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 6GGSee location on maps.ox**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