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Building and Destroying Urns, Graphs, and Trees
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2023 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, consisting of an introduction and four papers, different models in the mathematical area of combinatorial probability are investigated.

In Paper I, two operations for combining generalised Pólya urns, called disjoint union and product, are defined. This is then shown to turn the set of isomorphism classes of Pólya urns into a semiring, and we find that assigning to an urn its intensity matrix is a semiring homomorphism.

In paper II, a modification and generalisation of the random cutting model is introduced. For a finite graph with given source and target vertices, we remove vertices at random and discard all resulting components without a source node. The results concern the number of cuts needed to remove all target vertices and the size of the remaining graph, and suggest that this model interpolates between the traditional cutting model and site percolation.

In paper III, we define several polynomial invariants for rooted trees based on the modified cutting model in Paper II.We find recursive identities for these invariants and, using an approach via irreducibility of polynomials, prove that two specific invariants are complete, that is, they distinguish rooted trees up to isomorphism.

In paper IV, joint with Paul Thévenin, we consider an operation of concatenating t random perfect matchings on 2n vertices. Our analysis of the resulting random graph as t tends to infinity shows that there is a giant component if and only if n is odd, and that the size of this giant component as well as the number of components is asymptotically normally distributed.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2023. , p. 33
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 131
National Category
Probability Theory and Statistics Discrete Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-500978ISBN: 978-91-506-3009-1 (print)OAI: oai:DiVA.org:uu-500978DiVA, id: diva2:1753798
Public defence
2023-08-24, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2023-05-29 Created: 2023-04-28 Last updated: 2023-05-30
List of papers
1. A Semiring Structure for Generalised Pólya Urns
Open this publication in new window or tab >>A Semiring Structure for Generalised Pólya Urns
(English)In: Article in journal (Other academic) Submitted
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-500974 (URN)
Available from: 2023-04-28 Created: 2023-04-28 Last updated: 2023-04-29
2. A Modification of the Random Cutting Model
Open this publication in new window or tab >>A Modification of the Random Cutting Model
(English)In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064Article in journal (Refereed) Accepted
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-500975 (URN)
Available from: 2023-04-28 Created: 2023-04-28 Last updated: 2023-04-29
3. Obtaining Polynomial Invariants for Rooted Trees from their Random Destruction
Open this publication in new window or tab >>Obtaining Polynomial Invariants for Rooted Trees from their Random Destruction
(English)In: Article in journal (Other academic) Submitted
National Category
Discrete Mathematics Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-500976 (URN)
Available from: 2023-04-28 Created: 2023-04-28 Last updated: 2023-04-29
4. Concatenating Random Matchings
Open this publication in new window or tab >>Concatenating Random Matchings
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics Discrete Mathematics
Identifiers
urn:nbn:se:uu:diva-500977 (URN)
Available from: 2023-04-28 Created: 2023-04-28 Last updated: 2023-04-29

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Burghart, Fabian

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