Related papers: The Elephant random walk with gradually increasing…
We study how conditioning affects the long-term behavior of the Elephant Random Walk, focusing on its first return to the origin, namely, the time it takes to forget the training. When the walker is conditioned to be at position $n$ at time…
Elephant random walk is a kind of one-dimensional discrete-time random walk with infinite memory: For each step, with probability $\alpha$ the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she…
An analytic formulation of memory-possessing random walks introduced recently [Cressoni et al., Phys. Rev. Lett. 98, 070603 (2007) and Sch\"utz and Trimper, Phys. Rev. E 70, 045101 (2004)] for Alzheimer behavior and related phenomena is…
Random walks process on networks plays a fundamental role in understanding the importance of nodes and the similarity of them, which has been widely applied in PageRank, information retrieval, and community detection, etc. Individual's…
A step-reinforced random walk is a discrete-time stochastic process with long-range dependence. At each step, with a fixed probability $\alpha$, the so-called positively step-reinforced random walk repeats one of its previous steps, chosen…
In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant random walk (MARW). The scaling limit of the MARW is shown to exist in the diffusive, critical and superdiffusive regimes. We also establish the almost sure…
This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and…
We study a one-dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker. The feedback is quite efficient and lead to a non-diffusive walk.…
The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We…
The elephant random walk is a history-dependent random walk. We study a class of interacting elephant random walks. Our model includes the exclusion process as a special case. By means of Monte Carlo simulations and mean-field arguments, we…
Random walks (RWs) are fundamental stochastic processes with applications across physics, computer science, and information processing. A recent extension, the laser chaos decision-maker, employs chaotic time series from semiconductor…
Recently, the elephant random walk has attracted a lot of attentions. A wide range of literature is available for the asymptotic behavior of the process, such as the central limit theorems, functional limit theorems and the law of iterated…
We consider a two-elephant walking model in which the elephants interact dynamically. At each time step, each elephant determines its next move randomly based on its partner's past movements. We show that the asymptotic behavior of the…
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 study the long time behavior of the elephant random walk with stops, introduced by Kumar, Harbola and Lindenberg (2010), and establish the phase transition of the number of visited points up to time $n$, and the correlation between the…
Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where…
We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $\beta\in (-1,…
In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically…
The one-dimensional elephant random walk is a typical model of discrete-time random walk with step-reinforcement, and is introduced by Sch\"{u}tz and Trimper (2004). It has a parameter $\alpha \in (-1,1)$: The case $\alpha=0$ corresponds to…
In this paper, we study the number of moves in a multidimensional elephant random walk with stops. We establish several convergence results for the number of moves, including the law of large numbers and the law of iterated logarithm. Using…