Top eigenvalues of random trees
Let $T_n$ be a uniformly random tree with vertex set $[n]={1,…,n}$. Let $Delta_n$ be the largest vertex degree in $T_n$ and let $\lambda_n$ be the largest eigenvalue of $T_n$. We show that $|\lambda_n-\sqrt{\Delta_n}| \to 0$ in probability as $n \to \infty$. The key ingredients of our proof are (a) the trace method, (b) a rewiring lemma that allows us to “clean up” our tree without decreasing its top eigenvalue, and© some careful combinatorial arguments.

This is extremely slow joint work with Roberto Imbuzeiro Oliveira and Gabor Lugosi, but we hope to finally finish our write-up in the coming weeks.
Date: 22 January 2024, 14:00 (Monday, 2nd week, Hilary 2024)
Venue: Mathematical Institute, Woodstock Road OX2 6GG
Venue Details: L5
Speaker: Louigi Addario-Berry (McGill)
Organising department: Department of Statistics
Part of: Probability seminar
Booking required?: Not required
Audience: Public
Editors: James Martin, Julien Berestycki