Poisson–Dirichlet distribution for random Belyi surfaces

By Alex Gamburd. 2005

60 citations recorded by the source corpus.

Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a "typical" compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces associated with random cubic graphs; by a theorem of Belyi, these are "dense" in the space of compact Riemann surfaces. The question as to how these surfaces are distributed in the Teichmuller spaces depends on the study of oriented cycles in random cubic graphs with random orientation; Brooks and Makover conjectured that asymptotically normalized cycle lengths follow Poisson-Dirichlet distribution. We present a proof of this conjecture using representation theory of the symmetric group.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd86e9561939af19a8576d50eaf7adc87b642ddd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus (2010, background): “emann surfaces with largegenus by using trivalent graphs. In this model the expected value of the systole of a random Riemann surface turns out to be bounded (independent of the genus) [MM]. See also [Ga]. We will see in this note that a random Riemann surface with respect to the WeilPetersson volum”
Planar stochastic hyperbolic triangulations (2016, background): “Some results in the literature already indicate that large triangulations of high genus should be locally planar such as [24, 28] or the paper [5] in the case of unicellular maps.”
A simple solution to the k-core problem (2007, background): “Gamburd [ 7 ] studied long cycles in random oriented cubic graphs; here the connection graph is a directed 3-cycle at each vertex.”
Recent progress in coalescent theory (2009, unknown): “There is a natural distance on the space P, which is to take d(π, π′) to be equal to 1 over the largest n such that the restriction of π and π′ to {1, . . . , n} are identical.”