Random flat bundles and equidistribution

Abstract

Each signature $\underline{\lambda }(n)=({\lambda }_1(n),\dots ,{\lambda }_n(n))$, where ${\lambda }_1(n)\ge \cdots \ge {\lambda }_n(n)$ are integers, gives an irreducible representation ${\pi }_{\underline{\lambda }(n)}:U(n)\to \text{GL}(V_{\underline{\lambda }(n)})$ 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 $0\ne |\underline{\lambda }(n)|\le c\frac{\log n}{\log \log n}$ 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 \[π_1(X)\xrightarrowχ U(n)\xrightarrow{π_{\underlineλ(n)}}\text{GL}(V_{\underlineλ(n)})\] have the property that for every $\epsilon >0$ \[\mathbb{P}\left[χ:\inf\text{Spec}(Δ_{χ,\underlineλ(n)})\geq\frac{1}{4}-\varepsilon\right]\xrightarrow{n\rightarrow\infty}1,\] 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 $\chi :{\pi }_1(X)\to U(n)$ asymptotically almost surely as $n\to \infty$ 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 $n$.

arXiv commentExposition improved. Comments are very welcome