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On Directed Random Graphs and Greedy Walks on Point Processes UPPSALA DISSERTATIONS IN MATHEMATICS 97 On Directed Random Graphs and Greedy Walks on Point Processes Katja Gabrysch Department of Mathematics Uppsala University UPPSALA 2016 Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 9 December 2016 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Mountford (EPFL, Switzerland). Abstract Gabrysch, K. 2016. On Directed Random Graphs and Greedy Walks on Point Processes. Uppsala Dissertations in Mathematics 97. 28 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2608-7. 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 com- parable 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 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, na ), a<3/14, is ℤthe Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on 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 infiniteℤ 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 Pois- son 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. Keywords: Directed random graphs, Tracy-Widom distribution, Poisson-weighted infinite tree, Greedy walk, Point processes Katja Gabrysch, Department of Mathematics, Analysis and Probability Theory, Box 480, Uppsala University, SE-75106 Uppsala, Sweden. © Katja Gabrysch 2016 ISSN 1401-2049 ISBN 978-91-506-2608-7 urn:nbn:se:uu:diva-305859 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-305859) To Markus and Lukas List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Konstantopoulos, T. and Trinajstic,´ K. (2013). Convergence to the Tracy-Widom distribution for longest paths in a directed random graph. ALEA Lat. Am. J. Probab. Math. Stat. 10, 711–730. II Gabrysch, K. (2016). Convergence of directed random graphs to the Poisson-weighted infinite tree. J. Appl. Probab. 53, 463–474. III Gabrysch, K. (2016). Distribution of the smallest visited point in a greedy walk on the line. J. Appl. Probab. 53, 880–887. IV Gabrysch, K. (2016). Greedy walks on two lines. Submitted for publication. V Gabrysch, K. and Thörnblad, E. (2016). The greedy walk on an inhomogeneous Poisson process. Submitted for publication. Reprints were made with permission from the publishers. Contents 1 Introduction .................................................................................................. 9 1.1 Point processes ................................................................................. 9 1.2 Random graphs ............................................................................... 11 1.2.1 Directed random graphs .................................................. 11 1.2.2 Rooted geometric graphs ................................................ 12 1.3 The longest path and skeleton points ............................................ 13 1.4 The Tracy-Widom distribution ...................................................... 15 1.5 Greedy walks .................................................................................. 16 2 Summary of Papers .................................................................................... 20 2.1 Paper I ............................................................................................. 20 2.2 Paper II ............................................................................................ 20 2.3 Paper III .......................................................................................... 21 2.4 Paper IV .......................................................................................... 21 2.5 Paper V ........................................................................................... 23 3 Summary in Swedish ................................................................................. 24 Acknowledgements .......................................................................................... 26 References ........................................................................................................ 27 1. Introduction This thesis contributes to two models in probability theory: directed random graphs and greedy walks. All models studied in this thesis are related to point processes. As they play an important role in the proofs, we give a brief overview of point processes in Section 1.1. The first two papers included in the thesis study models of directed random graphs, which are introduced in Sec- tion 1.2. In Paper I we look at the longest path in a long and thin rectangle and prove that the length of such a path, properly rescaled and centered, converges to the Tracy-Widom distribution. To be able to show this, we observe that there are special points in the graphs, called skeleton points, which are defined in Section 1.3. The Tracy-Widom distribution is described in Section 1.4. The last three papers study greedy walks defined on various point processes. The greedy walk model is presented in Section 1.5. 1.1 Point processes In this section we define point processes in one dimension and explain the con- cept of a stationary and ergodic point process. We also present two examples of point processes. Point processes (or some models of point processes) are studied in many textbooks in probability theory and stochastic processes. For a broad survey of the theory of point processes we refer to [15]. Let E be a complete separable metric space and let B(E) be the Borel s-field generated by the open balls of E.A counting measure m on E is a measure on (E;B(E)) such that m(C) 2 f0;1;2;:::g [ f¥g for all C ⊂ B(E) and m(C) < ¥ for all bounded C ⊂ B(E). The counting measure m is simple if m(fxg) is 0 or 1 for all x 2 E. Let M be the set of all counting measures on E and let M be the s-field of M generated by the functions m 7−! m(C), C 2 B(E). A counting measure m can be expressed as m(·) = ∑ kidxi (·); i2N where ki 2 f0;1;2;:::g, fxi : i 2 Ng ⊂ E and dx denotes the Dirac measure. If m is a simple counting measure, then ki = 1 for all i 2 N. A point process is a measurable mapping from a probability space (W;F ;P) into (M;M ). The point process is simple if it is a simple counting measure with probability 1. Instead of thinking of a simple point process P as a ran- dom measure, we may think of P as a random discrete subset of E. We write jP\Aj for the number of points of P in the set A and x 2 P for jP\fxgj = 1. 9 A point process P is stationary if, for all k ≥ 1 and for all bounded Borel sets A1;A2;:::;Ak ⊂ B(E), the joint distribution fjP \ (A1 +t)j;jP \ (A2 +t)j;:::;jP \ (Ak +t)jg does not depend on the choice of t 2 E. For t 2 E, define the shift operator qt : M ! M by qtm(A) = m(A+t) for all −1 A 2 B(E). Let I be the set of all I 2 B(E) such that qt I = I for all t 2 E. We say that a stationary point process is ergodic if P(P 2 I) = 0 or 1 for any I 2 I . If E = R, then the rate of a stationary point process is defined as r = E P \ (0;1] (more generally, for a stationary point process on any E the rate is the expected number of points in a set of measure 1). If the rate m is finite, then the limit jP \ (0;x]j y = lim x!¥ x exist almost surely and Ey = r. If a stationary point process with finite rate m is ergodic, then P(y = r) = 1: Two standard examples of point processes appearing also in this thesis are Bernoulli processes and Poisson processes. Let fXigi2Z be a sequence of in- dependent random variables with Bernoulli distribution with parameter p, that is, for all i 2 Z, P(Xi = 1) = 1−P(Xi = 0) = p. The law of f = fi 2 Z : Xi = 1g is called Bernoulli process with parameter p. Let yn = fi=n : Xi = 1g be the rescaled Bernoulli process with parameter −1 n . The number of points of yn in any interval (a;b] has binomial distribu- tion with parameters bn(b − a)c and n−1 and this distribution converges to the Poisson distribution with mean (b −a). Moreover, the number of points of yn in disjoint sets are independent and this is preserved in the limit. The law of the limit of the processes yn is called the homogeneous Poisson process with rate 1. Both, a Bernoulli process and a homogeneous Poisson process, are stationary and ergodic point processes. In general, a Poisson process on R is defined as follows. A Poisson pro- cess P with mean measure m is a random countable subset of R such that the number of points in disjoint Borel sets A1;A2;:::;An are independent and so the number of points in a Borel set A has the Poisson distribution with mean m(A), that is, m(A)k (jP \ Aj = k) = e−m(A) ; k ≥ 0: P k! The mean density m of a Poisson process might be given also in terms of an intensity function l, where l : R ! [0;¥) is measurable, so that Z m(A) = l(x)dx; A 10 for any Borel set A ⊂ R.
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