arXiv subjects

Gregory Schehr

Publications and source records attributed to Gregory Schehr.

At least 181 records · Page 10Linked to original sources

arXiv2010

Area distribution and the average shape of a Lévy bridge

We consider a one dimensional Lévy bridge x_B of length n and index 0 < α< 2, i.e. a Lévy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \sum_{m=1}^n x_B(m) under such a Lévy bridge and show that, for large n, it has the scaling form P_B(A,n) \sim n^{-1-1/α} F_α(A/n^{1+1/α}), with the asymptotic behavior F_α(Y) \sim Y^{-2(1+α)} for large Y. For α=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions. We also compute the average profile < \tilde x_B (m) > at time m of a Lévy bridge with fixed area A. For large n and large m and A, one finds the scaling form < \tilde x_B(m) > = n^{1/α} H_α({m}/{n},{A}/{n^{1+1/α}}), where at variance with Brownian bridge, H_α(X,Y) is a non trivial function of the rescaled time m/n and rescaled area Y = A/n^{1+1/α}. Our analytical results are verified by numerical simulations.

cond-mat.stat-mech↗

arXiv2009

The longest excursion of fractional Brownian motion : numerical evidence of non-Markovian effects

We study, using exact numerical simulations, the statistics of the longest excursion l_{\max}(t) up to time t for the fractional Brownian motion with Hurst exponent 0 \propto Q_\infty t where Q_\infty \equiv Q_\infty(H) depends continuously on H, and in a non trivial way. These results are compared with exact analytical results obtained recently for a renewal process with an associated persistence exponent θ= 1-H. This comparison shows that Q_\infty(H) carries the clear signature of non-Markovian effects for H\neq 1/2. The pre-asymptotic behavior of < l_{\max}(t)> is also discussed.

cond-mat.stat-mech↗

arXiv2009

Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks

We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.

cond-mat.stat-mech↗

arXiv2009

Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution

We introduce an alternative definition of the relative height h^κ(x) of a one-dimensional fluctuating interface indexed by a continuously varying real paramater 0 \leq κ\leq 1. It interpolates between the height relative to the initial value (i.e. in x=0) when κ= 0 and the height relative to the spatially averaged height for κ= 1. We compute exactly the distribution P^κ(h_m,L) of the maximum h_m of these relative heights for systems of finite size L and periodic boundary conditions. One finds that it takes the scaling form P^κ(h_m,L) = L^{-1/2} f^κ(h_m L^{-1/2}) where the scaling function f^κ(x) interpolates between the Rayleigh distribution for κ=0 and the Airy distribution for κ=1, the latter being the probability distribution of the area under a Brownian excursion over the unit interval. For arbitrary κ, one finds that it is related to, albeit different from, the distribution of the area restricted to the interval [0, κ] under a Brownian excursion over the unit interval.

cond-mat.stat-mech↗

arXiv2009

Universal non stationary dynamics at the depinning transition

We study the non-stationary dynamics of an elastic interface in a disordered medium at the depinning transition. We compute the two-time response and correlation functions, found to be universal and characterized by two independent critical exponents. We find a good agreement between two-loop Functional Renormalization Group calculations and molecular dynamics simulations for the scaling forms, and for the response aging exponent $θ_R$. We also describe a dynamical dimensional crossover, observed at long times in the relaxation of a finite system. Our results are relevant for the non-steady driven dynamics of domain walls in ferromagnetic films and contact lines in wetting.

cond-mat.dis-nn↗

arXiv2009

The longest excursion of stochastic processes in nonequilibrium systems

We consider the excursions, i.e. the intervals between consecutive zeros, of stochastic processes that arise in a variety of nonequilibrium systems and study the temporal growth of the longest one l_{\max}(t) up to time t. For smooth processes, we find a universal linear growth < l_{\max}(t) > \simeq Q_{\infty} t with a model dependent amplitude Q_\infty. In contrast, for non-smooth processes with a persistence exponent θ, we show that < l_{\max}(t) > has a linear growth if θ< θ_c while < l_{\max}(t) > \sim t^{1-ψ} if θ> θ_c. The amplitude Q_{\infty} and the exponent ψare novel quantities associated to nonequilibrium dynamics. These behaviors are obtained by exact analytical calculations for renewal and multiplicative processes and numerical simulations for other systems such as the coarsening dynamics in Ising model as well as the diffusion equation with random initial conditions.

cond-mat.stat-mech↗

arXiv2009

Condensation of the roots of real random polynomials on the real axis

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance = e^{-k^α}, indexed by a real α\geq 0. We compute exactly the mean number of real roots for large n. As αis varied, one finds three different phases. First, for 0 \leq α< 1, one finds that \sim (\frac{2}π) \log{n}. For 1 < α< 2, there is an intermediate phase where < N_n > grows algebraically with a continuously varying exponent, < N_n > \sim \frac{2}π \sqrt{\frac{α-1}α} n^{α/2}. And finally for α> 2, one finds a third phase where \sim n. This family of real random polynomials thus exhibits a condensation of their roots on the real line in the sense that, for large n, a finite fraction of their roots /n are real. This condensation occurs via a localization of the real roots around the values \pm \exp{[\fracα{2}(k+{1/2})^{α-1} ]}, 1 \ll k \leq n.

math-ph↗

arXiv2008

Exact distribution of the maximal height of p vicious walkers

Using path integral techniques, we compute exactly the distribution of the maximal height H_p 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\ge 1. For large p, we show that < H_p > \sim \sqrt{2p} (excursions) whereas < H_p > \sim \sqrt{p} (bridges). Our exact results prove that previous numerical experiments only measured the pre-asymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.

cond-mat.stat-mech↗

arXiv2008

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n^{-θ(d)} where θ(d) > 0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n \gg 1 even, the probability that they have no real root on the full real axis decays like n^{-2(θ(2)+θ(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like \exp{(-2θ_{\infty}} \sqrt{n}) and \exp{(-πθ_{\infty} \sqrt{n})} where θ_{\infty} is such that θ(d) = 2^{-3/2} θ_{\infty} \sqrt{d} in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by \exp{(-N_{ab} \tilde ϕ(k/N_{ab}))} where N_{ab} is the mean number of real roots in [a,b] and \tilde ϕ(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent {-2}. These analytical results are confirmed by detailed numerical computations.

cond-mat.stat-mech↗

arXiv2007

Finite temperature behavior of strongly disordered quantum magnets coupled to a dissipative bath

We study the effect of dissipation on the infinite randomness fixed point and the Griffiths-McCoy singularities of random transverse Ising systems in chains, ladders and in two-dimensions. A strong disorder renormalization group scheme is presented that allows the computation of the finite temperature behavior of the magnetic susceptibility and the spin specific heat. In the case of Ohmic dissipation the susceptibility displays a crossover from Griffiths-McCoy behavior (with a continuously varying dynamical exponent) to classical Curie behavior at some temperature $T^*$. The specific heat displays Griffiths-McCoy singularities over the whole temperature range. For super-Ohmic dissipation we find an infinite randomness fixed point within the same universality class as the transverse Ising system without dissipation. In this case the phase diagram and the parameter dependence of the dynamical exponent in the Griffiths-McCoy phase can be determined analytically.

cond-mat.dis-nn↗

arXiv2007

Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1] decays as a power law n^{-θ(d)} where θ(d)>0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n^{-2(θ(d) + θ(2))}. For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n^{-\tilde ϕ(k/\log n)} where \tilde ϕ(x) is a universal large deviation function.

cond-mat.stat-mech↗

arXiv2007

Dynamic crossover in the global persistence at criticality

We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter (e.g., the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m_0 of the order parameter and the typical time at which it occurs diverges as m_0 vanishes. Monte-Carlo simulations of the two-dimensional Ising model with Glauber dynamics display clearly this crossover. The measured exponent of the ultimate algebraic decay is in rather good agreement with our theoretical predictions for the Ising universality class.

cond-mat.dis-nn↗

arXiv2007

Disordered free fermions and the Cardy Ostlund fixed line at low temperature

Using functional RG, we reexamine the glass phase of the 2D random-field Sine Gordon model. It is described by a line of fixed points (FP) with a super-roughening amplitude $\bar{(u(0)-u(r))^2} \sim A(T) \ln^2 r $ as temperature $T$ is varied. A speculation is that this line is identical to the one found in disordered free-fermion models via exact results from ``nearly conformal'' field theory. This however predicts $A(T=0)=0$, contradicting numerics. We point out that this result may be related to failure of dimensional reduction, and that a functional RG method incorporating higher harmonics and non-analytic operators predicts a non-zero $A(T=0)$ which compares reasonably with numerics.

cond-mat.dis-nn↗

arXiv2006

Super-Aging in two-dimensional random ferromagnets

We study the aging properties, in particular the two-time autocorrelations, of the two-dimensional randomly diluted Ising ferromagnet below the critical temperature via Monte-Carlo simulations. We find that the autocorrelation function displays additive aging $C(t,t_w)=C_{st}(t)+C_{ag}(t,t_w)$, where the stationary part $C_{st}$ decays algebraically. The aging part shows anomalous scaling $C_{ag}(t,t_w)={\cal C}(h(t)/h(t_w))$, where $h(u)$ is a non-homogeneous function excluding a $t/t_w$ scaling.

cond-mat.dis-nn↗

arXiv2006

Strong Randomness Fixed Point in the Dissipative Random Transverse Field Ising Model

The interplay between disorder, quantum fluctuations and dissipation is studied in the random transverse Ising chain coupled to a dissipative Ohmic bath with a real space renormalization group. A typically very large length scale, L*, is identified above which the physics of frozen clusters dominates. Below L* a strong disorder fixed point determines scaling at a pseudo-critical point. In a Griffiths-McCoy region frozen clusters produce already a finite magnetization resulting in a classical low temperature behavior of the susceptibility and specific heat. These override the confluent singularities that are characterized by a continuously varying exponent z and are visible above a temperature T* ~ L*^{-z}.

cond-mat.dis-nn↗

arXiv2006

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 $f'(x)$.

cond-mat.stat-mech↗

arXiv2005

Non Markovian persistence in the diluted Ising model at criticality

We investigate global persistence properties for the non-equilibrium critical dynamics of the randomly diluted Ising model. The disorder averaged persistence probability $\bar{{P}_c}(t)$ of the global magnetization is found to decay algebraically with an exponent $θ_c$ that we compute analytically in a dimensional expansion in $d=4-ε$. Corrections to Markov process are found to occur already at one loop order and $θ_c$ is thus a novel exponent characterizing this disordered critical point. Our result is thoroughly compared with Monte Carlo simulations in $d=3$, which also include a measurement of the initial slip exponent. Taking carefully into account corrections to scaling, $θ_c$ is found to be a universal exponent, independent of the dilution factor $p$ along the critical line at $T_c(p)$, and in good agreement with our one loop calculation.

cond-mat.dis-nn↗

arXiv2005

Universal aging properties at a disordered critical point

We investigate, analytically near the dimension $d_{uc}=4$ and numerically in $d=3$, the non equilibrium relaxational dynamics of the randomly diluted Ising model at criticality. Using the Exact Renormalization Group Method to one loop, we compute the two times $t,t_w$ correlation function and Fluctuation Dissipation Ratio (FDR) for any Fourier mode of the order parameter, of finite wave vector $q$. In the large time separation limit, the FDR is found to reach a non trivial value $X^{\infty}$ independently of (small) $q$ and coincide with the FDR associated to the the {\it total} magnetization obtained previously. Explicit calculations in real space show that the FDR associated to the {\it local} magnetization converges, in the asymptotic limit, to this same value $X^{\infty}$. Through a Monte Carlo simulation, we compute the autocorrelation function in three dimensions, for different values of the dilution fraction $p$ at $T_c(p)$. Taking properly into account the corrections to scaling, we find, according to the Renormalization Group predictions, that the autocorrelation exponent $λ_c$ is independent on $p$. The analysis is complemented by a study of the non equilibrium critical dynamics following a quench from a completely ordered state.

cond-mat.dis-nn↗