Related papers: Maxima of Two Random Walks: Universal Statistics o…
We address the question of distance record-setting by a random walker in the presence of measurement error, $\delta$, and additive noise, $\gamma$ and show that the mean number of (upper) records up to $n$ steps still grows universally as…
We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the…
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…
We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution \phi(\eta) displaying asymmetric power law tails (i.e. \phi(\eta) \sim c/\eta^{\alpha +1} for large positive jumps and…
We consider two continuous-time generalizations of conservative random walks introduced in [J.Englander and S.Volkov (2022)], an orthogonal and a spherically-symmetrical one; the latter model is known as {\em random flights}. For both…
Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of…
In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x_0=0, x_1,x_2.... x_n} up to n steps where x_i represents the position at step i of a random walker hopping on a…
For the perimeter length $L_n$ and the area $A_n$ of the convex hull of the first $n$ steps of a planar random walk, this thesis study $n \to \infty$ mean and variance asymptotics and establish distributional limits. The results apply to…
L\'evy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the "jump lengths"---are drawn from an $\alpha$-stable jump length distribution with long-tailed, power-law asymptote. As a…
Levy walk (LW) process has been used as a simple model for describing anomalous diffusion in which the mean squared displacement of the walker grows non-linearly with time in contrast to the diffusive motion described by simple random walks…
We consider the general branching random walk under minimal assumptions, which in particular guarantee that the empirical particle distribution admits an almost sure central limit theorem. For such a process, we study the large time decay…
We study the large deviations of one-dimensional excited random walks. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions.…
We compute exactly the full distribution of the number $m$ of local minima in a one-dimensional landscape generated by a random walk or a L\'evy flight. We consider two different ensembles of landscapes, one with a fixed number of steps $N$…
We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is…
We investigate the statistics of records in a random sequence $\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence $x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$ steps that starts…
We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the Nth is random and its length equals lambda^{N-1}, with lambda<1. As lambda increases past a critical value lambda_c, the endpoint…
We have studied a random walk model based on majority rule. At a given instant, the moving direction of a cargo is determined by motor coordination mediated by a tug-of-war mechanism between two kinds of competing motor proteins. We have…
We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments. We give explicit upper and lower bounds for the rate function of the…
In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker,…
We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity $\lambda\in\mathbb{R}$. For ergodic shift-invariant environments, we show that the limiting…