Distinguishability threshold for random geometric graphs

Abstract

The spherical random geometric graph $G(n,d,p)$ is obtained by sampling $n$ independent points uniformly on the unit sphere $\mathbb{S}^{d-1}\subseteq\mathbb{R}^d$ and joining pairs of points which are sufficiently close, where the threshold is chosen so that the edge probability is $p$. The central question related to this model, and to a broad class of other models, is the following: when does the underlying geometry affect the resulting graph in a way which makes it distinguishable from the Erdős--Rényi random graph $G(n,p)$, as measured in total variation distance? The precise answer to this question was conjectured by Bubeck, Ding, Eldan, and Rácz, who predicted that $G(n,d,p)$ and $G(n,p)$ are indistinguishable precisely when $d \gg n^3p^3(\log p^{-1})^3$, and provided a test for distinguishing these models in the low-dimensional regime. Although this conjecture attracted considerable attention from researchers in probability, theoretical computer science, and high-dimensional statistics, it was previously fully proved only in the constant-density case. In this paper, we resolve the distinguishability conjecture in the broad range $1/3 \geq p \geq n^{-1/5} \text{polylog}(n)$. The key ingredient of our proof is a stronger statement which gives a precise asymptotic formula for the probability that $G(n,d,p)$ realizes a prescribed graph $H$: above the conjectured threshold, this probability is at most $(1+o(1))$ times the corresponding probability for $G(n,p)$, with the signed triangle count of $H$ appearing as the leading correction term.

Disclosure

“ng us about Conjecture 1.1, Yuval Wigderson and Sahar Diskin for valuable comments which improved the presentation of this paper, and Nina Kamčev for insightful discussions about RGGs. Additionally, we would like to acknowledge the use of AI tools for polishing this paper, and note that all of the main proof ideas were entirely human-generated.”

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2026-08-23T19:05:10.691436Z

Structural counts