Cosmological Equations for a Thick Brane

Generalized Friedmann equations governing the cosmological evolution inside a thick brane embedded in a five-dimensional Anti-de Sitter spacetime are derived. These equations are written in terms of four-dimensional effective brane quantities obtained by integrating, along the fifth dimension, over the brane thickness. In the case of a Randall-Sundrum type cosmology, different limits of these effective quantities are considered yielding cosmological equations which interpolate between the thin brane limit (governed by unconventional brane cosmology), and the opposite limit of an ``infinite'' brane thickness corresponding to the familiar Kaluza-Klein approach. In the more restrictive case of a Minkowski bulk, it is shown that no effective four-dimensional reduction is possible in the regimes where the brane thickness is not small enough.Comment: 23 pages, Latex, 2 figure

Note on a diffraction-amplification problem

We investigate the solution of the equation \partial_t E(x,t)-iD\partial_x^2 E(x,t)= \lambda |S(x,t)|^2 E(x,t)$, for x in a circle and S(x,t) a Gaussian stochastic field with a covariance of a particular form. It is shown that the coupling \lambda_c at which diverges for t>=1 (in suitable units), is always less or equal for D>0 than D=0.Comment: REVTeX file, 8 pages, submitted to Journal of Physics

Testing the Instanton Approach to the Large Amplification Limit of a Diffraction-Amplification Problem

The validity of the instanton analysis approach is tested numerically in the case of the diffraction-amplification problem $\partial_z\psi -\frac{i}{2m}\nabla_{\perp}^2 \psi =g\vert S\vert^2\, \psi$ for $\ln U\gg 1$, where $U=\vert\psi(0,L)\vert^2$. Here, $S(x,z)$ is a complex Gaussian random field, $z$ and $x$ respectively are the axial and transverse coordinates, with $0\le z\le L$, and both $m\ne 0$ and $g>0$ are real parameters. To sample the rare and extreme amplification values of interest ($\ln U\gg 1$), we devise a specific biased sampling procedure by which $p(U)$, the probability distribution of $U$, is obtained down to values less than $10^{-2270}$ in the far right tail. We find that the agreement of our numerical results with the instanton analysis predictions in Mounaix (2023 {\it J. Phys. A: Math. Theor.} {\bf 56} 305001) is remarkable. Both the predicted algebraic tail of $p(U)$ and concentration of the realizations of $S$ onto the leading instanton are clearly confirmed, which validates the instanton analysis numerically in the large $\ln U$ limit.Comment: 19 pages, 9 figures, submitted to J. Phys. A: Math. Theo

Schr\"{o}dinger Equation Driven by the Square of a Gaussian Field: Instanton Analysis in the Large Amplification Limit

We study the tail of $p(U)$, the probability distribution of $U=\vert\psi(0,L)\vert^2$, for $\ln U\gg 1$, $\psi(x,z)$ being the solution to $\partial_z\psi -\frac{i}{2m}\nabla_{\perp}^2 \psi =g\vert S\vert^2\, \psi$, where $S(x,z)$ is a complex Gaussian random field, $z$ and $x$ respectively are the axial and transverse coordinates, with $0\le z\le L$, and both $m\ne 0$ and $g>0$ are real parameters. We perform the first instanton analysis of the corresponding Martin-Siggia-Rose action, from which it is found that the realizations of $S$ concentrate onto long filamentary instantons, as $\ln U\to +\infty$. The tail of $p(U)$ is deduced from the statistics of the instantons. The value of $g$ above which $\langle U\rangle$ diverges coincides with the one obtained by the completely different approach developed in Mounaix et al. 2006 {\it Commun. Math. Phys.} {\bf 264}~741. Numerical simulations clearly show a statistical bias of $S$ towards the instanton for the largest sampled values of $\ln U$. The high maxima -- or `hot spots' -- of $\vert S(x,z)\vert^2$ for the biased realizations of $S$ tend to cluster in the instanton region.Comment: 41 pages, 10 figures, submitted to J. Phys. A: Math. Theo

Survival Probability of Random Walks and L\'evy Flights on a Semi-Infinite Line

We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, $f(\eta)$, characterized by a L\'evy index $\mu \in (0,2]$, which includes standard random walks ($\mu=2$) and L\'evy flights ($0<\mu<2$). We study the survival probability, $q(x_0,n)$, representing the probability that the RW stays non-negative up to step $n$, starting initially at $x_0 \geq 0$. Our main focus is on the $x_0$-dependence of $q(x_0,n)$ for large $n$. We show that $q(x_0,n)$ displays two distinct regimes as $x_0$ varies: (i) for $x_0= O(1)$ ("quantum regime"), the discreteness of the jump process significantly alters the standard scaling behavior of $q(x_0,n)$ and (ii) for $x_0 = O(n^{1/\mu})$ ("classical regime") the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for $\mu =2$ this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in $q(x_0,n)$ occurs between the quantum and the classical regime as one increases $x_0$.Comment: 20 pages, 3 figures, revised and accepted versio