Title: Clustering, Percolation and directionally convex ordering of point processes
Speaker: Prof. D. Yogeshwaran Ecole Normale Superieure - INRIA
Date: 13 July 2011
Time: 2:30-3:30 p.m.
Venue: Department of Mathematics, Indian Institute of Science, Lecture Hall I
Heuristics indicate that point processes exhibiting clustering of points have larger critical radii for the percolation of their continuum percolation models than spatially homogeneous point processes. I will explain why the dcx ordering of point processes is suitable to compare their clustering tendencies. Hence, it is tempting to conjecture that the critical radius is increasing in dcx order. We will prove the conjecture for some non-standard critical radii; however it is false for the standard critical radii. I will discuss the implications of these results. A powerful implication is that point processes dcx-smaller than a homogeneous Poisson point process admit uniformly non-degenerate lower and upper bounds on their critical radii. In fact, all the above results hold under weaker assumptions of ordering of moment measures and void probabilities of the point processes. Examples of point processes comparable to Poisson point processes in this weaker sense include determinantal and permanental point processes with trace-class integral kernels. Perturbed lattices are the most general examples of dcx sub- and super-Poisson point processes. More generally, we show that point processes dcx-smaller than a homogeneous Poisson point process exhibit phase transitions in certain percolation models based on the level-sets of additive shot-noise fields of these point process. Examples of such models are k-percolation and SINR-percolation. This is a joint work with Bartek Blaszczyszyn.