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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
A sequential importance sampling algorithm for generating random graphs with prescribed degrees
, 2006
"... Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a ra ..."
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Cited by 46 (1 self)
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Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence.
Rainbow hamilton cycles in random regular graphs
, 2005
"... A rainbow subgraph of an edgecoloured graph has all edges of distinct colours. A random dregular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, provided d ≥ 8. ..."
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A rainbow subgraph of an edgecoloured graph has all edges of distinct colours. A random dregular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, provided d ≥ 8.
Random regular graphs and the systole of a random surface. ArXiv eprints
, 2013
"... 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 tr ..."
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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 goes 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 controling 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 for the number of vertices going to infinity and show that this holds for circuits of a length up to log2 of the number of vertices. 1.
Boundary Cycles in Random Triangulated Surfaces
 Department of Mathematics, Harvey Mudd College
"... available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copying is by permission of the author. To disseminate otherwise or to republish requires written permission from the author. Random triangulat ..."
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available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copying is by permission of the author. To disseminate otherwise or to republish requires written permission from the author. Random triangulated surfaces are created by taking an even number, n, of triangles and arbitrarily ”gluing ” together pairs of edges until every edge has been paired. The resulting surface can be described in terms of its number of boundary cycles, a random variable denoted by h. Building upon the work of Nicholas Pippenger and Kristin Schleich, and using a recent result from Alex Gamburd, we establish an improved approximation for the expectation of h for certain values of n. We use a computer simulation to exactly determine the distribution of h for small values of n, and present a method for calculating these probabilities. We also conduct an investigation into the related problem of creating one connected component out of n
FINITE LENGTH SPECTRA OF RANDOM SURFACES AND THEIR DEPENDENCE ON GENUS
"... Abstract. The main goal of this article 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 s ..."
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Abstract. The main goal of this article 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. 1.
fleming.moments.tex Large Deviations and Moments for the Euler Characteristic of a Random Surface
, 902
"... Abstract: We study random surfaces constructed by glueing together N/k filled kgons along their edges, with all (N − 1)!! = (N − 1)(N − 3) · · ·3 · 1 pairings of the edges being equally likely. (We assume that lcm{2, k} divides N.) The Euler characteristic of the resulting surface is related to ..."
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Abstract: We study random surfaces constructed by glueing together N/k filled kgons along their edges, with all (N − 1)!! = (N − 1)(N − 3) · · ·3 · 1 pairings of the edges being equally likely. (We assume that lcm{2, k} divides N.) The Euler characteristic of the resulting surface is related to the number of cycles in a certain random permutation of {1,...,N}. Gamburd has shown that when 2 lcm{2, k} divides N, the distribution of this random permutation converges to that of the uniform distribution on the alternating group AN in the totalvariation distance as N → ∞. We obtain largedeviations bounds for the number of cycles that, together with Gamburd’s result, allow us to derive sharp estimates for the moments of the number of cycles. These estimates allow us to confirm certain cases of conjectures made by Pippenger and Schleich. 1.