uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Convergence to the Tracy-Widom distribution for longest paths in a directed random graph
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2013 (English)In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 10, no 2, 711-730 p.Article in journal (Refereed) Published
Abstract [en]

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i1,i2) to (j1,j2), whenever i1 ≤ j1, i2 ≤ j2, with probability p, independently for each such pair of vertices. Let Ln,m denote the maximum length of all paths contained in an n×m rectangle. We show that there is a positive exponent a, such that, if m/na→1, as n→∞, then a properly centered/rescaled version of Ln,m converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.

Place, publisher, year, edition, pages
2013. Vol. 10, no 2, 711-730 p.
Keyword [en]
Random graph, last passage percolation, strong approximation, Tracy- Widom distribution
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-304258OAI: oai:DiVA.org:uu-304258DiVA: diva2:1014917
Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2017-11-30
In thesis
1. On Directed Random Graphs and Greedy Walks on Point Processes
Open this publication in new window or tab >>On Directed Random Graphs and Greedy Walks on Point Processes
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2016. 28 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 97
Keyword
Directed random graphs, Tracy-Widom distribution, Poisson-weighted infinite tree, Greedy walk, Point processes
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-305859 (URN)978-91-506-2608-7 (ISBN)
Public defence
2016-12-09, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2016-11-15 Created: 2016-10-23 Last updated: 2016-11-15

Open Access in DiVA

No full text

Other links

http://alea.impa.br/articles/v10/10-30.pdf

Search in DiVA

By author/editor
Konstantopoulos, TakisTrinajstić, Katja
By organisation
Analysis and Probability Theory
In the same journal
Latin American Journal of Probability and Mathematical Statistics
Probability Theory and Statistics

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 542 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf