2026/07/24 by Alexis Imbert
Mathematics · #math.PR #math.GR
Let Λ be a finitely generated abelian group and Γ=\mathbb Z*d, we study the Cayley graph of the wreath product G=Λ\wrΓ with natural set of generators and their inverse S. First, we establish a random matrix model XN=∑s X(s)N where the sum is indexed by the set S. As the size N of the matrices goes to infinity, the traffic distribution of the X(s)N's converges to that of the image of these generators in the reduced C^*-algebra of G. In particular, the spectral measure of XN converges toward that of the Cayley graph of G with generators S. Moreover, in the case Γ=\mathbb Z, we establish a central limit theorem for linear statistics of this random matrix model. Then, we exhibit a formula for the asymptotic R-transform and derive the second-order distribution of the limit of XN in terms of its limiting first-order distribution.