uu.seUppsala University Publications

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Random tournaments and random circuitsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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1999 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis , 1999. , p. 11
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 14
##### Keywords [en]

Mathematics, Poisson-Dirichlet distribution, random circuit decomposition, random tournament, subgraph count
##### Keywords [sv]

MATEMATIK
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-58ISBN: 91-506-1380-4 (print)OAI: oai:DiVA.org:uu-58DiVA, id: diva2:166421
##### Public defence

1999-12-17, Room 247, Building 2, Polacksbacken, Uppsala University, Uppsala, 13:15
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true}); Available from: 1999-11-26 Created: 1999-11-26Bibliographically approved

This thesis is devoted to two different topics in the area of probabilistic combinatorics: asymptotic behaviour of subgraph counts in a random tournament and random circuit decompositions of complete graphs.

Let *T*_{n} be a random tournament on n vertices, chosen uniformly from all 2(^{n}_{2}) such tournaments, and let *D* be an arbitrary directed graph. Then the number of copies of *D* in *T*_{n} is a random variable which, after normalization, converges in distribution as *n* tends to infinity. The limit distribution is determined, and it turns out to be normal for a typical *D*, but for some *D* it is a quadratic polynomial in normal variables. It is also shown that the variance of the number of copies of *D* in *T*_{n} is a polynomial in n and that the degree of this polynomial is, for a typical *D*, 2*v* - 3, where v is the number of vertices of D. However, examples are given for which this degree is as low as *v*.

Some directed graphs turn out to appear the same number of times in all tournaments with a fixed number of vertices. A partial characterization of these directed graphs is given.

In the complete undirected or directed graph on *n* vertices, choose a random decomposition of the set of all edges into circuits, uniformly from all such decompositions. Letting *L*_{k} be the fraction of edges contained in the *k*-th longest circuit in this decomposition, it is shown that (*L*_{1}, *L*_{2},...) converges to a Poisson-Dirichlet distribution as n tends to infinity. It is also shown that the numbersof circuits of given lengths converge jointly to independent Poisson variables, and that the expected number of circuits is log *n* + *O*(1) in the undirected case and 2 log *n* + *O*(1) in the directed case.

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