arXiv subjects

Perla Sousi

Publications and source records attributed to Perla Sousi.

40 records · Page 3Linked to original sources

arXiv2010

Mobile Geometric Graphs: Detection, Coverage and Percolation

We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of the graph be a Poisson point process in R^d with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes if their distance is at most r. We study three features in this model: detection (the time until a target point---fixed or moving---is within distance r from some node of the graph), coverage (the time until all points inside a finite box are detected by the graph), and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these features by combining ideas from stochastic geometry, coupling and multi-scale analysis.

math.PR↗

arXiv2010

Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

By the Cameron--Martin theorem, if a function $f$ is in the Dirichlet space $D$, then $B+f$ has the same a.s. properties as standard Brownian motion, $B$. In this paper we examine properties of $B+f$ when $f \notin D$. We start by establishing a general 0-1 law, which in particular implies that for any fixed $f$, the Hausdorff dimension of the image and the graph of $B+f$ are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function $f$ is Hölder$(1/2)$, then $B+f$ is intersection equivalent to $B$. Moreover, $B+f$ has double points a.s. in dimensions $d\le 3$, while in $d\ge 4$ it does not. We also give examples of functions which are Hölder with exponent less than $1/2$, that yield double points in dimensions greater than 4. Finally, we show that for $d \ge 2$, the Hausdorff dimension of the image of $B+f$ is a.s. at least the maximum of 2 and the dimension of the image of $f$.

math.PR↗

arXiv2010

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 $d \ge 19$ 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.

math.PR↗

arXiv2008

A stochastic epidemiological model and a deterministic limit for BitTorrent-like peer-to-peer file-sharing networks

In this paper, 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 prove that the fluid limits of a scaled Markov model of this system are of the coagulation form, special cases of which are well-known epidemiological (SIR) models. In addition, Lyapunov stability and settling-time results are explored. We derive conditions under which the BitTorrent incentives under consideration result in shorter mean file-acquisition times for peers compared to client-server (single chunk) systems. Finally, a diffusion approximation is given and some open questions are discussed.

math.PR↗