On the distribution of eigenvalues of increasing trees

TL;DR

This paper establishes a central limit theorem for the distribution of fixed eigenvalues in random recursive trees, providing explicit constants for zero eigenvalues and extending results to Laplacian eigenvalues and binary increasing trees.

Contribution

It proves a central limit theorem for eigenvalue multiplicities in random recursive trees and related structures, with explicit constants for certain eigenvalues.

Findings

Eigenvalue multiplicities follow a CLT with linear mean and variance.

Explicit constants are derived for the zero eigenvalue case.

Results extend to Laplacian eigenvalues and binary increasing trees.

Abstract

We prove that the multiplicity of a fixed eigenvalue $\alpha$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to $\mu_{\alpha} n$ and $\sigma^2_{\alpha} n$ respectively. It is also shown that $\mu_{\alpha}$ and $\sigma^2_{\alpha}$ are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue $0$, the constants $\mu_0$ and $\sigma^2_0$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.