Omer Bobrowski : Phase transitions in random Cech complexes

A simplicial complex is a collection of vertices, edges, triangles, and simplexes of higher dimensions, and one can think of it as a generalization of a graph. Given a random set of points P in a metric space and a real number r > 0, one can create a simplicial complex by looking at the balls of radius r around the points in P, and adding a k-dimensional face for every subset of k+1 balls that has a nonempty intersection. This construction produces a random topological space known as the Ã?ech complex - C(P,r). We wish to study the homology of this space, more specifically - its Betti numbers - the number of connected components and 'holes' or 'cycles'. In this talk we discuss the limiting behavior of the random Ã?ech complex as the number of points in P goes to infinity and the radius r goes to zero. We show that the limiting behavior exhibits multiple phase transitions at different levels, depending on the rate at which the radius goes to zero. We present the different regimes and phase transitions discovered so far, and observe the nicely ordered fashion in which cycles of different dimensions appear and vanish.

• Category: Probability
• Duration: 01:34:10
• Date:  March 20, 2014 at 4:15 PM
• Views: 106
• Tags:   seminar, Probability Seminar