2018/02/27 by Pierre‐Loïc Méliot, Méliot, Pierre-Loïc
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1802.10071
openalex publication_date 2018/02/27 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Let G be a compact Lie group, N\≥ 1 and L>0. The random geometric\ngraph on G is the random graph \Γ(N,L) whose vertices are N random\npoints g1,\…,gN chosen under the Haar measure of G, and whose edges\nare the pairs gi,gj with d(gi,gj)\≤ L, d being the distance\nassociated to the standard Riemannian structure on G. In this paper, we\ndescribe the asymptotic behavior of the spectrum of the adjacency matrix of\n\Γ(N,L), when N goes to infinity. If L is fixed and N \→ + \∞\n(Gaussian regime), then the largest eigenvalues of \Γ(N,L) converge after\nan appropriate renormalisation towards certain explicit linear combinations of\nvalues of Bessel functions. If L = O(N-\(1)/(\dim G)) and N \→\n+\∞ (Poissonian regime), then the random geometric graph \Γ(N,L)\nconverges in the local Benjamini-Schramm sense, which implies the weak\nconvergence in probability of the spectral measure of \Γ(N,L). In both\nsituations, the representation theory of the group G provides us with\ninformations on the limit of the spectrum, and conversely, the computation of\nthis limiting spectrum involves many classical tools from representation\ntheory: Weyl's character formula and the weight lattice in the Gaussian regime,\nand a degeneration of these objects in the Poissonian regime. The\nrepresentation theoretic approach allows one to understand precisely how the\ndegeneration from the Gaussian to the Poissonian regime occurs, and the article\nis written so as to highlight this degeneration phenomenon. In the Poissonian\nregime, this approach leads us to an algebraic conjecture on certain\nfunctionals of the irreducible representations of G.\n