On Directed Random Graphs and Greedy Walks on Point Processes

This thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes.           We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparabl...

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Main Author: Gabrysch, Katja
Format: Doctoral Thesis
Language:English
Published: Uppsala universitet, Analys och sannolikhetsteori 2016
Subjects:
Online Access:http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-305859
http://nbn-resolving.de/urn:isbn:978-91-506-2608-7
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spelling ndltd-UPSALLA1-oai-DiVA.org-uu-3058592016-11-16T05:07:14ZOn Directed Random Graphs and Greedy Walks on Point ProcessesengGabrysch, KatjaUppsala universitet, Analys och sannolikhetsteoriUppsala2016Directed random graphsTracy-Widom distributionPoisson-weighted infinite treeGreedy walkPoint processesThis thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes.           We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparable vertices present with probability p, independently of all other edges, and each edge is directed from the vertex with smaller label to the vertex with larger label. In Paper I we consider a directed random graph on ℤ2 with the vertices ordered according to the product order and we show that the limiting distribution of the centered and rescaled length of the longest path from (0,0) to (n, [na] ), a<3/14, is the Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on ℤ with edge probability n−1 converges in distribution to the Poisson-weighted infinite tree. Moreover, we derive limit theorems for the length of the longest path of the Poisson-weighted infinite tree.           The greedy walk is a deterministic walk on a point process that always moves from its current position to the nearest not yet visited point. Since the greedy walk on a homogeneous Poisson process on the real line, starting from 0, almost surely does not visit all points, in Paper III we find the distribution of the number of visited points on the negative half-line and the distribution of the index at which the walk achieves its minimum. In Paper IV we place homogeneous Poisson processes first on two intersecting lines and then on two parallel lines and we study whether the greedy walk visits all points of the processes. In Paper V we consider the greedy walk on an inhomogeneous Poisson process on the real line and we determine sufficient and necessary conditions on the mean measure of the process for the walk to visit all points. Doctoral thesis, comprehensive summaryinfo:eu-repo/semantics/doctoralThesistexthttp://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-305859urn:isbn:978-91-506-2608-7Uppsala Dissertations in Mathematics, 1401-2049 ; 97application/pdfinfo:eu-repo/semantics/openAccess
collection NDLTD
language English
format Doctoral Thesis
sources NDLTD
topic Directed random graphs
Tracy-Widom distribution
Poisson-weighted infinite tree
Greedy walk
Point processes
spellingShingle Directed random graphs
Tracy-Widom distribution
Poisson-weighted infinite tree
Greedy walk
Point processes
Gabrysch, Katja
On Directed Random Graphs and Greedy Walks on Point Processes
description This thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes.           We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparable vertices present with probability p, independently of all other edges, and each edge is directed from the vertex with smaller label to the vertex with larger label. In Paper I we consider a directed random graph on ℤ2 with the vertices ordered according to the product order and we show that the limiting distribution of the centered and rescaled length of the longest path from (0,0) to (n, [na] ), a<3/14, is the Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on ℤ with edge probability n−1 converges in distribution to the Poisson-weighted infinite tree. Moreover, we derive limit theorems for the length of the longest path of the Poisson-weighted infinite tree.           The greedy walk is a deterministic walk on a point process that always moves from its current position to the nearest not yet visited point. Since the greedy walk on a homogeneous Poisson process on the real line, starting from 0, almost surely does not visit all points, in Paper III we find the distribution of the number of visited points on the negative half-line and the distribution of the index at which the walk achieves its minimum. In Paper IV we place homogeneous Poisson processes first on two intersecting lines and then on two parallel lines and we study whether the greedy walk visits all points of the processes. In Paper V we consider the greedy walk on an inhomogeneous Poisson process on the real line and we determine sufficient and necessary conditions on the mean measure of the process for the walk to visit all points.
author Gabrysch, Katja
author_facet Gabrysch, Katja
author_sort Gabrysch, Katja
title On Directed Random Graphs and Greedy Walks on Point Processes
title_short On Directed Random Graphs and Greedy Walks on Point Processes
title_full On Directed Random Graphs and Greedy Walks on Point Processes
title_fullStr On Directed Random Graphs and Greedy Walks on Point Processes
title_full_unstemmed On Directed Random Graphs and Greedy Walks on Point Processes
title_sort on directed random graphs and greedy walks on point processes
publisher Uppsala universitet, Analys och sannolikhetsteori
publishDate 2016
url http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-305859
http://nbn-resolving.de/urn:isbn:978-91-506-2608-7
work_keys_str_mv AT gabryschkatja ondirectedrandomgraphsandgreedywalksonpointprocesses
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