The top eigenvalue of uniformly random trees

Let ${\mathbf{T}}_n$ be a uniformly random tree with vertex set $[n]=\{1,\dots ,n\}$, let ${\mathrm{\Delta }}_{{\mathbf{T}}_n}$ be the largest vertex degree in ${\mathbf{T}}_n$, and let ${\lambda }_1({\mathbf{T}}_n),\dots ,{\lambda }_n({\mathbf{T}}_n)$ be the eigenvalues of its adjacency matrix, arranged in decreasing order. We prove that $\mathrm{\mid }{\lambda }_1({\mathbf{T}}_n)-\sqrt{{\mathrm{\Delta }}_{{\mathbf{T}}_n}}\mathrm{\mid }\to 0$ in expectation as $n\to \mathrm{\infty }$, and additionally prove probability tail bounds for $\mathrm{\mid }{\lambda }_1({\mathbf{T}}_n)-\sqrt{{\mathrm{\Delta }}_{{\mathbf{T}}_n}}\mathrm{\mid }$. Writing $a_n$ for any median of ${\mathrm{\Delta }}_{{\mathbf{T}}_n}$, we also prove that $\mathrm{\mid }{\lambda }_k({\mathbf{T}}_n)-\sqrt{a_n}\mathrm{\mid }\to 0$ in expectation, uniformly over $1\le k\le e^{{\log }^{\beta }(n)}$, for any fixed $\beta \in (0,1\mathrm{/}2)$. The proof is based on the trace method and thus on counting closed walks in a random tree. To this end, we develop novel combinatorial tools for encoding walks in trees that we expect will find other applications. In order to apply these tools, we show that uniformly random trees – after appropriate “surgery” – satisfy, with high probability, the properties required for the combinatorial bounds to be effective.