Exact distribution of the maximal height of p vicious walkers

Bernoulli trial under restarts: A comparative study of resetting transitions

A Bernoulli trial describing the escape behavior of a lamb to a safe haven in pursuit by a lion is studied under restarts. The process ends in two ways: either the lamb makes it to the safe haven (success) or is captured by the lion (failure). We study the first passage properties of this Bernoulli trial and find that only mean first passage time exists. Considering Poisson and sharp resetting, we find that the success probability is a monotonically decreasing function of the restart rate. The mean time, however, exhibits a nonmonotonic dependence on the restart rate taking a minimal value at an optimal restart rate. Furthermore, for sharp restart, the mean time possesses a local and a global minima. As a result, the optimal restart rate exhibits a continuous transition for Poisson resetting while it exhibits a discontinuous transition for sharp resetting as a function of the relative separation of the lion and the lamb. We also find that the distribution of first passage times under sharp resetting exhibits a periodic behavior.

Dynamical phase transition in the occupation fraction statistics for noncrossing Brownian particles

We consider a system of N noncrossing Brownian particles in one dimension. We find the exact rate function that describes the long-time large deviation statistics of their occupation fraction in a finite interval in space. Remarkably, we find that, for any general N≥2, the system undergoes N-1 dynamical phase transitions of second order. The N-1 transitions are the boundaries of N phases that correspond to different numbers of particles which are in the vicinity of the interval throughout the dynamics. We achieve this by mapping the problem to that of finding the ground-state energy for N noninteracting spinless fermions in a square-well potential. The phases correspond to different numbers of single-body bound states for the quantum problem. We also study the process conditioned on a given occupation fraction and the large-N limiting behavior.

Capture of a diffusing lamb by a diffusing lion when both return home

A diffusing lion pursues a diffusing lamb when both of them are allowed to get back to their homes intermittently. Identifying the system with a pair of vicious random walkers, we study their dynamics under Poissonian and sharp resetting. In the absence of any resets, the location of intersection of the two walkers follows a Cauchy distribution. In the presence of resetting, the distribution of the location of annihilation is composed of two parts: one in which the trajectories cross without being reset (center) and the other where trajectories are reset at least once before they cross each other (tails). We find that the tail part decays exponentially for both the resetting protocols. The central part of the distribution, on the other hand, depends on the nature of the restart protocol, with Cauchy for Poisson resetting and Gaussian for sharp resetting. We find good agreement of the analytical results with numerical calculations.

Stretching of a Fractal Polymer around a Disc Reveals Kardar-Parisi-Zhang Scaling

While stretching of a polymer along a flat surface is hardly different from the classical Pincus problem of pulling chain ends in free space, the role of curved geometry in conformational statistics of the stretched chain is an exciting open question. We use scaling analysis and computer simulations to examine stretching of a fractal polymer chain around a disc in 2D (or a cylinder in 3D) of radius R. We reveal that the typical excursions of the polymer away from the surface and curvature-induced correlation length scale as Δ∼R^{β} and S^{*}∼R^{1/z}, respectively, with the Kardar-Parisi-Zhang (KPZ) growth β=1/3 and dynamic exponents z=3/2. Although probability distribution of excursions does not belong to KPZ universality class, the KPZ scaling is independent of the fractal dimension of the polymer and, thus, is universal across classical polymer models, e.g., SAW, randomly branching polymers, crumpled unknotted rings. Additionally, our Letter establishes a mapping between stretched polymers in curved geometry and the Balagurov-Vaks model of random walks among traps.

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.

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.

Harmonically Confined Particles with Long-Range Repulsive Interactions

We study an interacting system of N classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repel each other via pairwise interaction potential that behaves as a power law ∝∑[under i≠j][over N]|x_{i}-x_{j}|^{-k} (with k>-2) of their mutual distance. This is a generalization of the well-known cases of the one-component plasma (k=-1), Dyson's log gas (k→0^{+}), and the Calogero-Moser model (k=2). Because of the competition between harmonic confinement and pairwise repulsion, the particles spread over a finite region of space for all k>-2. We compute exactly the average density profile for large N for all k>-2 and show that while it is independent of temperature for sufficiently low temperature, it has a rich and nontrivial dependence on k with distinct behavior for -2<k<1, k>1 and k=1.

Anomalous one-dimensional fluctuations of a simple two-dimensional random walk in a large-deviation regime

The following question is the subject of our work: could a two-dimensional (2D) random path pushed by some constraints to an improbable "large-deviation regime" possess extreme statistics with one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though nonuniversal, since the fluctuations depend on the underlying geometry. We consider in detail two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span 〈d(t)〉∼t^{γ} of the trajectory of t steps above the top of the semicircle or the triangle. We show that γ=1/3, i.e., 〈d(t)〉 shares the KPZ statistics for the semicircle, while γ=0 for the triangle. We propose heuristic derivations of scaling exponents γ for different geometries, justify them by explicit analytic computations, and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.

Convex hulls of random walks in higher dimensions: A large-deviation study

The distribution of the hypervolume V and surface ∂V of convex hulls of (multiple) random walks in higher dimensions are determined numerically, especially containing probabilities far smaller than P=10^{-1000} to estimate large deviation properties. For arbitrary dimensions and large walk lengths T, we suggest a scaling behavior of the distribution with the length of the walk T similar to the two-dimensional case and behavior of the distributions in the tails. We underpin both with numerical data in d=3 and d=4 dimensions. Further, we confirm the analytically known means of those distributions and calculate their variances for large T.

Maximum of an Airy process plus Brownian motion and memory in Kardar-Parisi-Zhang growth

We obtain several exact results for universal distributions involving the maximum of the Airy_{2} process minus a parabola and plus a Brownian motion, with applications to the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic growth universality class. This allows one to obtain (i) the universal limit, for large time separation, of the two-time height correlation for droplet initial conditions, e.g., C_{∞}=lim_{t_{2}/t_{1}→+∞}h(t_{1})h(t_{2})[over ¯]^{c}/h(t_{1})^{2}[over ¯]^{c}, with C_{∞}≈0.623, as well as conditional moments, which quantify ergodicity breaking in the time evolution; (ii) in the same limit, the distribution of the midpoint position x(t_{1}) of a directed polymer of length t_{2}; and (iii) the height distribution in stationary KPZ with a step. These results are derived from the replica Bethe ansatz for the KPZ continuum equation, with a "decoupling assumption" in the large time limit. They agree and confirm, whenever they can be compared, with (i) our recent tail results for two-time KPZ with the work by de Nardis and Le Doussal [J. Stat. Mech. (2017) 0532121742-546810.1088/1742-5468/aa6bce], checked in experiments with the work by Takeuchi and co-workers [De Nardis et al., Phys. Rev. Lett. 118, 125701 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.125701] and (ii) a recent result of Maes and Thiery [J. Stat. Phys. 168, 937 (2017)JSTPBS0022-471510.1007/s10955-017-1839-2] on midpoint position.

Mutually avoiding paths in random media and largest eigenvalues of random matrices

Recently, it was shown that the probability distribution function (PDF) of the free energy of a single continuum directed polymer (DP) in a random potential, equivalently to the height of a growing interface described by the Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom distribution. The latter describes the fluctuations of the largest eigenvalue of a random matrix, drawn from the Gaussian unitary ensemble (GUE), and the result holds for a DP with fixed end points, i.e., for the KPZ equation with droplet initial conditions. A more general conjecture can be put forward, relating the free energies of N>1 noncrossing continuum DP in a random potential, to the sum of the Nth largest eigenvalues of the GUE. Here, using replica methods, we provide an important test of this conjecture by calculating exactly the right tails of both PDFs and showing that they coincide for arbitrary N.

Convex hulls of multiple random walks: A large-deviation study

We study the polygons governing the convex hull of a point set created by the steps of n independent two-dimensional random walkers. Each such walk consists of T discrete time steps, where x and y increments are independent and identically distributed Gaussian. We analyze area A and perimeter L of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below 10^{-900}. We find that the densities exhibit in the limit T→∞ a time-independent scaling behavior as a function of A/T and L/sqrt[T], respectively. As in the case of one walker (n=1), the densities follow Gaussian distributions for L and sqrt[A], respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for T→∞, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for n→∞ as found in the n=1 case. We also investigated the behavior of the averages as a function of the number of walks n and found good agreement with the predicted behavior.

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.

Dysonian dynamics of the Ginibre ensemble

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a nontrivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of a new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the nonanalytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics that we observe for the Ginibre ensemble is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

Slow kinetics of Brownian maxima

We study extreme-value statistics of Brownian trajectories in one dimension. We define the maximum as the largest position to date and compare maxima of two particles undergoing independent Brownian motion. We focus on the probability P(t) that the two maxima remain ordered up to time t and find the algebraic decay P ∼ t(-β) with exponent β = 1/4. When the two particles have diffusion constants D(1) and D(2), the exponent depends on the mobilities, β = (1/π) arctan sqrt[D(2)/D(1)]. We also use numerical simulations to investigate maxima of multiple particles in one dimension and the largest extension of particles in higher dimensions.

Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions

We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large-size matrices, the inverse Cole-Hopf transform of this polynomial obeys a nonlinear partial differential equation whose solutions exhibit shocks at the evolving edges of the eigenvalue spectrum. In a particular scenario one of those shocks hits the origin that plays the role of an impassable wall. To investigate the universal behavior in the vicinity of this wall, i.e., in the vicinity of a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.

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 with N independent random walkers. Using this mapping, we compute exactly their probability distributions P(N)(d)(S,t) and P(N)(c)(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)/√t∝2√(log N)+s/(2√(log N)) and W(N)(t)/√t∝w/N where s and w are random variables whose probability density functions are computed exactly and are found to be nontrivial. We verify our results through direct numerical simulations.

O'Connell's process as a vicious Brownian motion

Vicious Brownian motion is a diffusion scaling limit of Fisher's vicious walk model, which is a system of Brownian particles in one dimension such that if two motions meet they kill each other. We consider the vicious Brownian motions conditioned never to collide with each other and call it noncolliding Brownian motion. This conditional diffusion process is equivalent to the eigenvalue process of the Hermitian-matrix-valued Brownian motion studied by Dyson [J. Math. Phys. 3, 1191 (1962)]. Recently, O'Connell [Ann. Probab. (to be published)] introduced a generalization of the noncolliding Brownian motion by using the eigenfunctions (the Whittaker functions) of the quantum Toda lattice in order to analyze a directed polymer model in 1 + 1 dimensions. We consider a system of one-dimensional Brownian motions with a long-ranged killing term as a generalization of the vicious Brownian motion and construct the O'Connell process as a conditional process of the killing Brownian motions to survive forever.

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)].

How many eigenvalues of a Gaussian random matrix are positive?

We study the probability distribution of the index N(+), i.e., the number of positive eigenvalues of an N×N Gaussian random matrix. We show analytically that, for large N and large N(+) with the fraction 0≤c=N(+)/N≤1 of positive eigenvalues fixed, the index distribution P(N(+)=cN,N)~exp[-βN(2)Φ(c)] where β is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function Φ(c) is computed explicitly for all 0≤c≤1. It is independent of β and displays a quadratic form modulated by a logarithmic singularity around c=1/2. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance Δ(N) of index fluctuations growing as Δ(N)~lnN/βπ(2) for large N. For β=2, this result is independently confirmed against an exact finite-N formula, yielding Δ(N)=lnN/2π(2)+C+O(N(-1)) for large N, where the constant C for even N has the nontrivial value C=(γ+1+3ln2)/2π(2)≃0.185 248… and γ=0.5772… is the Euler constant. We also determine for large N the probability that the interval [ζ(1),ζ(2)] is free of eigenvalues. Some of these results have been announced in a recent letter [Phys. Rev. Lett. 103, 220603 (2009)].

Vicious Lévy flights

We study the statistics of encounters of Lévy flights by introducing the concept of vicious Lévy flights--distinct groups of walkers performing independent Lévy flights with the process terminating upon the first encounter between walkers of different groups. We show that the probability that the process survives up to time t decays as t-α at late times. We compute α up to the second order in ε expansion, where ε=σ-d, σ is the Lévy exponent, and d is the spatial dimension. For d=σ, we find the exponent of the logarithmic decay exactly. Theoretical values of the exponents are confirmed by numerical simulations. Our results indicate that walkers with smaller values of σ survive longer and are therefore more effective at avoiding each other.

Nonintersecting Brownian interfaces and Wishart random matrices

We study a system of N nonintersecting (1+1)-dimensional fluctuating elastic interfaces ("vicious bridges") at thermal equilibrium, each subject to periodic boundary condition in the longitudinal direction and in presence of a substrate that induces an external confining potential for each interface. We show that, for a large system and with an appropriate choice of the external confining potential, the joint distribution of the heights of the N nonintersecting interfaces at a fixed point on the substrate can be mapped to the joint distribution of the eigenvalues of a Wishart matrix of size N with complex entries (Dyson index beta=2), thus providing a physical realization of the Wishart matrix. Exploiting this analogy to random matrix, we calculate analytically (i) the average density of states of the interfaces, (ii) the height distribution of the uppermost and lowermost interfaces (extrema), and (iii) the asymptotic (large-N) distribution of the center of mass of the interfaces. In the last case, we show that the probability density of the center of mass has an essential singularity around its peak, which is shown to be a direct consequence of a phase transition in an associated Coulomb gas problem.

Large deviations of the maximum eigenvalue for wishart and Gaussian random matrices

We present a Coulomb gas method to calculate analytically the probability of rare events where the maximum eigenvalue of a random matrix is much larger than its typical value. The large deviation function that characterizes this probability is computed explicitly for Wishart and Gaussian ensembles. The method is general and applies to other related problems, e.g., the joint large deviation function for large fluctuations of top eigenvalues. Our results are relevant to widely employed data compression techniques, namely, the principal components analysis. Analytical predictions are verified by extensive numerical simulations.

Maximum distributions of bridges of noncolliding Brownian paths

One-dimensional Brownian motion starting from the origin at time t=0 , conditioned to return to the origin at time t=1 and to stay positive during time interval 0<t<1, is called the Bessel bridge with duration 1. We consider an N-particle system of such Bessel bridges conditioned never to collide with each other in 0<t<1, which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximum values of paths attained in the time interval t epsilon (0,1) are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane. For the outermost path, the distribution function of maximum value is exactly determined for general N. We show that the present N-path system of noncolliding Bessel bridges is realized as the positive-eigenvalue process of the 2Nx2N matrix-valued Brownian bridge in the symmetry class C. Using this fact, computer simulations are performed and numerical results on the N dependence of the maximum-value distributions of the inner paths are reported. The present work demonstrates that the extreme-value problems of noncolliding paths are related to random matrix theory, the representation theory of symmetry, and number theory.