  # Random Graph Geometry and Convergence

## Periodic Reporting for period 4 - RGGC (Random Graph Geometry and Convergence)

Reporting period: 2019-11-01 to 2021-04-30

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, and social & tranportation networks, this model treats all pairs of nodes equally. An important 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 deepen the understanding of the effect of the underlying geometry on the resulting random graphs.

The project proposal consisted of 4 themes. The second of those themes was the most relevant to GRG’s, and the one that turned out to be most active. This theme revolved around a new construction of GRG’s introduced by the PI, called `Group Walk Random Graphs’ (GWRG’s). These were extensively studied during the project. It was realised that some of those random graph models can also be obtained as an instance of (long-range) percolation on groups, a welcome connection to an already well-studied random graph model. The connection turned out to be very fruitful: by advancing techniques that we were using for the study of GWRG’s, we were able to solve one of the oldest open problems in percolation theory, namely a question of Kesten from 1981, asking whether the percolation density is an analytic function of the percolation parameter p. We answered this in the affirmative, using a combination of tools involving combinatorics, probability, and complex analysis. In particular, we introduced a new notion of `interfaces’ that has already found further applications.

The first theme of the project was also about random geometry, but the random graph models it considers are different: it considers a graph chosen uniformly at random among all graphs of a certain size with some restrictions, e.g. planarity or more generally absence of certain minors. One of the main objectives set out, namely the BS-convergence of the uniform planar map, was solved early on: we realised that this could be deduced from a combination of known results. The main technical result we obtained was that a uniform random graph chosen from a subcritical graph class BS-converges. Examples of such classes include well-studied classes defined via forbidden minors, e.g. cacti, series-parallel, and outplanar graphs. An unexpected spin-off of our collaboration with S. Wagner on this topic was a counterexample to Noy’s conjecture on subcritical graph classes: we proved that these cannot be characterized by forbidding a planar minor.

Other outcomes of the project (from Themes 3 & 4) include a construction of a diffusion on graph-like spaces, and extending the power-of-choice paradigm to random walks on graphs. Much of this work revolves around the Cover Time of random walk, a notion of great interest to both probabilists and computer scientists. Another unexpected outcome was an algorithm for computing a Riemann map from an arbitrary planar domain to a square. A visualisation of an example output (where the domain is a Julia set) it attached in Figures 1 & 2.

To summarize, the work carried out made important progress towards the original objectives, which also led to solving long-standing open problems and developing new techniques with further applications.
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 provided 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 one of the main open problems of percolation theory, dating back to 1981, asking whether, for Bernoulli percolation on the hypercubic lattice Z^d, the percolation density is an analytic function of the parameter. This is a question of both theoretical and physical importance, as it is related to the existence of phase transitions in statistical mechanics models. An analogous result for percolation on non-amenable Cayley-graphs has been obtained in later work by Hermon & Hutchcroft. Our result has attracted widespread attention as witnessed by many invitations to present it in international meetings, as well as the award of an LMS Whitehead Prize to the PI.

We proved that a uniform random graph chosen from a subcritical graph class converges in the Benjamini-Schramm sense.
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.

We have introduced the power-of-two-choices paradigm to random walk on a graph. We have shown that this leads to a significant speed-up of the cover time in several natural graph classes.

We have introduced a diffusion on graph-like spaces of finite total length, and proved that its cover time is finite.
All of the aforementioned results considerably advance the state of the art in percolation theory and enumerative combinatorics.