arXiv subjects

Gregory Schehr

Publications and source records attributed to Gregory Schehr.

At least 163 records · Page 9Linked to original sources

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↗

arXiv2013

Exact distributions of the number of distinct and common sites visited by N independent random walkers

We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{s}/(2 \sqrt{\log N}) and W_N(t)/\sqrt{t} \propto \widetilde{w}/N where \widetilde{s} and \widetilde{w} are random variables whose probability density functions (pdfs) are computed exactly and are found to be non trivial. We verify our results through direct numerical simulations.

cond-mat.stat-mech↗

arXiv2012

Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions

We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional XY model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e. below the critical temperature $T_c$, the disorder and thermally averaged correlation function $B(r)$ of the phase field $θ(x)$, $B(r) = \bar{<[θ(x) - θ(x+ r) ]^2>}$ behaves, for $r \gg a$, as $B(r) \simeq A(τ) \ln^2 (r/a)$ where $r = |r|$ and $a$ is a microscopic length scale. We derive the RG equations up to cubic order in $τ= (T_c-T)/T_c$ and predict the universal amplitude ${A}(τ) = 2τ^2-2τ^3 + {\cal O}(τ^4)$. The universality of $A(τ)$ results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute ${A}(τ)$ numerically and obtain a remarkable agreement with our analytical prediction, up to $τ\approx 0.5$.

cond-mat.dis-nn↗

arXiv2012

Large deviations of the top eigenvalue of large Cauchy random matrices

We compute analytically the probability density function (pdf) of the largest eigenvalue $λ_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($β= 2$), orthogonal ($β=1$) and symplectic ($β=4$) ensembles of such heavy-tailed random matrices. We show that a central non-Gaussian regime for $λ_{\max} \sim \mathcal{O}(N)$ is flanked by large deviation tails on both sides which we compute here exactly for any value of $β$. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors of the pdf in the central regime, which generalizes the Tracy-Widom distribution known for Gaussian ensembles, both at small and large arguments and for any $β$. Our analytical results are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2012

Reunion probability of N vicious walkers: typical and large fluctuations for large N

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyse the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for L\ll O(\sqrt{N}) and L\geq O(\sqrt{N}). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value L=L_c(N)\sim \sqrt{N}. This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L \sim L_c(N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L \gg L_c(N)] and the left [L \ll L_c(N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L \sim L_c(N) and atypical fluctuations in the right tail L \gg L_c(N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices.

cond-mat.stat-mech↗

arXiv2012

Record statistics and persistence for a random walk with a drift

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length ηdrawn from a continuous distribution f(η) which is symmetric around a constant drift c. We focus in particular on the case were f(η) is a symmetric stable law with a Lévy index 0 < μ\leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter μ. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and μ) of the jump distribution. We compute the asymptotic mean record number after n steps as well as its full distribution P(R,n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < μ\leq 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.

cond-mat.stat-mech↗

arXiv2012

On the joint distribution of the maximum and its position of the Airy2 process minus a parabola

The maximal point of the Airy2 process minus a parabola is believed to describe the scaling limit of the end-point of the directed polymer in a random medium, which was proved to be true for a few specific cases. Recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained, one by Moreno Flores, Quastel and Remenik, and the other by Schehr. The first formula is given in terms of the Airy function and an associated operator, and the second formula is expressed in terms of the Lax pair equations of the Painleve II equation. We give a direct proof that these two formulas are the same.

nlin.SI↗

arXiv2012

Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces

We compute the joint probability density function (jpdf) P_N(M, τ_M) of the maximum M and its position τ_M for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N \to \infty, this jpdf is peaked around M = \sqrt{2N} and τ_M = 1/2, while the typical fluctuations behave for large N like M - \sqrt{2N} \propto s N^{-1/6} and τ_M - 1/2 \propto w N^{-1/3} where s and w are correlated random variables. One obtains an explicit expression of the limiting jpdf P(s,w) in terms of the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory and a psi-function for the Hastings-McLeod solution to the Painlevé II equation. Our result yields, up to a rescaling of the random variables s and w, an expression for the jpdf of the maximum and its position for the Airy_2 process minus a parabola. This latter describes the fluctuations in many different physical systems belonging to the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimensions. In particular, the marginal probability density function (pdf) P(w) yields, up to a model dependent length scale, the distribution of the endpoint of the directed polymer in a random medium with one free end, at zero temperature. In the large w limit one shows the asymptotic behavior \log P(w) \sim - w^3/12.

math-ph↗

arXiv2012

Finite-temperature and finite-time scaling of the directed polymer free-energy with respect to its geometrical fluctuations

We study the fluctuations of the directed polymer in 1+1 dimensions in a Gaussian random environment with a finite correlation length ξ and at finite temperature. We address the correspondence between the geometrical transverse fluctuations of the directed polymer, described by its roughness, and the fluctuations of its free-energy, characterized by its two-point correlator. Analytical arguments are provided in favor of a generic scaling law between those quantities, at finite time, non-vanishing ξ and explicit temperature dependence. Numerical results are in good agreement both for simulations on the discrete directed polymer and on a continuous directed polymer (with short-range correlated disorder). Applications to recent experiments on liquid crystals are discussed.

cond-mat.stat-mech↗

arXiv2012

Lévy flights on the half line

We study the probability distribution function (pdf) of the position of a Lévy flight of index 0<α<2 in presence of an absorbing wall at the origin. The solution of the associated fractional Fokker-Planck equation can be constructed using a perturbation scheme around the Brownian solution (corresponding to α= 2), as an expansion in ε= 2 - α. We obtain an explicit analytical solution, exact at the first order in ε, which allows us to conjecture the precise asymptotic behavior of this pdf, including the first subleading corrections, for any α. Careful numerical simulations, as well as an exact computation for α= 1, confirm our conjecture.

cond-mat.stat-mech↗

arXiv2012

Record Statistics for Multiple Random Walks

We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance σ^2 of the jump distribution is finite and (II) when σ^2 is divergent as in the case of Lévy flights with index 0 < μ< 2. In both cases we find that the mean record number grows universally as \sim α_N \sqrt{n} for large n, but with a very different behavior of the amplitude α_N for N > 1 in the two cases. We find that for large N, α_N \approx 2 \sqrt{\log N} independently of σ^2 in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, α_N \approx 4/\sqrtπ, independently of 0<μ<2. For finite σ^2 we argue, and this is confirmed by our numerical simulations, that the full distribution of (R_{n,N}/\sqrt{n} - 2 \sqrt{\log N}) \sqrt{\log N} converges to a Gumbel law as n \to \infty and N \to \infty. In case II, our numerical simulations indicate that the distribution of R_{n,N}/\sqrt{n} converges, for n \to \infty and N \to \infty, to a universal nontrivial distribution, independently of μ. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poors 500 index.

cond-mat.stat-mech↗

arXiv2011

Universal Order Statistics of Random Walks

We study analytically the order statistics of a time series generated by the successive positions of a symmetric random walk of n steps with step lengths of finite variance σ^2. We show that the statistics of the gap d_{k,n}=M_{k,n} -M_{k+1,n} between the k-th and the (k+1)-th maximum of the time series becomes stationary, i.e, independent of n as n\to \infty and exhibits a rich, universal behavior. The mean stationary gap (in units of σ) exhibits a universal algebraic decay for large k, /σ\sim 1/\sqrt{2πk}, independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, Proba.(d_{k,\infty}=δ)\simeq (\sqrt{k}/σ) P(δ\sqrt{k}/σ), in the scaling regime when δ\sim \simeq σ/\sqrt{2πk}. The scaling function P(x) is universal and has an unexpected power law tail, P(x) \sim x^{-4} for large x. For δ\gg the scaling breaks down and the pdf gets cut-off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multi-scaling behavior.

cond-mat.stat-mech↗

arXiv2011

Maximum relative height of elastic interfaces in random media

The distribution of the maximal relative height (MRH) of self-affine one-dimensional elastic interfaces in a random potential is studied. We analyze the ground state configuration at zero driving force, and the critical configuration exactly at the depinning threshold, both for the random-manifold and random-periodic universality classes. These configurations are sampled by exact numerical methods, and their MRH distributions are compared with those with the same roughness exponent and boundary conditions, but produced by independent Fourier modes with normally distributed amplitudes. Using Pickands' theorem we derive an exact analytical description for the right tail of the latter. After properly rescaling the MRH distributions we find that corrections from the Gaussian independent modes approximation are in general small, as previously found for the average width distribution of depinning configurations. In the large size limit all corrections are finite except for the ground-state in the random-periodic class whose MRH distribution becomes, for periodic boundary conditions, indistinguishable from the Airy distribution. We find that the MRH distributions are, in general, sensitive to changes of boundary conditions.

cond-mat.stat-mech↗

arXiv2011

Two stock options at the races: Black-Scholes forecasts

Suppose one buys two very similar stocks and is curious about how much, after some time T, one of them will contribute to the overall asset, expecting, of course, that it should be around 1/2 of the sum. Here we examine this question within the classical Black and Scholes (BS) model, focusing on the evolution of the probability density function P(w) of a random variable w = a_T^{(1)}/(a_T^{(1)} + a_T^{(2)}) where a_T^{(1)} and a_T^{(2)} are the values of two (either European- or the Asian-style) options produced by two absolutely identical BS stochastic equations. We show that within the realm of the BS model the behavior of P(w) is surprisingly different from common-sense-based expectations. For the European-style options P(w) always undergoes a transition, (when T approaches a certain threshold value), from a unimodal to a bimodal form with the most probable values being close to 0 and 1, and, strikingly, w =1/2 being the least probable value. This signifies that the symmetry between two options spontaneously breaks and just one of them completely dominates the sum. For path-dependent Asian-style options we observe the same anomalous behavior, but only for a certain range of parameters. Outside of this range, P(w) is always a bell-shaped function with a maximum at w = 1/2.

q-fin.ST↗

arXiv2011

Non-intersecting Brownian walkers and Yang-Mills theory on the sphere

We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the Tracy-Widom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the Tracy-Widom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the Tracy-Widom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N non-intersecting Brownian excursions ("watermelons" with a wall) whose distribution in the asymptotic scaling limit is then described by GOE Tracy-Widom law. In addition, large deviation formulas for the maximum height are also computed.

math-ph↗

arXiv2011

Distribution of the time at which N vicious walkers reach their maximal height

We study the extreme statistics of N non-intersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum M and of the time τ_M at which this maximum is reached. We focus in particular on non-intersecting Brownian bridges ("watermelons without wall") and non-intersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelons configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium (DPRM) with one free end-point on the other hand. We also check our results using numerical simulations of Dyson's Brownian motion and confront them with numerical simulations of the Polynuclear Growth Model (PNG) and of a model of DPRM on a discrete lattice. Some of the results presented here were announced in a recent letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].

cond-mat.stat-mech↗

arXiv2010

Dynamic crossover in the persistence probability of manifolds at criticality

We investigate the persistence properties of critical d-dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d'-dimensional manifold. The persistence probability P(t) shows three distinct long-time decays depending on the value of the parameter ζ= (D-2+η)/z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d-d' and of the critical exponents z and η. We find that the asymptotic behavior of P(t) is exponential for ζ> 1, stretched exponential for 0 <= ζ<= 1, and algebraic for ζ< 0. Whereas the exponential and stretched exponential relaxations are not affected by the initial value of the order parameter, we predict and observe a crossover between two different power-law decays when the algebraic relaxation occurs, as in the case d'=d of the global order parameter. We confirm via Monte Carlo simulations our analytical predictions by studying the magnetization of a line and of a plane of the two- and three-dimensional Ising model, respectively, with Glauber dynamics. The measured exponents of the ultimate algebraic decays are in a rather good agreement with our analytical predictions for the Ising universality class. In spite of this agreement, the expected scaling behavior of the persistence probability as a function of time and of the initial value of the order parameter remains problematic. In this context, the non-equilibrium dynamics of the O(n) model in the limit n->\infty and its subtle connection with the spherical model is also discussed in detail.

cond-mat.stat-mech↗

arXiv2010

Extremal statistics of curved growing interfaces in 1+1 dimensions

We study the joint probability distribution function (pdf) of the maximum M of the height and its position X_M of a curved growing interface belonging to the universality class described by the Kardar-Parisi-Zhang equation in 1+1 dimensions. We obtain exact results for the closely related problem of p non-intersecting Brownian bridges where we compute the joint pdf P_p(M,τ_M) where τ_M is there the time at which the maximal height M is reached. Our analytical results, in the limit p \to \infty, become exact for the interface problem in the growth regime. We show that our results, for moderate values of p \sim 10 describe accurately our numerical data of a prototype of these systems, the polynuclear growth model in droplet geometry. We also discuss applications of our results to the ground state configuration of the directed polymer in a random potential with one fixed endpoint.

cond-mat.stat-mech↗