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

Let $E/\mathbb{Q}$ be a number field of degree $n$. We show that if $\operatorname{Reg}(E)\ll_n |\operatorname{Disc}(E)|^{1/4}$ 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 $\mathbf{PGL}_n(\mathbb{Z})\backslash\mathbf{PGL}_n(\mathbb{R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.