Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Wednesday, September 25, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Matan Harel (Courant Institute of Mathematical Sciences)

Localization in Random Geometric Graphs with Too Many Edges

Consider a random geometric graph G(n, r(n)), given by a connecting two vertices of a Poisson Point Process of intensity n on the unit torus whenever their distance is smaller than the parameter r(n). This model is conditioned on the rare event that the number of edges observed, |E|, is greater than [1 + delta (n)] times its expectation, producing a three parameter model. We show that, for delta constant or vanishing sufficiently slowly in n, with high probability, there exists a "giant clique" with almost all of the excess edges. Furthermore, if delta vanishes sufficiently quickly, the largest clique will be, at most, a constant multiple of the usual clique number of the unconditioned random geometric graph; roughly, all excess edges will result from the "standard" entropy-like effects of a vanishing Radon-Nykodyn derivative. Finally, we discuss progress in finding a phase transition function delta₀(n), so that when delta is much bigger than delta₀, the giant clique scenario holds, while delta much smaller than delta₀ implies the entropy scenario.

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