Random flat bundles and equidistribution

Abstract

Each signature $\underlineλ(n)=(λ_1(n),\dots,λ_n(n))$, where $λ_1(n)\geq\dots\geqλ_n(n)$ are integers, gives an irreducible representation $π_{\underlineλ(n)}:U(n)\rightarrow\text{GL}(V_{\underlineλ(n)})$ of the unitary group $U(n)$. Suppose $X$ is a finite-area cusped hyperbolic surface, $χ$ is a random surface representation in $\text{Hom}(π_1(X),U(n))$ equipped with a Haar unitary probability measure, and $(\underlineλ(n))_{n=1}^{\infty}$ is a sequence of signatures. Let $|\underlineλ(n)|:=\sum_i|λ_i(n)|$. We show that there is an absolute constant $c>0$ such that if $0\neq |\underlineλ(n)|\leq c\frac{\log n}{\log\log n}$ for sufficiently large $n$, then the Laplacians $Δ_{χ,\underlineλ(n)}$ 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 $\varepsilon>0$ \[\mathbb{P}\left[χ:\inf\text{Spec}(Δ_{χ,\underlineλ(n)})\geq\frac{1}{4}-\varepsilon\right]\xrightarrow{n\rightarrow\infty}1,\] where $\text{Spec}(Δ_{χ,\underlineλ(n)})$ is the spectrum of $Δ_{χ,\underlineλ(n)}$. A special case of this is that flat unitary bundles associated to $χ:π_1(X)\rightarrow U(n)$ 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 $χ$ of geodesics of lengths dependent on the rank $n$.