# Testing thresholds for sparse random geometric graphs

Sidhanth Mohanty The University of Chicago

### Abstract

In the random geometric graph model, we identify each of our vertices with an independently and uniformly sampled vector from a high-dimensional unit sphere, and we connect pairs of vertices whose vectors are sufficiently close.
A fundamental question is: when is a random geometric graph a faithful model for its underlying geometry? As the dimension grows relative to the number of vertices, the edges in the graph become increasingly independent, and the underlying geometry becomes less apparent. This talk will cover some recent progress on this question: we show that in sparse random geometric graphs, if the dimension is at least polylogarithmic in the number of vertices, then the graphs are statistically indistinguishable from Erdős-Rényi graphs, i.e. the underlying geometry disappears.
Based on joint work with Siqi Liu, Tselil Schramm, and Elizabeth Yang (https://arxiv.org/abs/2111.11316).

Date

May 11, 2022 12:30 PM — 1:30 PM