Random flat bundles and equidistribution

Each signature $\underset{‾}{\lambda }(n)=({\lambda }_1(n),\dots ,{\lambda }_n(n))$, where ${\lambda }_1(n)\ge \cdots \ge {\lambda }_n(n)$ are integers, gives an irreducible representation ${\pi }_{\underset{‾}{\lambda }(n)}:U(n)\to \text{GL}(V_{\underset{‾}{\lambda }(n)})$ of the unitary group $U(n)$. Suppose $X$ is a finite-area cusped hyperbolic surface, $\chi$ is a random surface representation in $\text{Hom}({\pi }_1(X),U(n))$ equipped with a Haar unitary probability measure, and $(\underset{‾}{\lambda }(n){)}_{n=1}^{\mathrm{\infty }}$ is a sequence of signatures. Let $\mathrm{\mid }\underset{‾}{\lambda }(n)\mathrm{\mid }:={\sum }_i\mathrm{\mid }{\lambda }_i(n)\mathrm{\mid }$. We show that there is an absolute constant $c>0$ such that if $0\mathrm{\ne }\mathrm{\mid }\underset{‾}{\lambda }(n)\mathrm{\mid }\le c\frac{\log n}{\log \log n}$ for sufficiently large $n$, then the Laplacians ${\mathrm{\Delta }}_{\chi ,\underset{‾}{\lambda }(n)}$ acting on sections of the flat unitary bundles associated to the surface representations $${\pi }_1(X)\stackrel{\chi }{\to }U(n)\stackrel{{\pi }_{\underset{‾}{\lambda }(n)}}{\to }\text{GL}(V_{\underset{‾}{\lambda }(n)})$$ have the property that for every $\epsilon >0$ $$\mathbb{P}[\chi :\inf \text{Spec}({\mathrm{\Delta }}_{\chi ,\underset{‾}{\lambda }(n)})\ge \frac{1}{4}-\epsilon ]\stackrel{n\to \mathrm{\infty }}{\to }1,$$ where $\text{Spec}({\mathrm{\Delta }}_{\chi ,\underset{‾}{\lambda }(n)})$ is the spectrum of ${\mathrm{\Delta }}_{\chi ,\underset{‾}{\lambda }(n)}$. A special case of this is that flat unitary bundles associated to $\chi :{\pi }_1(X)\to U(n)$ asymptotically almost surely as $n\to \mathrm{\infty }$ have least eigenvalue at least $\frac{1}{4}-\epsilon$, irrespective of the spectral gap of $X$ itself. This is proved using the Hide–Magee method. Using the spectral theorem above and proving a probabilistic prime geodesic theorem, we also obtain a probabilistic equidistribution theorem for the images under $\chi$ of geodesics of lengths dependent on the rank $n$.