Conjecture: Twisted stable equilibria in the Kuramoto model on random geometric graphs

Establish whether, with high probability as the number of nodes n tends to infinity under the scaling regime epsilon -> 0 and epsilon^{d+2} n / log n -> infinity, for each integer k and each coordinate index 1 <= ell <= d there exists a stable equilibrium of the homogeneous Kuramoto model on random geometric graphs on the d-dimensional torus—constructed from n i.i.d. uniformly distributed points with edges between pairs at torus distance less than epsilon and weights w_{ij} = K(epsilon^{-1}(x_j^i - x_i))—that is uniformly close to the continuous twisted state u_{k,ell}(x) = k x · e_{ell} (mod 2pi).

Background

The paper studies the homogeneous Kuramoto model on random geometric graphs formed by n i.i.d. uniform points on the d-dimensional torus, connecting points within distance epsilon and assigning weights via a compactly supported radial kernel K. Under the scaling condition epsilon^{d+2} n / log n -> infinity, the authors prove that the Kuramoto dynamics converge (on compact time intervals) to the heat equation on the torus, whose equilibria are the family of continuous twisted states u_{k,ell}(x) = k x * e_{ell} (mod 2pi).

Motivated by this scaling limit and the stability of twisted states in the heat equation, the authors conjecture that the discrete Kuramoto system on these random geometric graphs admits stable equilibria that are close to those twisted states, with high probability as n grows. They note that this conjecture has been proved in dimension d = 1 in prior work and present their main theorem as evidence supporting the conjecture in higher dimensions.