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Quantum walks are expected to provide useful algorithmic tools for quantum computation. This paper introduces absorbing probability and time of quantum walks and gives both numerical simulation results and theoretical analyses on Hadamard…
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 show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial…
Given a random walk a method is presented to produce a matrix of transition probabilities that is consistent with that random walk. The method is a kind of reverse application of the usual ergodicity and is tested by using a transition…
Random walks describe diffusion processes, where movement at every time step is restricted to only the neighbouring locations. We construct a quantum random walk algorithm, based on discretisation of the Dirac evolution operator inspired by…
In this paper, we investigate the distribution of $k$-free numbers in a class of $\alpha$-random walks on the integer lattice $\mathbb{Z}$. In these walks, the walker starts from a non-negative integer $r$ and moves to the right by $a$…
We consider the simple random walk (or P\'olya walk) on the one-dimensional lattice subject to stochastic resetting to the origin with probability $r$ at each time step. The focus is on the joint statistics of the numbers…
We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an…
On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the rectangle. For each integer $k$ we ask for…
One of the main goals of design theory is to classify, characterize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete…
We study the isolated partial Hadamard matrices, under the assumption that the entries are roots of unity, or more generally, under the assumption that the combinatorics comes from vanishing sums of roots of unity. We first review the…
We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided…
In random walk theory, it is customary to assume that a given walk is irreducible and/or aperiodic. While these prevailing assumptions make particularly tractable the analysis of random walks and help to highlight their diffusive nature,…
The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the…
We provide a new strategy to compute the exponential growth constant of enumeration sequences counting walks in lattice path models restricted to the quarter plane. The bounds arise by comparison with half-planes models. In many cases the…
A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random waker moves. On the other hand, the return probability of…
We define a random walk on the set of primitive points of $\mathbb{Z}^d$. We prove that for walks generated by measures satisfying mild conditions these walks are recurrent in a strong sense. That is, we show that the associated Markov…
We present numerical study of a model of quantum walk in periodic potential on the line. We take the simple view that different potentials affect differently the way the coin state of the walker is changed. For simplicity and definiteness,…
Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to…
Random walk on the set of irreducible representations of a finite group is investigated. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cutoff phenomenon is proved. As related results, an…