Perla Sousi

Collisions of random walks

A recurrent graph G has the infinite collision property if two independent random walks on G, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in Z d with d ≥ 19 and the uniform spanning tree in Z 2 all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property.

A Stochastic Epidemiological Model and a Deterministic Limit for BitTorrent-Like Peer-to-Peer File-Sharing Networks

We propose a stochastic model for a file-sharing peer-to-peer network which resembles the popular BitTorrent system: large files are split into chunks and a peer can download or swap from another peer only one chunk at a time. We exhibit the fluid and diffusion limits of a scaled Markov model of this system and look at possible uses of them to draw practical conclusions.

Sensitivity of Mixing Times in Eulerian Digraphs

Let

Minkowski dimension of Brownian motion with drift

X

Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

be a lazy random walk on a graph

Total variation cutoff in a tree

G

Intersection and mixing times for reversible chains

. If

Martingale defocusing and transience of a self-interacting random walk

G

Random walks colliding before getting trapped

is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle

Mobile geometric graphs: detection, coverage and percolation

\mathbb{Z}_n