Non-vanishing of Class Group L-functions for Number Fields with a Small Regulator

Let $E\mathrm{/}\mathbb{Q}$ be a number field of degree $n$. We show that if $\mathrm{Reg}(E){\ll }_n\mathrm{\mid }\mathrm{Disc}(E){\mathrm{\mid }}^{1\mathrm{/}4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg_{n,\varepsilon} |\operatorname{Disc}|^{-1/4-\varepsilon}$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in ${\mathbf{P}\mathbf{G}\mathbf{L}}_n(\mathbb{Z})\mathrm{\backslash }{\mathbf{P}\mathbf{G}\mathbf{L}}_n(\mathbb{R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.