Poisson approximation of the length spectrum of random surfaces

The main goal of this paper is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number of appearances of fixed finite sets of combinatorial types of curves. Of any such set we determine the asymptotics of the probability distribution. It turns out that these distributions are independent of the genus in an appropriate sense. As an application of our results we study the probability distribution of the systole of random surfaces in a hyperbolic and a more general Riemannian setting. In the hyperbolic setting we are able to determine the limit of the probability distribution for the number of triangles tending to infinity and in the Riemannian setting we derive bounds.

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Fractal dimensions of dynamically triangulated random surfaces

Journal de Physique I ◽

10.1051/jp1:1992275 ◽

1992 ◽

Vol 2 (12) ◽

pp. 2181-2190 ◽

Cited By ~ 4

Author(s):

Christian Münkel ◽

Dieter W. Heermann

Keyword(s):

Fractal Dimensions ◽

Random Surfaces

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Extreme Value for Dependent Sequences Via the Stein-Chen Method of Poisson Approximation.

10.21236/ada190325 ◽

1987 ◽

Author(s):

Richard L. Smith

Keyword(s):

Poisson Approximation ◽

Extreme Value

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Short geodesic loops and $$L^p$$ norms of eigenfunctions on large genus random surfaces

Geometric and Functional Analysis ◽

10.1007/s00039-021-00556-6 ◽

2021 ◽

Author(s):

Clifford Gilmore ◽

Etienne Le Masson ◽

Tuomas Sahlsten ◽

Joe Thomas

Keyword(s):

Random Surfaces ◽

Large Genus ◽

Short Geodesic ◽

Geodesic Loops

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On improvements of the order of approximation in the Poisson limit theorem

Journal of Applied Probability ◽

10.1017/s0021900200103808 ◽

1996 ◽

Vol 33 (01) ◽

pp. 146-155 ◽

Cited By ~ 1

Author(s):

K. Borovkov ◽

D. Pfeifer

Keyword(s):

Limit Theorem ◽

Random Variables ◽

Poisson Approximation ◽

Order Of Approximation ◽

Operator Semigroups ◽

Bernoulli Random Variables ◽

Usual Rate ◽

Poisson Limit ◽

Poisson Measures ◽

Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

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