Random Construction of Riemann Surfaces

By R. Brooks, Eran Makover. 2001

113 citations recorded by the source corpus.

In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi surfaces. And in this construction we can control the geometry of the compact Riemann surface by the geometry of the graph. We show that almost all such surfaces have large first eigenvalue and large Cheeger constant.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f251fadb6ee1a56da719fa6d6a42bf89d06ca62

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Pants Decompositions of Random Surfaces (2010, background): “asses form a kind of combinatorial \moduli space". We put the uniform measure on this nite set, so we can speak of a random combinatorial surface. (This denition goes back to Brooks and Makover [BM04], who studied the rst eigenvalue of the Laplacian and the systole of random combinatorial surfa”
Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus (2010, methodology): “om hyperbolic surfaces; in particular, we are interested in the length of the shortest simple closed geodesic of a given combinatorial type, diameter and the Cheeger constant of a random surface. See [BM] for more in the case of random hyperbolic surfaces constructed by random trivalent graphs. 4.1 ”
A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces (2019, background): “itive answer. A version of the rst part of the rst problem appears as [WX18, Conjecture 5]. Mirzakhani was also interested discrete models of random surfaces, resulting from gluing together triangles [BM04], and in the collection of all covers of a xed surface. Problem 10.6. Fix X2M 2 and ">”
Lengths of closed geodesics on random surfaces of large genus (2017, background): “joint factorial moments and prove the convergence we are after. Finally, Theorem 5.1 follows from Theorem 4.1 together with a dominated convergence argument. 4 MARYAMMIRZAKHANIANDBRAMPETRI Remark. In [BM04], Brooks and Makover deﬁned a more combinatorial model for random hyperbolic surfaces based on”
Poisson–Dirichlet distribution for random Belyi surfaces (2005, methodology): “Here we note just two immediate corollaries which substantially refine results in [14, 28, 43].”