Related papers: Solving the Tower of Hanoi with Random Moves

When we want to simulate the realization of a symmetric simple random walk on $\mathbb Z^d$, we use $(2d)$-side fair dice to decide to which neighbor it jumps at each step if $d\geq 2$ or we simply use a fair coin when $d=1$. Assume that…

We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a 3-body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on…

We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions $f(\eta)$, including the case of L\'evy flights. We study the expected maximum ${\mathbb E}[M_n]$ of bridge RWs, i.e.,…

In the present work, we study random walks on complex networks subject to stochastic resetting when the resetting probability is node-dependent. Using a renewal approach, we derive the exact expressions of the stationary occupation…

Diffusion is modeled on the recently proposed Hanoi networks by studying the mean- square displacement of random walks with time, <r^2>~t^{2/d_w}. It is found that diffusion - the quintessential mode of transport throughout Nature -…

In certain instances an electric network transforms in natural ways by the addition or removal of an edge. This can have interesting consequences for random walks, in light of the known relationships between electric resistance and random…

This letter treats the quantum random walk on the line determined by a 2 times 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P, Q, R and S given by U. The…

This paper presents a more complete version than hitherto published of our explanation of a transition from regular to irregular motions and more generally of the nature of a certain kind of deterministic chaos. To this end we introduced a…

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular…

In this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step towards this objective, the following question is being…

We analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex, a.k.a. excited random walk. We show that it is recurrent with an expected number of returns of square-root log n.

This paper explores the mixing time of the random transposition walk on the symmetric group. While it has long been known that this walk mixes in order n*log(n) time, this result has not previously been attained using coupling. A coupling…

We study certain self-interacting walks on the set of integers, that choose to jump to the right or to the left randomly but influenced by the number of times they have previously jumped along the edges in the finite neighbourhood of their…

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study…

We investigate the hitting times of random walks on graphs, where a hitting time is defined as the number of steps required for a random walker to move from one node to another. While much of the existing literature focuses on calculating…

Quantum walks are well-known for their ballistic dispersion, traveling $\Theta(t)$ away in $t$ steps, which is quadratically faster than a classical random walk's diffusive spreading. In physical implementations of the walk, however, the…

In this paper, we present an overview of different types of random walk strategies with local and non-local transitions on undirected connected networks. We present a general approach to analyzing these strategies by defining the dynamics…

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing…

We use simple deterministic dynamical systems as coins in studying quantum walks. These dynamical systems can be chosen to display, in the classical limit, a range of behaviors from the integrable to chaotic, or deterministically random. As…

We introduce weighted Markovian graphs, a random walk model that decouples the transition dynamics of a Markov chain from (random) edge weights representing the cost of traversing each edge. This decoupling allows us to study the…