arXiv subjects

Yotam Maoz

Publications and source records attributed to Yotam Maoz.

2 recordsLinked to original sources

arXiv2025

Asymptotic independence for random permutations from surface groups

Let $X$ be an orientable hyperbolic surface of genus $g\geq 2$ with a marked point $o$, and let $Γ$ be an orientable hyperbolic surface group isomorphic to $π_{1}(X,o)$. Consider the space $\text{Hom}(Γ,S_{n})$ which corresponds to $n$-sheeted covers of $X$ with labeled fiber. Given $γ\inΓ$ and a uniformly random $ϕ\in\text{Hom}(Γ,S_{n})$, what is the expected number of fixed points of $ϕ(γ)$? Formally, let $F_{n}(γ)$ denote the number of fixed points of $ϕ(γ)$ for a uniformly random $ϕ\in\text{Hom}(Γ,S_{n})$. We think of $F_{n}(γ)$ as a random variable on the space $\text{Hom}(Γ,S_{n})$. We show that an arbitrary fixed number of products of the variables $F_{n}(γ)$ are asymptotically independent as $n\to\infty$ when there are no obvious obstructions. We also determine the limiting distribution of such products. Additionally, we examine short cycle statistics in random permutations of the form $ϕ(γ)$ for a uniformly random $ϕ\in\text{Hom}(Γ,S_{n})$. We show a similar asymptotic independence result and determine the limiting distribution.

math.GR↗

arXiv2025

Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces -- A central limit theorem and almost sure RMT statistics

We study smooth linear spectral statistics of twisted Laplacians on random $n$-covers of a fixed compact hyperbolic surface $X$. We consider two aspects of such statistics. The first is the fluctuations of such statistics in a small energy window around a fixed energy level when averaged over the space of all degree $n$ covers of $X$. The second is the energy variance of a typical surface. In the first case, we show a central limit theorem. Specifically, we show that the distribution of such fluctuations tends to a Gaussian with variance given by the corresponding quantity for the Gaussian Orthogonal/Unitary Ensemble (GOE/GUE). In the second case, we show that the energy variance of a typical random $n$-cover is that of the GOE/GUE. In both cases, we consider a double limit where first we let $n$, the covering degree, go to $\infty$ then let $L\to \infty$ where $1/L$ is the window length.

math.SP↗