Random flat bundles and equidistribution

Abstract

Each signature $\underline{\lambda}(n)=(\lambda_1(n),\dots,\lambda_n(n))$, where $\lambda_1(n)\geq\dots\geq\lambda_n(n)$ are integers, gives an irreducible representation $\pi_{\underline{\lambda}(n)}:U(n)\rightarrow\text{GL}(V_{\underline{\lambda}(n)})$ 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 $(\underline{\lambda}(n))_{n=1}^{\infty}$ is a sequence of signatures. Let $|\underline{\lambda}(n)|:=\sum_i|\lambda_i(n)|$. We show that there is an absolute constant $c>0$ such that if $0\neq |\underline{\lambda}(n)|\leq c\frac{\log n}{\log\log n}$ for sufficiently large $n$, then the Laplacians $\Delta_{\chi,\underline{\lambda}(n)}$ acting on sections of the flat unitary bundles associated to the surface representations \[\pi_1(X)\xrightarrow{\chi} U(n)\xrightarrow{\pi_{\underline{\lambda}(n)}}\text{GL}(V_{\underline{\lambda}(n)})\] have the property that for every $\varepsilon>0$ \[\mathbb{P}\left[\chi:\inf\text{Spec}(\Delta_{\chi,\underline{\lambda}(n)})\geq\frac{1}{4}-\varepsilon\right]\xrightarrow{n\rightarrow\infty}1,\] where $\text{Spec}(\Delta_{\chi,\underline{\lambda}(n)})$ is the spectrum of $\Delta_{\chi,\underline{\lambda}(n)}$. A special case of this is that flat unitary bundles associated to $\chi:\pi_1(X)\rightarrow U(n)$ asymptotically almost surely as $n\rightarrow\infty$ have least eigenvalue at least $\frac{1}{4}-\varepsilon$, 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 $n$.