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On Directed Random Graphs and Greedy Walks on Point ProcessesPrimeFaces.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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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Description

##### Abstract [en]

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

Uppsala: Department of Mathematics, 2016. , 28 p.
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 97
##### Keyword [en]

Directed random graphs, Tracy-Widom distribution, Poisson-weighted infinite tree, Greedy walk, Point processes
##### National Category

Probability Theory and Statistics
##### Identifiers

URN: urn:nbn:se:uu:diva-305859ISBN: 978-91-506-2608-7 (print)OAI: oai:DiVA.org:uu-305859DiVA: diva2:1039330
##### Public defence

2016-12-09, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Opponent

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##### Supervisors

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#####

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Available from: 2016-11-15 Created: 2016-10-23 Last updated: 2016-11-15
##### List of papers

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 ℤ^{2} 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*, [*n ^{a}*] ),

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.

1. Convergence to the Tracy-Widom distribution for longest paths in a directed random graph$(function(){PrimeFaces.cw("OverlayPanel","overlay1014917",{id:"formSmash:j_idt562:0:j_idt566",widgetVar:"overlay1014917",target:"formSmash:j_idt562:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Convergence of directed random graphs to the Poisson-weighted infinite tree$(function(){PrimeFaces.cw("OverlayPanel","overlay950354",{id:"formSmash:j_idt562:1:j_idt566",widgetVar:"overlay950354",target:"formSmash:j_idt562:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Distribution of the smallest visited point in a greedy walk on the line$(function(){PrimeFaces.cw("OverlayPanel","overlay1039178",{id:"formSmash:j_idt562:2:j_idt566",widgetVar:"overlay1039178",target:"formSmash:j_idt562:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Greedy walks on two lines$(function(){PrimeFaces.cw("OverlayPanel","overlay1039187",{id:"formSmash:j_idt562:3:j_idt566",widgetVar:"overlay1039187",target:"formSmash:j_idt562:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. The greedy walk on an inhomogeneous Poisson process$(function(){PrimeFaces.cw("OverlayPanel","overlay1039198",{id:"formSmash:j_idt562:4:j_idt566",widgetVar:"overlay1039198",target:"formSmash:j_idt562:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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