Related papers: Self-similarity and random walks
Self-similar sets with open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples…
A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and…
A new method, called the method of self-similar approximants, and its recent developments are described. The method is based on the ideas of renormalization group theory and optimal control theory. It allows for the effective extrapolation…
Self-projective sets are natural fractal sets which describe the action of a semigroup of matrices on projective space. In recent years there has been growing interest in studying the dimension theory of self-projective sets, as well as…
The fourfold research proposal regards in particular: critical oriented percolation; random walk limit laws; neural networks with long-range connections; the ant in a labyrinth.
Fractals are self-similar recursive structures that have been used in modeling several real world processes. In this work we study how "fractal-like" processes arise in a prediction game where an adversary is generating a sequence of bits…
In this paper we consider finitary symmetric random walks on groups. We construct new possible asymptotics for the drift. We show that the drift can be very close to linear ant yet sublinear. We also give estimates for entropy growth of…
Quantum walks on graphs can model physical processes and serve as efficient tools in quantum information theory. Once we admit random variations in the connectivity of the underlying graph, we arrive at the problem of percolation, where the…
This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking…
We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov…
The purpose of this paper is to suggest additional aspects of social psychology that could help when making sense of autonomous agile teams. To make use of well-tested theories in social psychology and instead see how they replicated and…
The notion of the abundance of fractals is critically re-examined in light of surprising data regarding the scaling range in empirical reports on fractality.
We present a new approach of topology biased random walks for undirected networks. We focus on a one parameter family of biases and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of…
In this note, we give an original convergence result for products of independent random elements of motion group. Then we consider dynamic random walks which are inhomogeneous Markov chains whose transition probability of each step is, in…
There have been extensive studies of a random walk among a field of immobile traps (or obstacles), where one is interested in the probability of survival as well as the law of the random walk conditioned on its survival up to time $t$. In…
We give a precise definition of ``generic-case complexity'' and show that for a very large class of finitely generated groups the classical decision problems of group theory - the word, conjugacy and membership problems - all have…
This book is concerned with analytic approaches of studying groups and their actions. Much attention is devoted to the study of amenability and Kazhdan's property (T), which are perhaps the most important analytic properties of a group, but…
We consider self-avoiding walks (SAWs) on the backbone of percolation clusters in space dimensions d=2, 3, 4. Applying numerical simulations, we show that the whole multifractal spectrum of singularities emerges in exploring the…
This is a review of results obtained by the author concerning the relation between conformally invariant random loops and conformal field theory. This review also attempts to provide a physical context in which to interpret these results by…
A particle subject to successive, random displacements is said to execute a random walk (in position or some other coordinate). The mathematical properties of random walks have been very thoroughly investigated, and the model is used in…