid for both the Edwards–Wilkinson (EW) and the Kardar–Parisi–Zhang interfaces. For the free boundary case, the same scaling holds P(hm,L)=L−1/2F(hmL−1/2), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the EW interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [S.N. Majumdar and A. Comtet, Phys. Rev. Lett. 92: 225501 (2004)].

Exact maximal height distribution of fluctuating interfaces.

We present an exact solution for the distribution P(h(m),L) of the maximal height h(m) (measured with respect to the average spatial height) in the steady state of a fluctuating Edwards-Wilkinson interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h(m),L)=L(-1/2)f(h(m)L(-1/2)) for all L>0, where the function f(x) is the Airy distribution function that describes the probability density of the area under a Brownian excursion over a unit interval. For the free boundary case, the same scaling holds, but the scaling function is different from that of the periodic case. Numerical simulations are in excellent agreement with our analytical results. Our results provide an exactly solvable case for the distribution of extremum of a set of strongly correlated random variables.

Exact distribution of the maximal height of p vicious walkers

Using path-integral techniques, we compute exactly the distribution of the maximal height Hp of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions p watermelons with a wall, and bridges p watermelons without a wall, for all integer p>or=1. For large p, we show that approximately square root 2p (excursions) whereas approximately square root p (bridges). Our exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of N random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy’s formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of n independent random walks. In the continuum time limit this reduces to n independent planar Brownian trajectories for which we compute exactly, for all n, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. 103:140602, 2009].

Spectral Determinant on Quantum Graphs

We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of the Laplacian in terms of either a V_V vertex matrix or a 2B_2B link matrix that couples the arcs (oriented bonds) together. This latter expression allows us to rewrite the spectral determinant as an infinite product of contributions of periodic orbits on the graph. We also present a diagrammatic method that permits us to write the spectral determinant in terms of a finite number of periodic orbit contributions. These results are generalized to the case of graphs in a magnetic field. Several examples illustrating this formalism are presented and its application to the thermodynamic and transport properties of weakly disordered and coherent mesoscopic networks is discussed. 2000 Academic Press

Statistical aspects of the anyon model

The second virial coefficient for a gas of anyons is computed (i) by discretising the two-particle spectrum through the introduction of a harmonic potential regulator and (ii) by considering the problem in the continuum directly through heat kernel methods. In both cases the result of Arovas et al. (1985) is recovered.

Diffusion in a one-dimensional random medium and hyperbolic Brownian motion

Classical diffusion in a random medium involves an exponential functional of Brownian motion. This functional also appears in the study of Brownian diffusion on a Riemann surface of constant negative curvature. We analyse in detail this relationship and study various distributions using stochastic calculus and functional integration.

Functionals of Brownian motion, localization and metric graphs

We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed: some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schrodinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of planar Brownian motion.