Exact maximal height distribution of fluctuating interfaces

Full-record statistics of one-dimensional random walks

We develop a comprehensive framework for analyzing full-record statistics, covering record counts M(t_{1}),M(t_{2}),..., their corresponding attainment times T_{M(t_{1})},T_{M(t_{2})},..., and the intervals until the next record. From this multiple-time distribution, we derive general expressions for various observables related to record dynamics, including the conditional number of records given the number observed at a previous time and the conditional time required to reach the current record given the occurrence time of the previous one. Our formalism is exemplified by a variety of stochastic processes, including biased nearest-neighbor random walks, asymmetric run-and-tumble dynamics, and random walks with stochastic resetting.

Dynamically emergent correlations between particles in a switching harmonic trap

We study a one dimensional gas of N noninteracting diffusing particles in a harmonic trap, whose stiffness switches between two values μ_{1} and μ_{2} with constant rates r_{1} and r_{2}, respectively. Despite the absence of direct interaction between the particles, we show that strong correlations between them emerge in the stationary state at long times, induced purely by the dynamics itself. We compute exactly the joint distribution of the positions of the particles in the stationary state, which allows us to compute several physical observables analytically. In particular, we show that the extreme value statistics (EVS), i.e., the distribution of the position of the rightmost particle, has a nontrivial shape in the large N limit. The scaling function characterizing this EVS has a finite support with a tunable shape (by varying the parameters). Remarkably, this scaling function turns out to be universal. First, it also describes the distribution of the position of the kth rightmost particle in a 1d trap. Moreover, the distribution of the position of the particle farthest from the center of the harmonic trap in d dimensions is also described by the same scaling function for all d≥1. Numerical simulations are in excellent agreement with our analytical predictions.

Exact extreme, order, and sum statistics in a class of strongly correlated systems

Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations. We consider a set of N independent and identically distributed random variables {X_{1},X_{2},...,X_{N}} whose common distribution has a parameter Y (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter Y, the X_{i} variables are independent and we call them conditionally independent and identically distributed. However, once integrated over the distribution of the parameter Y, the X_{i} variables get strongly correlated yet retain a solvable structure for various observables, such as for the sum and the extremes of X_{i}^{'}s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where N particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via simultaneous resetting to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.

Extremal statistics for a resetting Brownian motion before its first-passage time

We study the extreme value statistics of a one-dimensional resetting Brownian motion (RBM) till its first passage through the origin starting from the position x_{0} (>0). By deriving the exit probability of RBM in an interval [0,M] from the origin, we obtain the distribution P_{r}(M|x_{0}) of the maximum displacement M and thus gives the expected value 〈M〉 of M as functions of the resetting rate r and x_{0}. We find that 〈M〉 decreases monotonically as r increases, and tends to 2x_{0} as r→∞. In the opposite limit, 〈M〉 diverges logarithmically as r→0. Moreover, we derive the propagator of RBM in the Laplace domain in the presence of both absorbing ends, and then leads to the joint distribution P_{r}(M,t_{m}|x_{0}) of M and the time t_{m} at which this maximum is achieved in the Laplace domain by using a path decomposition technique, from which the expected value 〈t_{m}〉 of t_{m} is obtained explicitly. Interestingly, 〈t_{m}〉 shows a nonmonotonic dependence on r, and attains its minimum at an optimal r^{*}≈2.71691D/x_{0}^{2}, where D is the diffusion coefficient. Finally, we perform extensive simulations to validate our theoretical results.

Extreme value statistics and arcsine laws for heterogeneous diffusion processes

Heterogeneous diffusion with a spatially changing diffusion coefficient arises in many experimental systems such as protein dynamics in the cell cytoplasm, mobility of cajal bodies, and confined hard-sphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in a power-law way, i.e., D(x)∼|x|^{-α} with the exponent α>-1. This model is known to exhibit anomalous scaling of the mean-squared displacement (MSD) of the form ∼t^{2/2+α} and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M(t) and arg-maximum t_{m}(t) (i.e., the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time t_{r}(t) and the last-passage time t_{ℓ}(t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α≠0) displays many rich and contrasting features compared to that of the standard Brownian motion (BM). For example, while for BM (α=0), the distributions of t_{m}(t),t_{r}(t), and t_{ℓ}(t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for nonzero α. Another interesting property of t_{r}(t) is the existence of a critical α (which we denote by α_{c}=-0.3182) such that the distribution exhibits a local maximum at t_{r}=t/2 for α<α_{c} whereas it has minima at t_{r}=t/2 for α≥α_{c}. The underlying reasoning for this difference hints at the very contrasting natures of the process for α≥α_{c} and α<α_{c} which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.

Generating discrete-time constrained random walks and Lévy flights

We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method is based on an effective jump distribution that implicitly accounts for the bridge constraint. It is illustrated on various jump distributions and is shown to be very efficient in practice. In addition, we show how to generalize the method to other types of constrained random walks such as generalized bridges, excursions, and meanders.

Extremal statistics for stochastic resetting systems

While averages and typical fluctuations often play a major role in understanding the behavior of a nonequilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance, and ecology. Herein, we study extreme value statistics (EVS) of stochastic resetting systems, which have recently gained significant interest due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes, and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum, i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains a uniform distribution independent of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting, namely, simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process, and random acceleration process in one dimension. Rigorous results are presented for the first two setups, while the latter two are supported with heuristic and numerical analysis.

Weak scaling of the contact distance between two fluctuating interfaces with system size

A pair of flat parallel surfaces, each freely diffusing along the direction of their separation, will eventually come into contact. If the shapes of these surfaces also fluctuate, then contact will occur when their centers-of-mass remain separated by a nonzero distance ℓ. An example of such a situation is the motion of interfaces between two phases at conditions of thermodynamic coexistence, and in particular the annihilation of domain wall pairs under periodic boundary conditions. Here we present a general approach to calculate the probability distribution of the contact distance ℓ and determine how its most likely value ℓ^{*} depends on the surfaces' lateral size L. Using the Edward-Wilkinson equation as a model for interfaces, we demonstrate that ℓ^{*} scales weakly with system size, i.e., the dependence of ℓ^{*} on L for both (1+1)- and (2+1)-dimensional interfaces is such that lim_{L→∞}(ℓ^{*}/L)=0. In particular, for (2+1)-dimensional interfaces ℓ^{*} is an algebraic function of logL, a result that is confirmed by computer simulations of slab-shaped domains formed under periodic boundary conditions. This weak scaling implies that such domains remain topologically intact until ℓ becomes very small compared to the lateral size of the interface, contradicting expectations from equilibrium thermodynamics.

Variational approach to KPZ: Fluctuation theorems and large deviation function for entropy production

Motivated by the time behavior of the functional arising in the variational approach to the Kardar-Parisi-Zhang (KPZ) equation, and in order to study fluctuation theorems in such a system, we have adapted a path-integral scheme that adequately fits to this kind of study dealing with unstable systems. As the KPZ system has no stationary probability distribution, we show how to proceed for obtaining detailed as well as integral fluctuation theorems. This path-integral methodology, together with the variational approach, in addition to allowing analyze fluctuation theorems, can be exploited to determine a large deviation function for entropy production.

Distribution of the time between maximum and minimum of random walks

We consider a one-dimensional Brownian motion of fixed duration T. Using a path-integral technique, we compute exactly the probability distribution of the difference τ=t_{min}-t_{max} between the time t_{min} of the global minimum and the time t_{max} of the global maximum. We extend this result to a Brownian bridge, i.e., a periodic Brownian motion of period T. In both cases, we compute analytically the first few moments of τ, as well as the covariance of t_{max} and t_{min}, showing that these times are anticorrelated. We demonstrate that the distribution of τ for Brownian motion is valid for discrete-time random walks with n steps and with a finite jump variance, in the limit n→∞. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of τ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "τ=n" is exactly 1/(2n) for any finite n, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of (1+1)-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size L. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.200201].

Time Between the Maximum and the Minimum of a Stochastic Process

We present an exact solution for the probability density function P(τ=t_{min}-t_{max}|T) of the time difference between the minimum and the maximum of a one-dimensional Brownian motion of duration T. We then generalize our results to a Brownian bridge, i.e., a periodic Brownian motion of period T. We demonstrate that these results can be directly applied to study the position difference between the minimal and the maximal heights of a fluctuating (1+1)-dimensional Kardar-Parisi-Zhang interface on a substrate of size L, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.

Roughening of k-mer-growing interfaces in stationary regimes

We discuss the steady-state dynamics of interfaces with periodic boundary conditions arising from body-centered solid-on-solid growth models in 1+1 dimensions involving random aggregation of extended particles (dimers, trimers, ...,k-mers). Roughening exponents as well as width and maximal height distributions can be evaluated directly in stationary regimes by mapping the dynamics onto an asymmetric simple exclusion process with k-type of vacancies. Although for k≥2 the dynamics is partitioned into an exponentially large number of sectors of motion, the results obtained in some generic cases strongly suggest a universal scaling behavior closely following that of monomer interfaces.

Facets on the convex hull of d-dimensional Brownian and Lévy motion

For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d≥3, we establish an exact formula for the average number of (d-1)-dimensional facets that can be defined by d points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when d=3, a case which is of particular interest for applications in biophysics, chemistry, and polymer science. We also show that the asymptotical average number of facets behaves as 〈F_{T}^{(d)}〉∼2[ln(T/Δt)]^{d-1}, where T is the total duration of the motion and Δt is the minimum time lapse separating points that define a facet.

Spatial Competition: Roughening of an Experimental Interface

Limited dispersal distance generates spatial aggregation. Intraspecific interactions are then concentrated within clusters, and between-species interactions occur near cluster boundaries. Spread of a locally dispersing invader can become motion of an interface between the invading and resident species, and spatial competition will produce variation in the extent of invasive advance along the interface. Kinetic roughening theory offers a framework for quantifying the development of these fluctuations, which may structure the interface as a self-affine fractal, and so induce a series of temporal and spatial scaling relationships. For most clonal plants, advance should become spatially correlated along the interface, and width of the interface (where invader and resident compete directly) should increase as a power function of time. Once roughening equilibrates, interface width and the relative location of the most advanced invader should each scale with interface length. We tested these predictions by letting white clover (Trifolium repens) invade ryegrass (Lolium perenne). The spatial correlation of clover growth developed as anticipated by kinetic roughening theory, and both interface width and the most advanced invader’s lead scaled with front length. However, the scaling exponents differed from those predicted by recent simulation studies, likely due to clover’s growth morphology.

Size distribution of ring polymers

We present an exact solution for the distribution of sample averaged monomer to monomer distance of ring polymers. For non-interacting and local-interaction models these distributions correspond to the distribution of the area under the reflected Bessel bridge and the Bessel excursion respectively, and are shown to be identical in dimension d ≥ 2, albeit with pronounced finite size effects at the critical dimension, d = 2. A symmetry of the problem reveals that dimension d and 4 - d are equivalent, thus the celebrated Airy distribution describing the areal distribution of the d = 1 Brownian excursion describes also a polymer in three dimensions. For a self-avoiding polymer in dimension d we find numerically that the fluctuations of the scaled averaged distance are nearly identical in dimension d = 2, 3 and are well described to a first approximation by the non-interacting excursion model in dimension 5.

Width and extremal height distributions of fluctuating interfaces with window boundary conditions

We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size l, for interfaces in several universality classes, in substrate dimensions d_{s}=1 and 2. We show that their cumulants follow a Family-Vicsek-type scaling, and, at early times, when ξ≪l (ξ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their nth cumulant scaling as (ξ/l)^{(n-1)d_{s}}. This gives rise to an interesting temporal scaling for such cumulants as 〈w_{n}〉_{c}∼t^{γ_{n}}, with γ_{n}=2nβ+(n-1)d_{s}/z=[2n+(n-1)d_{s}/α]β. This scaling is analytically proved for the Edwards-Wilkinson (EW) and random deposition interfaces and numerically confirmed for other classes. In general, it is featured by small corrections, and, thus, it yields exponents γ_{n} (and, consequently, α,β and z) in good agreement with their respective universality class. Thus, it is a useful framework for numerical and experimental investigations, where it is usually hard to estimate the dynamic z and mainly the (global) roughness α exponents. The stationary (for ξ≫l) SLRDs and LEHDs of the Kardar-Parisi-Zhang (KPZ) class are also investigated, and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidence of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large l. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.

Extreme fluctuations in stochastic network coordination with time delays

We study the effects of uniform time delays on the extreme fluctuations in stochastic synchronization and coordination problems with linear couplings in complex networks. We obtain the average size of the fluctuations at the nodes from the behavior of the underlying modes of the network. We then obtain the scaling behavior of the extreme fluctuations with system size, as well as the distribution of the extremes on complex networks, and compare them to those on regular one-dimensional lattices. For large complex networks, when the delay is not too close to the critical one, fluctuations at the nodes effectively decouple, and the limit distributions converge to the Fisher-Tippett-Gumbel density. In contrast, fluctuations in low-dimensional spatial graphs are strongly correlated, and the limit distribution of the extremes is the Airy density. Finally, we also explore the effects of nonlinear couplings on the stability and on the extremes of the synchronization landscapes.

Pulled Polymer Loops as a Model for the Alignment of Meiotic Chromosomes

During recombination, the DNA of parents exchange their genetic information to give rise to a genetically unique offspring. For recombination to occur, homologous chromosomes need to find each other and align with high precision. Fission yeast solves this problem by folding chromosomes in loops and pulling them through the viscous nucleoplasm. We propose a theory of pulled polymer loops to quantify the effect of drag forces on the alignment of chromosomes. We introduce an external force field to the concept of a Brownian bridge and thus solve for the statistics of loop configurations in space.

Spatial extent of branching Brownian motion

We study the one-dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost (X(max)≥0) and leftmost (X(min)≤0) visited sites up to time t. At each time step the existing particles in the system either diffuse (with diffusion constant D), die (with rate a), or split into two particles (with rate b). We focus on the regime b≤a where these two extreme values X(max) and X(min) are strongly correlated. We show that at large time t, the joint probability distribution function (PDF) of the two extreme points becomes stationary P(X,Y,t→∞)→p(X,Y). Our exact results for p(X,Y) demonstrate that the correlation between X(max) and X(min) is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF p(ζ) of the (dimensionless) span ζ=(X(max)-X(min))/√[D/b], which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior p(ζ)∼1/2(1+Δ)ζ for small spans, with Δ=(a/b-1). In the critical case (Δ=0) this distribution has a nontrivial power law tail p(ζ)∼8π√[3]/ζ(3) for large spans. On the other hand, in the subcritical case (Δ>0), we show that the span distribution decays exponentially as p(ζ)∼(A(2)/2)ζexp(-√[Δ]ζ) for large spans, where A is a nontrivial function of Δ, which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between X(max) and X(min). Finally we verify our results via direct Monte Carlo simulations.

Freezing transitions and extreme values: random matrix theory, ζ (1/2 + it) and disordered landscapes

We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N → ∞. In addition, we argue that it leads to multi-fractal-like behaviour in the total length μN(x) of the intervals in which |pN(θ)|>N(x), x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(s) over stretches of the critical line s = 1/2 + it of given constant length and present the results of numerical computations of the large values of ζ(1/2 + it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

Near-extreme statistics of Brownian motion

We study the statistics of near-extreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states near the maximum, ρ(r,t), which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function of ρ(r,t) and obtain an explicit expression for the moments <[ρ(r,t)]k> for arbitrary integer k. We also study near extremes of constrained BM, like the Brownian bridge. Finally we also present numerical simulations to check our analytical results.

Universal fluctuations and extreme statistics of avalanches near the depinning transition

We derive exact predictions for universal scaling exponents and scaling functions associated with the statistics of maximum velocities v(m) during avalanches described by the mean-field theory of the interface depinning transition. In particular, we find a robust power-law regime in the statistics of maximum events that can explain the observed distribution of the peak amplitudes in acoustic emission experiments of crystal plasticity. Our results are expected to be broadly applicable to a broad range of systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.

Positive feedback produces broad distributions in maximum activation attained within a narrow time window in stochastic biochemical reactions

How do single cell fate decisions induced by activation of key signaling proteins above threshold concentrations within a time interval are affected by stochastic fluctuations in biochemical reactions? We address this question using minimal models of stochastic chemical reactions commonly found in cell signaling and gene regulatory systems. Employing exact solutions and semi-analytical methods we calculate distributions of the maximum value (N) of activated species concentrations (P(max)(N)) and the time (t) taken to reach the maximum value (P(max)(t)) within a time interval in the minimal models. We find, the presence of positive feedback interactions make P(max)(N) more spread out with a higher "peakedness" in P(max)(t). Thus positive feedback interactions may help single cells to respond sensitively to a stimulus when cell decision processes require upregulation of activated forms of key proteins to a threshold number within a time window.

Distribution of maximum velocities in avalanches near the depinning transition

We report exact predictions for universal scaling exponents and scaling functions associated with the distribution of the maximum collective avalanche propagation velocities v(m) in the mean field theory of the interface depinning transition. We derive the extreme value distribution P(v(m)|T) for the maximum velocities in avalanches of fixed duration T and verify the results by numerical simulation near the critical point. We find that the tail of the distribution of maximum velocity for an arbitrary avalanche duration, v(m), scales as P(v(m))~v(m)(-2) for large v(m). These results account for the observed power-law distribution of the maximum amplitudes in acoustic emission experiments of crystal plasticity and are also broadly applicable to other systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.

Theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices

Recently, anomalous superdiffusion of ultracold 87Rb atoms in an optical lattice has been observed along with a fat-tailed, Lévy type, spatial distribution. The anomalous exponents were found to depend on the depth of the optical potential. We find, within the framework of the semiclassical theory of Sisyphus cooling, three distinct phases of the dynamics as the optical potential depth is lowered: normal diffusion; Lévy diffusion; and x∼t(3/2) scaling, the latter related to Obukhov's model (1959) of turbulence. The process can be formulated as a Lévy walk, with strong correlations between the length and duration of the excursions. We derive a fractional diffusion equation describing the atomic cloud, and the corresponding anomalous diffusion coefficient.

Universal order statistics of random walks

We study analytically the order statistics of a time series generated by the 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 kth and the (k+1)th maximum of the time series becomes stationary, i.e., independent of n as n → ∞ and exhibits a rich, universal behavior. The mean stationary gap exhibits a universal algebraic decay for large k, ~d(k,∞)-/σ 1/sqrt[2πk], independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, Pr(d(k,∞) = δ) ~/= (sqrt[k]/σ)P(δsqrt[k]/σ), in the regime δ~ (d(k,∞)). The scaling function P(x) is universal and has an unexpected power law tail, P(x) ~ x(-4) for large x. For δ>> (d(k,∞)) the scaling breaks down and the pdf gets cut off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multiscaling behavior.

Order statistics of 1/fα signals

Order statistics of periodic, Gaussian noise with 1/f(α) power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d(k) = (x(k) -x(k) + 1) between the kth and (k+1)st largest values of the signal. The result d(k) k(-1), known for independent, identically distributed variables, remains valid for 0 ≤ α < 1. Nontrivial, α-dependent scaling exponents, d(k) k((α-3)/2), emerge for 1 < α < 5, and, finally, α-independent scaling, d(k) ~ k, is obtained for α > 5. The spectra of average ordered values ε(k) =(x(1) - x(k))~ k(β) is also examined. The exponent β is derived from the gap scaling as well as by relating ε(k) to the density of near-extreme states. Known results for the density of near-extreme states combined with scaling suggest that β(α = 2) = 1/2, β(4) = 3/2, and β(∞) = 2 are exact values. We also show that parallels can be drawn between ε(k) and the quantum mechanical spectra of a particle in power-law potentials.

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.

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

We study the extreme statistics of N nonintersecting 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 nonintersecting Brownian bridges ("watermelons without wall") and nonintersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelon 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)].

Ballistic deposition patterns beneath a growing Kardar-Parisi-Zhang interface

We consider a (1+1)-dimensional ballistic deposition process with next-nearest-neighbor interactions, which belongs to the Kardar-Parisi-Zhang (KPZ) universality class. The focus of our analysis is on the properties of structures appearing in the bulk of a growing aggregate: a forest of independent clusters separated by "crevices." Competition for growth (mutual screening) between different clusters results in "thinning" of this forest, i.e., the number density c(h) of clusters decreases with the height h of the pattern. For the discrete stochastic equation describing the process we introduce a variational formulation similar to that used for the randomly forced continuous Burgers equation. This allows us to identify the "clusters" and crevices with minimizers and shocks in the Burgers turbulence. Capitalizing on the ideas developed for the latter process, we find that c(h) ∼ h(-α) with α=2/3. We compute also scaling laws that characterize the ballistic deposition patterns in the bulk: the law of transversal fluctuations of cluster boundaries and the size distribution of clusters. It turns out that the intercluster interface is superdiffusive: the corresponding exponent is twice as large as the KPZ exponent for the surface of the aggregate. Finally we introduce a probabilistic concept of ballistic growth, dubbed the "hairy" Airy process in view of its distinctive geometric features. Its statistical properties are analyzed numerically.

Renormalization-group theory for finite-size scaling in extreme statistics

We present a renormalization-group (RG) approach to explain universal features of extreme statistics applied here to independent identically distributed variables. The outlines of the theory have been described in a previous paper, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point, leading to the computation of stable perturbations as eigenfunctions. Here we show details of the RG theory which exhibit remarkable similarities to the RG known in statistical physics. Besides the fixed points explaining universality, and the least stable eigendirections accounting for convergence rates and shape corrections, the similarities include marginally stable perturbations which turn out to be generic for the Fisher-Tippett-Gumbel class. Distribution functions containing unstable perturbations are also considered. We find that, after a transitory divergence, they return to the universal fixed line at the same or at a different point depending on the type of perturbation.

Extreme value statistics and return intervals in long-range correlated uniform deviates

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider (e.g., maximum relative to minimum) are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself (i.e., a Weibull distribution), reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.

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 <Hp> approximately square root 2p (excursions) whereas <Hp> 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.

Finite-size scaling in extreme statistics

We study the deviations from the limit distributions in extreme value statistics arising due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. It is found that, for the correlated systems of subcritical percolation and 1/f;(alpha) stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f;(alpha) noise, the shape correction is obtained in terms of the limit distribution itself.

Maximal- and minimal-height distributions of fluctuating interfaces

Maximal- and minimal-height distributions (MAHD, MIHD) of two-dimensional interfaces grown with the nonlinear equations of Kardar-Parisi-Zhang (KPZ, second order) and of Villain-Lai-Das Sarma (VLDS, fourth order) are shown to be different. Two universal curves may be MAHD or MIHD of each class depending on the sign of the relevant nonlinear term, which is confirmed by results of several lattice models in the KPZ and VLDS classes. The difference between MAHD and MIDH is connected with the asymmetry of the local height distribution. A simple, exactly solvable deposition-erosion model is introduced to illustrate this feature. The average extremal heights scale with the same exponent of the average roughness. In contrast to other correlated systems, generalized Gumbel distributions do not fit those MAHD and MIHD, nor those of Edwards-Wilkinson growth.

Extreme statistics for time series: distribution of the maximum relative to the initial value

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/falpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha , the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

Extreme fluctuations in noisy task-completion landscapes on scale-free networks

We study the statistics and scaling of extreme fluctuations in noisy task-completion landscapes, such as those emerging in synchronized distributed-computing networks, or generic causally constrained queuing networks, with scale-free topology. In these networks the average size of the fluctuations becomes finite (synchronized state) and the extreme fluctuations typically diverge only logarithmically in the large system-size limit ensuring synchronization in a practical sense. Provided that local fluctuations in the network are short tailed, the statistics of the extremes are governed by the Gumbel distribution. We present large-scale simulation results using the exact algorithmic rules, supported by mean-field arguments based on a coarse-grained description.

Universal extremal statistics in a freely expanding Jepsen gas

We study the extremal dynamics emerging in an out-of-equilibrium one-dimensional Jepsen gas of (N+1) hard-point particles. The particles undergo binary elastic collisions, but move ballistically in between collisions. The gas is initially uniformly distributed in a box [-L,0] with the "leader" (or the rightmost particle) at X=0 , and a random positive velocity, independently drawn from a distribution phi(V) , is assigned to each particle. The gas expands freely at subsequent times. We compute analytically the distribution of the leader's velocity at time t , and also the mean and the variance of the number of collisions that are undergone by the leader up to time t . We show that in the thermodynamic limit and at fixed time t>>1 (the so-called "growing regime"), when interactions are strongly manifest, the velocity distribution exhibits universal scaling behavior of only three possible varieties, depending on the tail of phi(V) . The associated scaling functions are entirely different from the usual extreme-value distributions of uncorrelated random variables. In this growing regime the mean and the variance of the number of collisions of the leader up to time t increase logarithmically with t , with universal prefactors that are computed exactly. The implications of our results in the context of biological evolution modeling are pointed out.

Extreme times for volatility processes

Extreme times techniques, generally applied to nonequilibrium statistical mechanical processes, are also useful for a better understanding of financial markets. We present a detailed study on the mean first-passage time for the volatility of return time series. The empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern. The empirical mean first-passage time to a certain level L is fairly different from that of the Wiener process showing a dissimilar behavior depending on whether L is higher or lower than the average volatility. All of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value. We thus present the mean first-passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description. We discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential Ornstein-Uhlenbeck model.

Maximal height statistics for 1/f(alpha) signals

Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/f(alpha) power spectrum. For 0<or=alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha-->infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha>1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.

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) approximately radical(12w(L))(-1) f(h(m)radical(12w(L))(-1), with w(L) the width of the fluctuating interface f(x) and 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 the 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 nonuniversal amplitude, and simply given by the derivative of the Airy distribution function f'(x).

Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the "sampling interval" used in the measurement for both "steady-state" and "finite" initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A "deterministic approximation" is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.

Extreme value statistics in records with long-term persistence

Many natural records exhibit long-term correlations characterized by a power-law decay of the autocorrelation function, C(s) approximately s-gamma, with time lag s and correlation exponent 0<gamma<1. We study how the presence of such correlations affects the statistics of the extreme events, i.e., the maximum values of the signal within time segments of the fixed duration R. We find numerically that (i) the integrated distribution function of the maxima converges to a Gumbel distribution for large R similar to uncorrelated signals, (ii) the deviations for finite R depend on the initial distribution of the records and on their correlation properties, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. The last item implies that conditional mean maxima and conditional maxima distributions (with the value of the previous maximum as condition) should be considered for an improved extreme event prediction. We provide indications that this dependence of the mean maxima on the previous maximum occurs also in observational long-term correlated records.

Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces

We investigate the statistics of the maximal fluctuation of two-dimensional Gaussian interfaces. Its relation to the entropic repulsion between rigid walls and a confined interface is used to derive the average maximal [EQUATION: SEE TEXT]and the asymptotic behavior of the whole distribution for [EQUATION: SEE TEXT] for m finite with N2 and K the interface size and tension, respectively. The standardized form of P(m) does not depend on N or K, but shows a good agreement with Gumbel's first asymptote distribution with a particular noninteger parameter. The effects of the correlations among individual fluctuations on the extreme value statistics are discussed in our findings.