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Reinforced Galton-Watson processes have been introduced in arxiv:2306.02476 as population models with non-overlapping generations, such that reproduction events along genealogical lines can be repeated at random. We investigate here some of…
Under suitable moment assumptions, we show that a genuinely d-dimensional step-reinforced random walk undergoes a phase transition between recurrence and transience in dimensions $d=1,2$, and that it is transient for all reinforcement…
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 study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., non-random) case, we allow any…
Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for…
We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the…
We consider asymptotics for the maximum of a modulated random walk whose increments $\xi_n^{X_n}$ are heavy-tailed. Of particular interest is the case where the modulating process $X$ is regenerative. Here we study also the maximum of the…
We present a continuous time generalization of a random walk with complete memory of its history [Phys. Rev. E 70, 045101(R) (2004)] and derive exact expressions for the first four moments of the distribution of displacement when the number…
Let $\{\mm_n, n=0,1,...\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,...$ let $W_n$ be the moment generating function of $\mm_n$ normalized by its mean.…
We provide Monte Carlo estimates of the scaling of the length $L_{n}$ of the longest increasing subsequences of $n$-steps random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate…
We present a model for a random walk with memory, phenomenologically inspired in a biological system. The walker has the capacity to remember the time of the last visit to each site and the step taken from there. This memory affects the…
We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16].…
We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_n\}$ and $\{W_n\}$ be two centered, weakly dependent random walks. We establish that…
Random walkers characterized by random positions and random velocities lead to normal diffusion. A random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a…
We study the mixing time of a non-Markovian process, the step-reinforced random walk (SRRW) on a finite group. This process differs from a classical random walk in that at each integer time, with probability $\alpha$ the next step is chosen…
A noise reinforced Brownian motion is a centered Gaussian process $\hat B=(\hat B(t))_{t\geq 0}$ with covariance $E(\hat B(t)\hat B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t,$ where $p\in(0,1/2)$ is a reinforcement…
We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which…
We study an exactly solvable random walk model with long-range memory on arbitrary networks. The walker performs unbiased random steps to nearest-neighbor nodes and intermittently resets to previously visited nodes in a preferential way,…
We present the first rigorous quantitative analysis of once-reinforced random walks (ORRW) on general graphs, based on a novel change of measure formula.~This enables us to prove large deviations estimates for the range of the walk to have…
The random walk process underlies the description of a large number of real world phenomena. Here we provide the study of random walk processes in time varying networks in the regime of time-scale mixing; i.e. when the network connectivity…