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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
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
