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Closed geodesics in homology classes on random hyperbolic surfaces of large genus

This paper demonstrates that on random hyperbolic surfaces of large genus, the variance of the weighted counting function for closed geodesics in homology classes modulo $q$ scales asymptotically as $X\log X$ (with a factor of two for $q=2$ ), a result that diverges from Hooley's conjecture for primes and is explained through an analogy with function fields over finite fields.

Original authors: Zeev Rudnick

Published 2026-07-08

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Zeev Rudnick

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world made of hyperbolic surfaces. Think of these not as flat sheets of paper, but as complex, saddle-shaped landscapes with many "holes" or "handles" (like a pretzel with many loops). Mathematicians call the number of these holes the genus. In this paper, the author, Ze'ev Rudnick, is interested in what happens when these surfaces have a massive number of holes (a "large genus").

Here is the story of the paper, broken down into simple concepts:

1. The Players: Geodesics and Homology

The Geodesics: On these bumpy surfaces, the shortest paths between points are called geodesics. If you walk along one and eventually return to your starting point without turning, you've traced a closed geodesic. Think of these as the "prime numbers" of the surface world.
The Homology (The "Address"): Every time a geodesic loops around the surface, it winds around the holes. Mathematicians can assign a "homology class" to each geodesic, which is essentially a coordinate or an "address" describing how many times it winds around each handle.
The Modulus ( $q$ ): The author asks: "If we group all these geodesics by their addresses, do they distribute evenly?" Specifically, if we look at addresses modulo a number $q$ (like looking at remainders when dividing by 3, 4, or 5), how are the geodesics spread out?

2. The Experiment: Rolling the Dice on Surfaces

Instead of studying just one specific surface, the author studies random surfaces. Imagine a giant bag filled with every possible shape of a hyperbolic surface with $g$ holes. The author picks one at random (using a specific mathematical probability rule called the Weil–Petersson measure) and counts the geodesics.

The goal is to measure the variance (or "fluctuation").

The Ideal Scenario: If geodesics were perfectly random, they would be distributed exactly evenly among all the different "addresses" (homology classes).
The Reality: There are always bumps and wiggles. Some addresses get a few extra geodesics, others get a few fewer. The paper calculates how big these wiggles are when the genus $g$ becomes very large.

3. The Big Discovery: The "X log X" Surprise

The author compares the size of these wiggles (the variance) against the total number of geodesics ( $X$ ).

The Expectation (The "Prime" Analogy): In the world of regular numbers (arithmetic), there is a famous guess called Hooley's Conjecture. It suggests that if you look at prime numbers in different remainders (like primes ending in 1, 3, 7, or 9), the size of the wiggles should be roughly proportional to $X \log q$ (where $q$ is the modulus).
The Result: Rudnick finds that for these random hyperbolic surfaces, the wiggles are much bigger.
- For most moduli ( $q > 2$ ), the variance is proportional to $X \log X$ .
- For the specific case of $q=2$ , it's $2X \log X$ .

The Analogy:
Imagine you are counting marbles in jars.

Hooley's Conjecture (Number Theory): If you sort marbles by color, the "noise" or unevenness in the counts depends on how many colors you have ( $q$ ). More colors = slightly more noise.
Rudnick's Finding (Surfaces): On these random surfaces, the noise depends on the total number of marbles ( $X$ ), not just the number of colors. It's as if the sheer volume of marbles creates a chaotic storm that makes the distribution much wilder than anyone expected.

4. Why is it different? (The Explanation)

The paper asks: Why don't these surfaces behave like regular numbers?

The author suggests the answer lies in the "dimension" of the problem.

Regular Numbers: Think of them as a 1-dimensional line.
Function Fields (Polynomials): These are like 2-dimensional grids.
Hyperbolic Surfaces: The author argues that as the genus (number of holes) gets huge, the surface behaves like a system with infinite dimensions.

In the world of high-dimensional mathematics (specifically involving something called "GL(d)" where $d$ is the dimension), as the dimension goes to infinity, the "noise" stops caring about the modulus ( $q$ ) and starts caring about the total size ( $X$ ). The paper suggests that random hyperbolic surfaces of large genus are the geometric equivalent of these infinite-dimensional systems.

Summary

The paper proves that on random, highly complex surfaces, the distribution of closed loops (geodesics) is much more chaotic than the distribution of prime numbers in regular arithmetic. While prime numbers have a predictable amount of "clumping" based on the modulus, these surface loops clump based on their total volume. The author explains this by comparing the surface to a high-dimensional mathematical object where the rules of "clumping" change entirely.

Technical Summary: Closed Geodesics in Homology Classes on Random Hyperbolic Surfaces of Large Genus

Problem Statement
This paper investigates the statistical distribution of closed geodesics within specific homology classes on random hyperbolic surfaces of large genus $g$ . Specifically, the author studies the fluctuations of a weighted counting function, $\Psi_M(X; q, \alpha)$ , which sums the von Mangoldt weights of oriented closed geodesics $\gamma$ with norm $N(\gamma) \leq X$ whose homology class $[\gamma]$ is congruent to a fixed class $\alpha$ modulo an integer $q$ .

The central object of study is the variance of this distribution, denoted $G_M(X, q)$ , averaged over all homology classes $\alpha \pmod q$ and over the moduli space $\mathcal{M}_g$ of hyperbolic surfaces of genus $g$ , equipped with the Weil–Petersson probability measure. The primary goal is to determine the asymptotic behavior of this variance in the limit as the genus $g \to \infty$ .

Methodology
The analysis relies on the analogy between the Prime Geodesic Theorem for hyperbolic surfaces and the Prime Number Theorem for arithmetic progressions. The methodology proceeds through the following steps:

Expansion of the Variance: The variance $G_M(X, q)$ is expanded into a double sum over pairs of geodesics $(\gamma_1, \gamma_2)$ satisfying the condition $[\gamma_1] \equiv [\gamma_2] \pmod q$ . This is expressed as a difference between a joint counting term $J_M(X, q)$ and the square of the total count.
Classification of Geodesic Pairs: The sum over pairs is decomposed into four distinct categories based on the geometric and topological properties of the geodesics:
- Diagonal pairs: $\gamma_1 = \gamma_2$ (primitive, simple, non-separating).
- Anti-diagonal pairs: $\gamma_2 = \gamma_1^{-1}$ (primitive, simple, non-separating).
- SNS pairs: Pairs of distinct, disjoint, simple, non-separating geodesics whose union leaves a connected complement.
- All other pairs: Including intersecting geodesics, non-simple geodesics, or those with disconnected complements.
Integration over Moduli Space: To evaluate the expected value of these sums over the moduli space $\mathcal{M}_g$ as $g \to \infty$ , the author employs Mirzakhani's integration formulas. These formulas allow the computation of expected values of sums over simple closed geodesics by converting them into integrals involving the length spectrum and the function $\left(\frac{\sinh(\ell/2)}{\ell/2}\right)^2$ .
Asymptotic Analysis: The contributions of each category are estimated. The diagonal and anti-diagonal terms are shown to dominate, while the contributions from SNS pairs and "all other pairs" are proven to be negligible in the large genus limit (either $O(X)$ or $O(1/g)$ ).

Key Contributions and Results
The paper establishes the following asymptotic behavior for the expected variance as $X \to \infty$ and $g \to \infty$ :

$\lim_{g \to \infty} \mathbb{E}^{\text{WP}}_g (G_M(X, q)) \sim \begin{cases} X \log X & \text{if } q > 2, \\ 2X \log X & \text{if } q = 2. \end{cases}$

Key technical findings include:

Dominance of Diagonal Terms: The variance is asymptotically determined entirely by the "diagonal" contributions (where $\gamma_1 = \gamma_2$ ). For $q=2$ , the anti-diagonal contribution ( $\gamma_2 = \gamma_1^{-1}$ ) is of the same order and doubles the coefficient. For $q > 2$ , the anti-diagonal term is negligible.
Negligibility of Off-Diagonal Terms: The contributions from pairs of distinct geodesics (SNS pairs) and pairs involving non-simple or intersecting geodesics vanish relative to the $X \log X$ term in the large genus limit.
Contrast with Arithmetic: The result contrasts with Hooley's conjecture for primes in arithmetic progressions, which predicts a variance of $X \log q$ . Here, the variance scales with $X \log X$ regardless of the modulus $q$ (for $q>2$ ).

Significance and Explanation
The paper suggests a heuristic explanation for the discrepancy between the hyperbolic surface result ( $X \log X$ ) and the arithmetic result ( $X \log q$ ). By drawing an analogy to the theory of $L$ -functions, the author compares the problem to the variance of primes in arithmetic progressions for higher-degree $L$ -functions (associated with $GL(d)$).

Citing work by Hall, Keating, and Roditty-Gershon, the paper notes that as the degree $d$ of the $L$ -function increases, the range of moduli where the variance is governed by $X \log X$ expands. In the limit $d \to \infty$ , only the $X \log X$ regime survives. The author posits that the theory of closed geodesics on random hyperbolic surfaces of large genus behaves analogously to the theory of automorphic forms on $GL(d)$ in the limit $d \to \infty$ . This provides a conceptual framework for why the variance depends on $\log X$ rather than $\log q$ .

The paper concludes that the large genus limit effectively "washes out" the dependence on the specific modulus $q$ , leaving a universal variance behavior driven by the length spectrum of the surface, distinct from the behavior observed in fixed-genus or classical number-theoretic settings.