Large deviations for the CRT

Abstract

We establish large deviation principles both for the CRT in the Gromov-Hausdorff topology and for the CRT weighted by its mass measure in the Gromov-Hausdorff-Prohorov topology. In the first case, the rate function of a tree $\mathcal{T}$ is half the square of the length of $\mathcal{T}$. In the second case, the rate function of a pair $(\mathcal{T},μ)$ is half the integral with respect to the length measure of $\mathcal{T}$ of the inverse of the absolutely continuous part of $μ$.