A Central Limit Theorem for the Number of Degree-k Vertices in Random Maps

Drmota, Michael; Παναγιώτου, Κωνσταντίνος

doi:10.1007/s00453-013-9751-x

Cited by 9 publications

(12 citation statements)

References 15 publications

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“…For example, asymptotic results have been obtained for the probability that P (n) will contain given components and subgraphs [20,28], for the number of vertices of given degree [13], and for the size of the maximum degree and largest face [12,27]. In addition, clever algorithms for generating and sampling planar graphs have been designed [6,15], and random planar maps have been studied [14,16,17].…”

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confidence: 99%

The Evolution of Random Graphs on Surfaces

Dowden

Kang

Sprüssel

2018

SIAM J. Discrete Math.

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For integers g, m ≥ 0 and n > 0, let Sg(n, m) denote the graph taken uniformly at random from the set of all graphs on {1, 2, . . . , n} with exactly m = m(n) edges and with genus at most g. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of Sg(n, m), finding that there is often different asymptotic behaviour depending on the ratio m n . In our main results, we show that the probability that Sg(n, m) contains any given non-planar component converges to 0 as n → ∞ for all m(n); the probability that Sg(n, m) contains a copy of any given planar graph converges to 1 as n → ∞ if lim inf m n > 1; the maximum degree of Sg(n, m) is Θ(ln n) with high probability if lim inf m n > 1; and the largest face size of Sg(n, m) has a threshold around m n = 1 where it changes from Θ(n) to Θ(ln n) with high probability.

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confidence: 99%

The Evolution of Random Graphs on Surfaces

Dowden

Kang

Sprüssel

2018

SIAM J. Discrete Math.

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“…It is furthermore expected of classes of random maps that, for fixed size, number of faces of fixed degree k is a normally distributed statistic [DP13]. In Figure 15 below, we compare distributions of different k-gon ratios for n = 40 and n = 100 crossings.…”

Section: Face Degrees In Plane Curvesmentioning

confidence: 99%

A Markov Chain Sampler for Plane Curves

Chapman

Rechnitzer

2019

Experimental Mathematics

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A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sampling, however that method is inefficient at even modest sizes. We introduce Markov chains that sample from the space of plane curves using local moves based on Reidemeister moves. By then mapping vertices on those curves to crossings we produce random knot diagrams. Combining this chain with flat histogram methods we achieve an efficient sampler of plane curves and knot diagrams. By analysing data from this chain we are able to estimate the number of knot diagrams of a given size and also compute knotting probabilities and so investigate their asymptotic behaviour.

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“…Another open question is to make the pattern count asymptotics of Theorem 1.1 more precise. Actually a central limit theorem is expected (as given, for example, in [10] for random quadrangulations and 2-connected triangulations or in [9] for vertices of degree k in random maps or 2-connected maps).…”

Section: Open Problemsmentioning

confidence: 99%