English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometricgraph on $G$ is the random graph $\Gamma(N,L)$ whose vertices are $N$ randompoints $g_1,\ldots,g_N$ chosen under the Haar measure of $G$, and whose edgesare the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distanceassociated to the standard Riemannian structure on $G$. In this paper, wedescribe the asymptotic behavior of the spectrum of the adjacency matrix of$\Gamma(N,L)$, when $N$ goes to infinity. If $L$ is fixed and $N \to + \infty$(Gaussian regime), then the largest eigenvalues of $\Gamma(N,L)$ converge afteran appropriate renormalisation towards certain explicit linear combinations ofvalues of Bessel functions. If $L = O(N^{-\frac{1}{\dim G}})$ and $N \to+\infty$ (Poissonian regime), then the random geometric graph $\Gamma(N,L)$converges in the local Benjamini-Schramm sense, which implies the weakconvergence in probability of the spectral measure of $\Gamma(N,L)$. In bothsituations, the representation theory of the group $G$ provides us withinformations on the limit of the spectrum, and conversely, the computation ofthis limiting spectrum involves many classical tools from representationtheory: Weyl's character formula and the weight lattice in the Gaussian regime,and a degeneration of these objects in the Poissonian regime. Therepresentation theoretic approach allows one to understand precisely how thedegeneration from the Gaussian to the Poissonian regime occurs, and the articleis written so as to highlight this degeneration phenomenon. In the Poissonianregime, this approach leads us to an algebraic conjecture on certainfunctionals of the irreducible representations of $G$.

Asymptotic representation theory and the spectrum of a random geometric graph on a compact Lie group