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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 3 - RGGC (Random Graph Geometry and Convergence)

Teaser

Summary

This is a project in pure mathematics, concentrating mainly on geometric random graphs and related objects. There is already a vast literature on random graphs, but it focuses almost exclusively on a particular model (Erdos-Renyi). Unlike what happens in real-world networks, e.g. computer grids, social & tranportation networks, this model treats all pairs of nodes equally. A big challenge is to come up with appropriate models that simulate the behaviour of real-world networks, and one of the ideas is that such networks have spatial behaviour: nearby nodes are more likely to be connected. Geometric random graph (GRG) models are the state-of-the-art way to take spatial behaviour into account. Appart from the practical importance, GRG\'s can be thought of as a theoretical tool in mathematics as well. For example, percolation on groups is a very active modern topic studying the interaction of probability and geometry. One of the main aims of the project is to provide a unified treatment of a wide variety of GRG models. Another is to deapen the understanding of the effect of the underlying geometry on the resulting random graphs.

Work performed

We have proved that the model of geometric random graphs introduced by the project can capture percolation models on groups as a special case. In the particular case where the underlying geometry is a tree, we provide upper and lower bounds for the expected size of the random graphs.

A far-reaching side outcome of the study of said expected size, was to settle, in the 2-dimensional case, one of the main open problems of percolation theory, dating back to 1982, asking whether the percolation density is an analytic function of the parameter. This is a question of physical importance, as it is related to the existence of phase transitions in statistical mechanics models.

A further important outcome was to disprove the conjecture of Noy on subcritical graph families, one of the best known recent problems in enumerative combinatorics.

Final results

The aforementioned successes considerably advance the state of the art in percolation theory and enumerative combinatorics. In the near future we expect to be able to extend our analyticity results beyond the 2-dimensional case. Moreover, we developed new techniques that apply complex analysis in the study of random graphs, and we expect them to find further applications. We also expect to narrow the gap between upper and lower bounds in the aforementioned random graphs. We expect to find further percolation models that arise as limits of convergent sequences of geometric random graphs.