arXiv subjects

Philippe Mounaix

Publications and source records attributed to Philippe Mounaix.

At least 19 recordsLinked to original sources

arXiv2026

Finite-Time Transition to Intermittency for a Stochastic Heat Equation Driven by the Square of a Gaussian Field

In this paper, we study the spatial behavior of the solution $\psi(x,t)$ to the stochastic heat equation $\partial_t\psi(x,t)-\frac{1}{2}\partial^2_{x^2} \psi(x,t)=g\, S(x,t)^2\, \psi(x,t)$, with $0\le t\le T$, $x\in\mathbb{R}$, and $\psi(x,0)=1$. Here, $g>0$ is a coupling constant and $S(x,t)$ is a stationary, homogeneous, and ergodic Gaussian field. Focusing on $\mathcal{E}(x,g)\equiv \psi(x,T)$ at a finite time $T>0$, we identify the critical coupling $g_c(T)$ above which the average of $\mathcal{E}(0,g)$ diverges. We show that in the subcritical regime $g g_c(T)$ it becomes spatially intermittent and loses ergodicity. Our results differ from the extensively studied case where $S(x,t)^2$ is replaced by $S(x,t)$, in which intermittency appears only asymptotically as $T\to +\infty$, with no finite-time intermittency.

math-ph↗

arXiv2025

From the Rose-DuBois Ansatz of Hot Spot Fields to the Instanton Solution: a Pedestrian Presentation

This paper gives a pedestrian presentation of some technical results recently published in mathematical physics with non-trivial implications for laser-plasma interaction. The aim is to get across the main results without going into the details of the calculations, nor offering a specialist's user guide, but by focusing conceptually on how these results modify the commonly-held description -- in terms of laser hot spot fields -- of backscattering instabilities with a spatially smoothed laser beam. The intended readers are plasma physicists as well as graduate students interested in laser-plasma interaction. No prior knowledge of scattering instabilities is required. Step by step, we explain how the laser hot spots are gradually replaced with other structures, called instantons, as the amplification of the scattered light increases. In the amplification range of interest for laser-plasma interaction, instanton--hot spot complexes tend to appear in the laser field (in addition to the expected hot spots), with a non-negligible probability. For even larger amplifications and systems longer than a hot spot length, the hot spot field description is clearly invalidated by the instanton takeover.

physics.plasm-ph↗

arXiv2024

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 $\partial_zψ-\frac{i}{2m}\partial^2_{x^2} ψ=g\vert S\vert^2\, ψ$ for $\ln U\gg 1$, where $U=\vertψ(0,L)\vert^2$. 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. We consider a class of $S$, called the `one-max class', for which we devise a specific biased sampling procedure. As an application, $p(U)$, 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 for $S$ in the one-max class.

cond-mat.stat-mech↗

arXiv2023

Schrödinger 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 $U=\vertψ(0,L)\vert^2$, for $\ln U\gg 1$, $ψ(x,z)$ being the solution to $\partial_zψ-\frac{i}{2m}\nabla_{\perp}^2 ψ=g\vert S\vert^2\, ψ$, 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.

cond-mat.stat-mech↗

arXiv2021

Record statistics for random walks and Lévy flights with resetting

We compute exactly the mean number of records $\langle R_N \rangle$ for a time-series of size $N$ whose entries represent the positions of a discrete time random walker on the line. At each time step, the walker jumps by a length $η$ drawn independently from a symmetric and continuous distribution $f(η)$ with probability $1-r$ (with $0\leq r < 1$) and with the complementary probability $r$ it resets to its starting point $x=0$. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for $r=0$) and an uncorrelated time-series (for $(1-r) \ll 1$). Remarkably, we found that for every fixed $r \in [0,1[$ and any $N$, the mean number of records $\langle R_N \rangle$ is completely universal, i.e., independent of the jump distribution $f(η)$. In particular, for large $N$, we show that $\langle R_N \rangle$ grows very slowly with increasing $N$ as $\langle R_N \rangle \approx (1/\sqrt{r})\, \ln N$ for $0<r <1$. We also computed the exact universal crossover scaling functions for $\langle R_N \rangle$ in the two limits $r \to 0$ and $r \to 1$. Our analytical predictions are in excellent agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2021

Universal record statistics for random walks and Lévy flights with a nonzero staying probability

We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability $0\leq p \leq 1$, while with the complementary probability $1-p$, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution $f_0(η)$. We have shown that, for arbitrary $p$, the statistics of records up to step $N$ is completely universal, i.e., independent of $f_0(η)$ for any $N$. We also compute the connected two-time correlation function $C_p(m_1, m_2)$ of the record-breaking events at times $m_1$ and $m_2$ and show it is also universal for all $p$. Moreover, we demonstrate that $C_p(m_1, m_2)< C_0(m_1, m_2)$ for all $p>0$, indicating that a nonzero $p$ induces additional anti-correlations between record events. We further show that these anti-correlations lead to a drastic reduction in the fluctuations of the record numbers with increasing $p$. This is manifest in the Fano factor, i.e. the ratio of the variance and the mean of the record number, which we compute explicitly. We also show that an interesting scaling limit emerges when $p \to 1$, $N \to \infty$ with the product $t = (1-p)\, N$ fixed. We compute exactly the associated universal scaling functions for the mean, variance and the Fano factor of the number of records in this scaling limit. .

cond-mat.stat-mech↗

arXiv2020

Statistics of the Number of Records for Random Walks and Lévy Flights on a ${1D}$ Lattice

We study the statistics of the number of records $R_n$ for a symmetric, $n$-step, discrete jump process on a $1D$ lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution $P(R_n)$ of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival probability of a random walk on a line, with discrete or continuous jump distributions. For the discrete jump process, we then derive the asymptotic large $n$ behavior of $P(R_n)$ as well as of the average number of records $E(R_n)$. We show that unlike the case of random walks with symmetric and continuous jump distributions where the record statistics is strongly universal (i.e., independent of the jump distribution for all $n$), the record statistics for lattice walks depends on the jump distribution for any fixed $n$. However, in the large $n$ limit, we show that the distribution of the scaled record number $R_n/E(R_n)$ approaches a universal, half-Gaussian form for any discrete jump process. The dependence on the jump distribution enters only through the scale factor $E(R_n)$, which we also compute in the large $n$ limit for arbitrary jump distributions. We present explicit results for a few examples and provide numerical checks of our analytical predictions.

cond-mat.stat-mech↗

arXiv2019

Smoluchowski flux and Lamb-Lion Problems for Random Walks and Lévy Flights with a Constant Drift

We consider non-interacting particles (or lions) performing one-dimensional random walks or Lévy flights (with Lévy index $1 < μ\leq 2$) in the presence of a constant drift $c$. Initially these random walkers are uniformly distributed over the positive real line $z\geq 0$ with a density $ρ_0$. At the origin $z=0$ there is an immobile absorbing trap (or a lamb), such that when a particle crosses the origin, it gets absorbed there. Our main focus is on (i) the flux of particles $Φ_c(n)$ out of the system (the "Smoluchowski problem") and (ii) the survival probability $S_c(n)$ of the trap or lamb (the "lamb-lion problem") until step $n$. We show that both observables can be expressed in terms of the average maximum $\mathbb{E}[M_c(n)]$ of a single random walk or Lévy flight after $n$ steps. This allows us to obtain the precise asymptotic behavior of both $Φ_c(n)$ and $S_c(n)$ analytically for large $n$ in the two problems, for any value of $1<μ\leq 2$ and $c \in {\mathbb{R}}$. In particular, for $c>0$, we show the rather counterintuitive result that for $1< μ< 2$, $S_{c>0}(n \to \infty)$ vanishes as $S_{c>0}(n \to \infty) \approx \exp\left(-λ\, n^{2-μ}\right)$, where $λ$ is a $μ$-dependent positive constant, while for standard random walks (i.e., with $μ= 2$), $S_{c>0}(n \to \infty) \to K_{RW} > 0$, as expected. Our analytical results are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2018

Asymptotics for the Expected Maximum of Random Walks and Lévy Flights with a Constant Drift

In this paper, we study the large $n$ asymptotics of the expected maximum of an $n$-step random walk/Lévy flight (characterized by a Lévy index $1<μ\leq 2$) on a line, in the presence of a constant drift $c$. For $0<μ\leq 1$, the expected maximum is infinite, even for finite values of $n$. For $1<μ\leq 2$, we obtain all the non-vanishing terms in the asymptotic expansion of the expected maximum for large $n$. For $c<0$ and $μ=2$, the expected maximum approaches a non-trivial constant as $n$ gets large, while for $1<μ< 2$, it grows as a power law $\sim n^{2-μ}$. For $c>0$, the asymptotic expansion of the expected maximum is simply related to the one for $c<0$ by adding to the latter the linear drift term $cn$, making the leading term grow linearly for large $n$, as expected. Finally, we derive a scaling form interpolating smoothly between the cases $c=0$ and $c\ne 0$. These results are borne out by numerical simulations in excellent agreement with our analytical predictions.

cond-mat.stat-mech↗

arXiv2017

Almost Sure Uniform Convergence of a Random Gaussian Field Conditioned on a Large Linear Form to a Non Random Profile

We investigate the realizations of a random Gaussian field on a finite domain of ${\mathbb R}^d$ in the limit where a given linear functional of the field is large. We prove that if its variance is bounded, the field converges uniformly and almost surely to a non random profile depending only on the covariance and the considered linear functional of the field. This is a significant improvement of the weaker $L^2$-convergence in probability previously obtained in the case of conditioning on a large quadratic functional.

math.PR↗

arXiv2017

Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line

We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, $f(η)$, characterized by a Lévy index $μ\in (0,2]$, which includes standard random walks ($μ=2$) and Lévy flights ($0<μ<2$). 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 $x_0 = O(n^{1/μ})$ ("classical regime") the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for $μ=2$ 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 $x_0$.

cond-mat.stat-mech↗

arXiv2016

First Gap Statistics of Long Random Walks with Bounded Jumps

We study one-dimensional discrete as well as continuous time random walks, either with a fixed number of steps (for discrete time) $n$ or on a fixed time interval $T$ (for continuous time). In both cases, we focus on symmetric probability distribution functions (PDF) of jumps with a finite support $[-g_{max}, g_{max}]$. For continuous time random walks (CTRWs), the waiting time $τ$ between two consecutive jumps is a random variable whose probability distribution (PDF) has a power law tail $Ψ(τ) \propto τ^{-1-γ}$, with $0<γ<1$. We obtain exact results for the joint statistics of the gap between the first two maximal positions of the random walk and the time elapsed between them. We show that for large $n$ (or large time $T$ for CTRW), this joint PDF reaches a stationary joint distribution which exhibits an interesting concentration effect in the sense that a gap close to its maximum possible value, $g\approx g_{max}$, is much more likely to be achieved by two successive jumps rather than by a long walk between the first two maxima. Our numerical simulations confirm this concentration effect.

cond-mat.stat-mech↗

arXiv2015

On the Gap and Time Interval between the First Two Maxima of Long Continuous Time Random Walks

We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval $T$ where at each time step the walker waits a random time $τ$, before performing a jump drawn from a symmetric continuous probability distribution function (PDF) $f(η)$, of Lévy index $0 < μ\leq 2$. Our study includes the case where the waiting time PDF $Ψ(τ)$ has a power law tail, $Ψ(τ) \propto τ^{-1 - γ}$, with $0< γ< 1$, such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if $γ< μ/2$ (and super-diffusive if $γ> μ/2$). We investigate the joint PDF of the gap $g$ between the first two highest positions of the CTRW and the time $t$ separating these two maxima. We show that this PDF reaches a stationary limiting joint distribution $p(g,t)$ in the limit of long CTRW, $T \to \infty$. Our exact analytical results show a very rich behavior of this joint PDF in the $(γ, μ)$ plane, which we study in great detail. Our main results are verified by numerical simulations. This work provides a non trivial extension to CTRWs of the recent study in the discrete time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).

cond-mat.stat-mech↗

arXiv2014

On the Gap and Time Interval between the First Two Maxima of Long Random Walks

In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker, starting from $x_0 = 0$, after $n$ time steps (taking the $x$-axis vertical). The jumps $η_i = x_i - x_{i-1}$ are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), $f(η)$, the Fourier transform of which has the small $k$ behavior $1 - \hat f(k) \propto |k|^μ$, with $0 < μ\leq 2$. For $μ=2$, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For $0<μ<2$, the RW is a Lévy flight of index $μ$. We show that the joint PDF of $G_n$ and $L_n$ converges to a well defined stationary bi-variate distribution $p(g,l)$ as the RW duration $n$ goes to infinity. We present a thorough analytical study of the limiting joint distribution $p(g,l)$, as well as of its associated marginals $p_{\rm gap}(g)$ and $p_{\rm time}(l)$, revealing a rich variety of behaviors depending on the tail of $f(η)$ (from slow decreasing algebraic tail to fast decreasing super-exponential tail). We also address the problem for a random bridge where the RW starts and ends at the origin after $n$ time steps. We show that in the large $n$ limit, the PDF of $G_n$ and $L_n$ converges to the {\it same} stationary distribution $p(g,l)$ as in the case of the free-end RW. Finally, we present a numerical check of our analytical predictions. Some of these results were announced in a recent letter [S. N. Majumdar, Ph. Mounaix, G. Schehr, Phys. Rev. Lett. {\bf 111}, 070601 (2013)].

cond-mat.stat-mech↗

arXiv2014

Quasi-deterministic properties of random Gaussian fields constrained by a large quadratic form

Completing the study initiated by Mounaix and Collet [J. Stat. Phys. {\bf 143}, 139-147 (2011)], we investigate the realizations of a Gaussian random field in the limit where a given (general) quadratic form of the field is large. Concentration in $L^2$ and in probability is proved under mild conditions and the resulting quasi-deterministic behavior of the field is given. Applications to a large {\it local} quadratic form are considered in two specific cases. In particular, the quasi-deterministic structure of a Gaussian random flow with a large local helicity at some given point is determined explicitly.

math-ph↗

arXiv2013

Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks

We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps η_i's of the RW, f(η), is symmetric and its Fourier transform has the small k behavior 1-\hat{f}(k)\sim| k|^μwith 0 < μ\leq 2. We compute the joint probability density function (pdf) P_n(g,l) of G_n and L_n and show that, when n \to \infty, it approaches a limiting pdf p(g,l). The corresponding marginal pdf of the gap, p_{\rm gap}(g), is found to behave like p_{\rm gap}(g) \sim g^{-1 - μ} for g \gg 1 and 0<μ< 2. We show that the limiting marginal distribution of L_n, p_{\rm time}(l), has an algebraic tail p_{\rm time}(l) \sim l^{-γ(μ)} for l \gg 1 with γ(1<μ\leq 2) = 1 + 1/μ, and γ(0<μ<1) = 2. For l, g \gg 1 with fixed l g^{-μ}, p(g,l) takes the scaling form p(g,l) \sim g^{-1-2μ} \tilde p_μ(l g^{-μ}) where \tilde p_μ(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.

cond-mat.stat-mech↗

arXiv2012

Bose-Einstein Condensation of a Gaussian Random Field in the Thermodynamic Limit

We derive the criterion for the Bose-Einstein condensation (BEC) of a Gaussian field $ϕ$ (real or complex) in the thermodynamic limit. The field is characterized by its covariance function and the control parameter is the intensity $u=\|ϕ\|_2^2/V$, where $V$ is the volume of the box containing the field. We show that for any dimension $d$ (including $d=1$), there is a class of covariance functions for which $ϕ$ exhibits a BEC as $u$ is increased through a critical value $u_c$. In this case, we investigate the probability distribution of the part of $u$ contained in the condensate. We show that depending on the parameters characterizing the covariance function and the dimension $d$, there can be two distinct types of condensate: a Gaussian distributed "normal" condensate with fluctuations scaling as $1/\sqrt{V}$, and a non Gaussian distributed "anomalous" condensate. A detailed analysis of the anomalous condensate is performed for a one-dimensional system ($d=1$). Extending this one-dimensional analysis to exactly the point of transition between normal and anomalous condensations, we find that the condensate at the transition point is still Gaussian distributed but with anomalously large fluctuations scaling as $\sqrt{\ln(L)/L}$, where $L$ is the system length. The conditional spectral density of $ϕ$, knowing $u$, is given for all the regimes (with and without BEC).

cond-mat.stat-mech↗

arXiv2011

Linear Amplifier Breakdown and Concentration Properties of a Gaussian Field Given that its $\bm{L^2}$-Norm is Large

In the context of linear amplification for systems driven by the square of a Gaussian noise, we investigate the realizations of a Gaussian field in the limit where its $L^2$-norm is large. Concentration onto the eigenspace associated with the largest eigenvalue of the covariance of the field is proved. When the covariance is trace class, the concentration is in probability for the $L^2$-norm. A stronger concentration, in mean for the sup-norm, is proved for a smaller class of Gaussian fields, and an example of a field belonging to that class is given. A possible connection with Bose-Einstein condensation is briefly discussed.

math-ph↗