On the genus of a random Riemann surface

Gamburd, Alexander; Makover, Eran

doi:10.1090/conm/311/05451

Cited by 15 publications

(19 citation statements)

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“…cross-metric surfaces as equal if some self-homeomorphism of the surface maps one to the other. This refines the model introduced by Gamburd and Makover [24]. Here is our precise result:…”

Section: Shortest Cellular Graphs With Prescribed Combinatorial Mapssupporting

confidence: 84%

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Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

Verdière

Hubard

Mesmay

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and viceversa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions.Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g 3/2 n 1/2 ) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition.Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations. * Supported by the French ANR Blanc project ANR-12-BS02-005 (RDAM

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“…cross-metric surfaces as equal if some self-homeomorphism of the surface maps one to the other. This refines the model introduced by Gamburd and Makover [24]. Here is our precise result:…”

Section: Shortest Cellular Graphs With Prescribed Combinatorial Mapssupporting

confidence: 84%

“…We remark that beyond the extremal qualities that concern us, random surfaces and their geometry have been heavily studied recently [24,45] in connection to quantum gravity [49] and Belyi surfaces [3].…”

Section: Short Pants Decompositionsmentioning

confidence: 99%

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

Verdière

Hubard

Mesmay

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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“…Using this fact, one can estimate the expected value of the genus. This was done by Gamburd and Makover in [10], Brooks and Makover in [6], Pippenger and Schleich in [16] and Gamburd in [9]. We state the version of Brooks and Makover here.…”

Section: 21mentioning

confidence: 99%

Random regular graphs and the systole of a random surface

Petri

2017

Journal of Topology

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Abstract. We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using ideal hyperbolic triangles and the second one using triangles that carry a given Riemannian metric.In the hyperbolic case we compute the limit of the expected value of the systole when the number of triangles tends to infinity (approximately 2.484). We also determine the asymptotic probability distribution of the number of curves of any finite length. This turns out to be a Poisson distribution.In the Riemannian case we give an upper bound to the limit supremum and a lower bound to the limit infimum of the expected value of the systole depending only on the metric on the triangle. We also show that this upper bound is sharp in the sense that there is a sequence of metrics for which the limit infimum comes arbitrarily close to the upper bound.The main tool we use is random regular graphs. One of the difficulties in the proof of the limits is controlling the probability that short closed curves are separating. To do this, we first prove that the probability that a random cubic graph has a short separating circuit tends to 0 as the number of vertices tends to infinity and show that this holds for circuits of a length up to log 2 of the number of vertices.

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“…This suggests that the injectivity radius around a typical point is growing to infinity. Gamburd and Makover [GM02] showed that as N grows the genus will converge to N/4 and using the Euler's characteristic the average degree will grow to infinity. Take a uniform measure on triangulations with N triangles conditioned on the genus to be CN for some fixed C < 1/4, then we conjecture that as N grows to infinity the random surface will locally converge in the sense of [BS01b] (see section 5 above) to a random triangulation of the hyperbolic plane with average degree 6 1−4C .…”

Section: Circle Packingmentioning

confidence: 99%