Quasi-random walks show similar features as standard random walks, but with much less randomness. We utilize this established model from discrete mathematics and show how agents carrying out quasi-random walks can be used for image transition and animation. The key idea is to generalize the notion of quasi-random walks and let a set of autonomous agents perform quasi-random walks painting an im...

We show how to apply the canonical path method to a non-reversible Markov chain with no holding probability: a random walk used in Pollard’s Rho algorithm for discrete logarithm. This is used to show that the Pollard Rho method for finding the discrete logarithm on a cyclic group G requires O( √ |G| (log |G|)3/2) steps until a collision occurs and discrete logarithm is possibly found, not far f...

We consider discrete random magnetic Laplacians in the plane and discrete random electromagnetic Laplacians in higher dimensions. The existence of these objects relies on a theorem of Feldman-Moore which was generalized by Lind to the nonabelian case. For example, it allows to realize ergodic Schrr odinger operators with stationary independent magnetic elds on discrete two dimensional lattices ...

We analyze the classical Pollard’s Rho algorithm for finding the discrete logarithm in a cyclic group G. We prove that, with high probability, a collision occurs and the discrete logarithm is potentially found in O( √ |G| log |G| log log |G|) steps, not far from the widely conjectured value of Θ( √ |G|). This improves upon a recent result of Miller–Venkatesan which showed an upper bound of O( √...

Let S, S be independent simple random walks in Z (d = 2, 3) started at the origin. We construct two-sided random walk paths conditioned that S[0,∞) ∩ S[1,∞) = ∅ by showing the existence of the following limit: lim n→∞ P (· | S[0, τ(n)] ∩ S[1, τ(n)] = ∅), where τ (n) = inf{k ≥ 0 : |S(k)| ≥ n}. Moreover, we give upper bounds of the rate of the convergence. These are discrete analogues of results ...

We study a system of simple random walks on graphs, known as frog model. This model can be described as follows: There are active and sleeping particles living on some graph G. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1 − p. When an active particle hits a sleeping particle, the latter becomes active. Phase transit...

The loop-erased random walk (LERW) in Zd, d ≥ 2, dimensions is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.