Related papers: Planar maps, random walks and circle packing
Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and…
This is a report of a scientific project carried out at Ecole Centrale Casablanca in 2019. This work is an entry into the world of Random Walk in a Random Environment (RWRE). We will discuss some intuitive concepts such as recurrence,…
We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared…
We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common…
These lecture notes were prepared for the 2018 Summer School on `Active Matter and Non-equilibrium Statistical Physics' at l'\'{E}cole de Physique des Houches. They survey metabolic activity across a wide range of living organisms, and…
These are lectures notes for a 4h30 mini-course held in Ulaanbaatar, National University of Mongolia, August 5-7th 2015, at the summer school "Stochastic Processes and Applications". It aims at presenting an introduction to basic results of…
A random walk is known as a random process which describes a path including a succession of random steps in the mathematical space. It has increasingly been popular in various disciplines such as mathematics and computer science.…
Simple random walks are a basic staple of the foundation of probability theory and form the building block of many useful and complex stochastic processes. In this paper we study a natural generalization of the random walk to a process in…
Expected urban population doubling calls for a compelling theory of the city. Random walks and diffusions defined on spatial city graphs spot hidden areas of geographical isolation in the urban landscape going downhill. First--passage time…
We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are…
These are the notes for a series of lectures given at the Clay Mathematical Institute Summer School in Buzios, July 11 - August 7, 2010. We review some of the recent aspects of scaling limits of random trees and planar maps, in particular…
Motivated by a problem arising from pharmaceutical science [B. Baeumer et al., Discr. Contin. Dyn. Sys. B 12], we study random walks on the contact graph of a bidisperse random sphere packing. For a random walk on the unweighted graph that…
In a previous paper we introduced examples of Hamiltonian mappings with phase space structures resembling circle packings. We now concentrate on one particular mapping and present numerical evidence which supports the conjecture that the…
We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited.…
Many seemingly disparate Markov chains are unified when viewed as random walks on the set of chambers of a hyperplane arrangement. These include the Tsetlin library of theoretical computer science and various shuffling schemes. If only…
Lecture Notes. Minicourse given at the workshop "Activated Random Walks, DLA, and related topics" at IM\'eRA-Marseille, March 2015.
These lecture notes are a guided tour through the fascinating world of polymer chains interacting with themselves and/or with their environment. The focus is on the mathematical description of a number of physical and chemical phenomena,…
This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general…
A complex web of roads, walkways and public transport systems can hide areas of geographical isolation very difficult to analyze. Random walks are used to spot the structural details of urban fabric.
This is an introductory level survey of some topics from a new branch of fractal analysis -- the theory of self-similar groups. We discuss recent works on random walks on self-similar groups and their applications to the problem of…