A central limit theorem for prime geodesics on random surfaces of large genus

Technical Summary: A Central Limit Theorem for Prime Geodesics on Random Surfaces of Large Genus

Problem Statement
The paper addresses the statistical distribution of prime geodesics on closed hyperbolic surfaces of large genus ($g \geq 2$) sampled from the moduli space $\mathcal{M}_g$ with respect to the Weil–Petersson probability measure. Specifically, it investigates the behavior of the weighted count of prime geodesics, denoted $\Psi_M(X; H)$, within an interval $(X, X+H]$.

While the Prime Geodesic Theorem establishes the asymptotic mean $\Psi_M(X) \sim X$, the distribution of fluctuations in short or long intervals on random surfaces remains a central question. Previous work by Rudnick [Rud26] established the mean and variance of $\Psi_M(X; H)$ as $g \to \infty$ followed by $X \to \infty$, showing the mean is $\sim H$ and the variance is $\sim 2H \log X$. However, the distributional limit (e.g., Gaussian vs. Poisson) for intervals where $H$ is not restricted to the logarithmic scale ($H \asymp \log X$) was an open problem. The paper aims to prove a Central Limit Theorem (CLT) for $\Psi_M(X; H)$ in the regime where $H \leq X$ and $H / \log X \to \infty$.

Methodology
The proof adapts the argument used by Rudnick and Wigman [RW23] for the Laplace spectrum of random surfaces, applying it to the length spectrum. The methodology proceeds in three main stages:

1. Functional Representation:
   The weighted count $\Psi_M(X; H)$ is expressed as a functional $F_{X;H}$ of the length spectrum. Let $L_g(M)$ be the point process of lengths of primitive, unoriented geodesics on a surface $M$. The author defines a function $f_{X;H}(\ell)$ such that:
   $$\Psi_M(X; H) = F_{X;H}(L_g(M)) = \sum_{\ell \in L_g(M)} f_{X;H}(\ell)$$
   where $f_{X;H}$ is constructed using floor functions of $\log X$ and $\log(X+H)$.

2. Poisson Approximation:
   The paper relies on the convergence in distribution of the length spectrum of random surfaces to the Mirzakhani–Petri point process, $L_{MP}$, as the genus $g \to \infty$ [MP19]. The intensity measure of this limiting process is:
   $$\nu_{MP}(dx) = \frac{1}{2} \frac{(e^{x/2} - e^{-x/2})^2}{x} dx$$
   Using the Continuous Mapping Theorem, the author establishes that $\Psi_M(X; H)$ converges in distribution to a random variable $\Sigma_{MP}(X; H)$, defined as the functional $F_{X;H}$ applied to the Poisson point process $L_{MP}$.

3. Moment Estimation and Characteristic Function:
   To prove the Gaussian limit for $\Sigma_{MP}(X; H)$, the author calculates the first three moments of the function $f_{X;H}$ with respect to the intensity measure $\nu_{MP}$.

   • First Moment: $\int f_{X;H} d\nu_{MP} = H + O(HX^{-1/2})$.
   • Second Moment: $\int f_{X;H}^2 d\nu_{MP} = 2H \log X + O(\dots)$.
   • Third Moment: $\int f_{X;H}^3 d\nu_{MP} \ll H \log^2 X$.

   Using Campbell's formula for the characteristic function of a Poisson functional, the author expands the logarithm of the characteristic function of the normalized variable. By applying the Taylor expansion $e^{iu} = 1 + iu - u^2/2 + O(|u|^3)$ and substituting the moment estimates, the higher-order error terms are shown to vanish under the condition $H / \log X \to \infty$.

Key Results
The main result is Theorem 1.1, which states that for $1 \leq H \leq X$ satisfying $H / \log X \to \infty$ as $X \to \infty$, the normalized weighted count of prime geodesics converges in distribution to a standard Gaussian:
$$\frac{\Psi_M(X; H) - H}{\sqrt{2H \log X}} \xrightarrow{d} \mathcal{N}(0, 1)$$
as $g \to \infty$ followed by $X \to \infty$.

This result confirms the prediction that the distribution transitions from Poisson (observed in intervals of length $\asymp \log X$) to Gaussian as the interval length $H$ grows larger relative to $\log X$.

Significance and Claims
The paper claims to establish an analogue of the Montgomery–Soundararajan central limit theorem for primes [MS04] within the context of random hyperbolic surfaces.

• Extension of Range: Unlike previous results on random surfaces which were limited to intervals of length $\asymp \log X$ (yielding Poisson limits), this work extends the analysis to intervals that are not necessarily short, provided $H$ grows faster than $\log X$.
• Variance Factor: The paper notes that the variance term contains a factor of 2 ($2H \log X$) compared to the prime number case. The author attributes this to the two possible orientations of a closed geodesic, whereas primes lack such a distinction.
• Methodological Adaptation: The work demonstrates that the techniques developed for the Laplace spectrum (Rudnick–Wigman) are directly applicable to the length spectrum of prime geodesics, unifying the statistical understanding of these two spectral properties on random surfaces.

The paper does not propose new experimental applications or future conjectures beyond confirming the Gaussian limit in the specified regime and noting that a similar argument could theoretically recover the prime number CLT under specific uniform variants of the Hardy–Littlewood conjectures (Remark 1.3).