2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72134Consider $n$ points distributed uniformly in $[0,1]^d$. Form a graph by connecting two points if their mutual distance is no greater than $r(n)$. This gives a random geometric graph, $\gnrn$, which is connected for appropriate $r(n)$. We show that the spectral measure of the transition matrix of the simple random walk (\abbr{srw}) on $\gnrn$ is concentrated, and in fact converges to that of the graph on the deterministic grid.ProbabilityStatistical MechanicsSpectral Theory60D05; 34L20text