Related papers: Deterministic walk in an excited random environmen…
Random walks serve as important tools for studying complex network structures, yet their dynamics in cases where transition probabilities are not static remain under explored and poorly understood. Here we study nonlinear random walks that…
We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed L\'evy process. In analogy to the…
Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks…
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.…
Random walks are powerful tools to analyze spatial-temporal patterns produced by living organisms ranging from cells to humans. At the same time, it is evident that these patterns are not completely random but are results of a convolution…
Coined discrete-time quantum walks are studied using simple deterministic dynamical systems as coins whose classical limit can range from being integrable to chaotic. It is shown that a Loschmidt echo like fidelity plays a central role and…
The following random process on $\Z^4$ is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk…
Very recently, a fundamental observable has been introduced and analyzed to quantify the exploration of random walks: the time $\tau_k$ required for a random walk to find a site that it never visited previously, when the walk has already…
Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during…
Many studies on animal and human movement patterns report the existence of scaling laws and power-law distributions. Whereas a number of random walk models have been proposed to explain observations, in many situations individuals actually…
Excited random walk is a process that has a drift to the right whenever it encounters a new vertex. The paper shows that in two dimensions it drifts to the right linearly in time.
There are presently two models for quantum walks on graphs. The "coined" walk uses discrete time steps, and contains, besides the particle making the walk, a second quantum system, the coin, that determines the direction in which the…
Random walks are the simplest way to explore or search a graph, and have revealed a very useful tool to investigate and characterize the structural properties of complex networks from the real world, e.g. they have been used to identify the…
We consider a random walk with death in $[-N,N]$ moving in a time dependent environment. The environment is a system of particles which describes a current flux from $N$ to $-N$. Its evolution is influenced by the presence of the random…
We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum…
A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution…
Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on $\mathbb{Z}^d$. The random walk on random walks model refers to a particular stochastic process on $\mathbb{Z}^d$…
The random walk in Dirichlet environment is a random walk in random environment where the transition probabilities are independent Dirichlet random variables. This random walk exhibits a property of statistical invariance by time-reversal…
Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in…
Motivated by the psychological literature on the "peak-end rule" for remembered experience, we perform an analysis within a random walk framework of a discrete choice model where agents' future choices depend on the peak memory of their…