On the Rigidity of Random Graphs in high-dimensional spaces

TL;DR

This paper investigates the maximum dimension for which Erdős-Rényi random graphs are rigid, revealing two distinct regimes separated by a critical probability related to the graph's degree distribution.

Contribution

It establishes the asymptotic behavior of the rigidity dimension in different probability regimes, confirming a conjecture for the high-probability regime.

Findings

For p below the critical threshold, rigidity dimension equals the minimum degree.

For p above the threshold but below n^{-1/2}, rigidity dimension is approximately half the expected degree.

The results identify two regimes of rigidity separated by a critical probability p_c.

Abstract

We study the maximum dimension $d=d(n,p)$ for which an Erd\H{o}s-R\'enyi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point where the graph's minimum degree exceeds half its average degree. We show that if $p < (1-\varepsilon)p_c $, then $d(n,p)$ is asymptotically almost surely (a.a.s.) equal to the minimum degree of $G(n,p)$. In contrast, if $p_c \leq p = o(n^{-1/2}) $ then $d(n,p) $ is a.a.s. equal to $(1/2 + o(1))np$. The second result confirms, in this regime, a conjecture of Krivelevich, Lew, and Michaeli.