Brooks-Makover Model: Random Hyperbolic Surfaces

• Brooks-Makover Model is a probabilistic construction defining random closed hyperbolic surfaces by gluing ideal triangles with zero shear.
• It bridges combinatorial maps with Belyi surfaces, offering a framework to analyze geometric, spectral, and topological statistics in large genus.
• Applications include estimating diameters and spectral gaps of surfaces, with asymptotic results that analogize high-girth expanders.

The Brooks–Makover model is a canonical probabilistic construction for random closed hyperbolic surfaces of large genus, based on the uniform gluing of ideal hyperbolic triangles. This model, introduced by Brooks and Makover, provides a striking combinatorial framework that realizes the class of Belyi (dessin d’enfant) surfaces and enables precise analysis of geometric and spectral statistics in the large-genus regime. Its definition and properties have established deep connections with random 3-regular graphs, Belyi maps, moduli space, and spectral theory.

1. Model Definition and Construction

Fix an integer $n \gg 1$. The Brooks–Makover model begins with $2n$ copies of the ideal hyperbolic triangle $\Delta \subset \mathbb{H}^2$, each of area $\pi$ and with all vertices on the boundary at infinity, $\partial\mathbb{H}^2$. The $6n$ sides are labeled arbitrarily and then paired uniformly at random into $3n$ unordered pairs, with two constraints: the resulting surface is orientable, and the shear across the gluing is zero. Orientation matching is preserved in each side-pairing.

Once all gluings are performed, one obtains a complete, typically connected (with probability $1-O(1/n)$), hyperbolic surface $S_n^o$ of area $2\pi n$ (Budzinski et al., 2019). This surface has a (random) number $s$ of cusps, where $s \sim \log n$ in probability. The combinatorial dual of the glued complex is a random trivalent map $T_n$ on $s$ vertices (cusps), $2n$ faces (triangles), and $3n$ edges.

Filling in the cusps yields the compact closed Riemann surface $S_n^c$, whose genus is given by the relation

$g = \frac{n + 2 - s}{2} \sim \frac{n}{2}.$

This is a probabilistic realization of closed hyperbolic surfaces of large genus, and the construction can also be described via random oriented 3-regular graphs on $2n$ vertices endowed with a cyclic orientation at each vertex (Shen et al., 4 Nov 2025).

2. Enumeration, Dessins d’Enfant, and Belyi Surfaces

A dessin d’enfant on a compact Riemann surface $X$ is equivalent to a holomorphic Belyi map $f: X \to \mathbb{P}^1$ branched only over $\{0,1,\infty\}$. The fibers over $0$ and $1$ can be colored, inducing a trivalent map whose faces correspond to preimages of $\infty$. Conversely, any trivalent map arises in this way from a Belyi map of degree equal to the number of faces.

In the Brooks–Makover model, one samples genus-varying Belyi maps of degree $2n$ whose dessin is a triangulation with ideal hyperbolic geometry and zero shear. The number of possible gluings is

$\frac{(6n)!}{2^{3n}\,(3n)!}$

which matches the count of labeled Belyi maps with three critical values. The model is thus in bijection with the set of Belyi (dessin d’enfant) surfaces of degree $2n$ (Budzinski et al., 2019, Shen et al., 2022).

3. Topological and Geometric Properties

After gluing and compactification, the surface $S_n^c$ realizes a closed hyperbolic metric, and the genus satisfies

$$g = 1 + \frac{n - \mathrm{LHT}(\Gamma, \mathcal{O})}{2}$$

where $\mathrm{LHT}$ denotes the number of left-hand-turn cycles in the oriented graph (which also equals the number of cusps prior to compactification) (Shen et al., 2022). With the uniform measure, $\mathbb{E}_n[\mathrm{LHT}] \sim O(\log n)$, so typically $g \approx n/2$.

Each ideal triangle's geodesic midpoints and horocycle segments have explicit hyperbolic geometry (short horocycle segments between midpoints have length exactly $1$). The tiling by the action of $G = \langle L, R \rangle \subset PSL_2(\mathbb{Z})$, with $R(z) = z+1$, $L(z) = \frac{z}{z+1}$, exhibits $\mathbb{H}$ as a union of copies of $T$.

4. Asymptotic Diameter of Random Surfaces

The diameter of the closed surface $S_n^c$ is asymptotic to $2\log n$ in probability as $n \to \infty$: $\mathrm{diam}\, S_n^c = 2 \log n + o_p(\log n)$ The lower bound arises from the existence of two high-degree cusps whose post-compactification hyperbolic disks are disjoint and have centers separated by at least $2(1-\epsilon)\log n$ (Budzinski et al., 2019). The upper bound follows by constructing short connecting paths using three combinatorial estimates on random trivalent maps:

• (A) For large-degree vertices, common faces allow corner distances substantially shorter than maximal graph diameter.
• (B) If both degrees are moderate, either the vertices are directly adjacent or mediating vertices provide short connections.
• (C) Any vertex is close (distance $\leq 6$) to some high-degree vertex.

Combining compactification estimates, bilipschitz bounds, and hyperbolic geometry yields the asymptotic result. This suggests that Brooks–Makover surfaces behave analogously to high-girth expanders, yet with explicit geometric control.

5. Spectral Gap and Eigenvalue Distribution

One writes the spectrum of the Laplace–Beltrami operator $\Delta$ on a closed hyperbolic surface $X_g$ as $$0 = \lambda_0(X_g) < \lambda_1(X_g) \leq \lambda_2(X_g) \leq \ldots$$. Classical results (Huber–Cheng) show $$\limsup_{g\to\infty} \lambda_1(X_g) \leq \frac{1}{4}$$, and a conjecture of Buser asserts the existence of surfaces with $\lambda_1 \to \frac{1}{4}$ (“nearly Ramanujan”).

In random surface models, including the Brooks–Makover model, it is now established that for any $\epsilon>0$,

$$\lim_{n\to\infty} \mathbb{P}_{BM}^n\left\{(\Gamma, \mathcal{O}) \in \mathcal{F}_n^*: \lambda_1(S^c(\Gamma, \mathcal{O})) > \frac{1}{4} - \epsilon\right\} = 1$$

This confirms the nearly-optimal spectral gap conjecture in this model (Shen et al., 4 Nov 2025). The proof proceeds by:

• Reducing to the spectrum of the non-compact (cusped) surface $S^o$, showing large cusps exist and validating the transfer of spectral gap under compactification using the Buser–Mok–Otal comparison.
• Employing strong operator-norm convergence—the permutation-minustrivial representation $\mathrm{std}\circ\Phi(\sigma, \tau)$ of $PSL_2(\mathbb{Z})$ converges to the left-regular representation as $n\to\infty$, implying no new eigenvalues below $\frac{1}{4} - \epsilon$ appear.

This suggests Brooks–Makover surfaces are almost Ramanujan, but whether a positive proportion are genuinely Ramanujan (i.e., $\lambda_1 \geq \frac{1}{4}$) remains an open question.

6. Cheeger Constants of Random Brooks–Makover Surfaces

For any closed hyperbolic surface $S$, the Cheeger constant is defined by

$$h(S) = \inf_{E \subset S} \;\frac{\ell(E)}{\min\{\mathrm{Area}(A),\,\mathrm{Area}(B)\}}$$

where $E$ ranges over all embedded 1-complexes separating $S$ into two regions $A,B$.

In the Brooks–Makover model, for every $\epsilon>0$, as $n \to \infty$, a generic random surface satisfies

$h(S^c) < \frac{3}{2\pi} + \epsilon$

with probability tending to $1$ (Shen et al., 2022). The proof decomposes the surface into three types of regions (large cusp-neighborhood disks, small horocyclic triangles, and annular sectors), constructs a partition with favorable area and boundary length ratios via probabilistic coloring, and shows the Cheeger constant converges almost surely to this bound as the number of triangles increases.

7. Comparative Context and Open Problems

The Brooks–Makover model is one of three "canonical" models for random closed hyperbolic surfaces of large genus, alongside the Weil–Petersson model and the random covering model. All three have now been shown to satisfy the nearly-optimal spectral gap property as $g \to \infty$ (Shen et al., 4 Nov 2025). The Brooks–Makover surfaces can be identified with finite covers of the modular orbifold $\mathbb{H}/PSL_2(\mathbb{Z})$ and, via their connection to trivalent maps and Belyi maps, admit combinatorial and arithmetic characterization.

Among the remaining open questions: whether a positive proportion of Brooks–Makover surfaces are genuinely Ramanujan (i.e., have $\lambda_1 \geq \frac{1}{4}$), or whether any infinite family of closed hyperbolic surfaces with $\lambda_1 \geq \frac{1}{4}$ exists. The analogy with random $d$-regular graphs (cf. Friedman’s theorem) motivates further inquiry into extremal spectral properties in this and related models.