The distribution of the number of small cuts in a random planar triangulation

International audienceWe enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with $n+3$ vertices is asymptotically normal with mean $(10/27)n$ and variance $(320/729)n$, and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with $n+2$ vertices is asymptotically normal with mean $(8/27)n$ and variance $(152/729)n$. We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with $n+2$ vertices is asymptotically normal with mean $n/3$ and variance $\frac{8}{ 27}n$ 

On properties of random dissections and triangulations

In this work we study properties of random graphs that are drawn uniformly at random from the class consisting of biconnected outerplanar graphs, or equivalently dissections of large convex polygons. We obtain very sharp concentration results for the number of vertices of any given degree, and for the number of induced copies of a given fixed graph. Our method gives similar results for random graphs from the class of triangulations of convex polygon

Random stable laminations of the disk

We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $\theta=2$, we recover Aldous' Brownian triangulation. However, if $\theta\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive L\'{e}vy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.Comment: Published in at http://dx.doi.org/10.1214/12-AOP799 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random combinatorial structures and randomized search heuristics

This thesis is concerned with the probabilistic analysis of random combinatorial structures and the runtime analysis of randomized search heuristics. On the subject of random structures, we investigate two classes of combinatorial objects. The first is the class of planar maps and the second is the class of generalized parking functions. We identify typical properties of these structures and show strong concentration results on the probabilities that these properties hold. To this end, we develop and apply techniques based on exact enumeration by generating functions. For several types of random planar maps, this culminates in concentration results for the degree sequence. For parking functions, we determine the distribution of the defect, the most characteristic parameter. On the subject of randomized search heuristics, we present, improve, and unify different probabilistic methods and their applications. In this, special focus is given to potential functions and the analysis of the drift of stochastic processes. We apply these techniques to investigate the runtimes of evolutionary algorithms. In particular, we show for several classical problems in combinatorial optimization how drift analysis can be used in a uniform way to give bounds on the expected runtimes of evolutionary algorithms.Diese Dissertationsschrift beschäftigt sich mit der wahrscheinlichkeitstheoretischen Analyse von zufälligen kombinatorischen Strukturen und der Laufzeitanalyse randomisierter Suchheuristiken. Im Bereich der zufälligen Strukturen untersuchen wir zwei Klassen kombinatorischer Objekte. Dies sind zum einen die Klasse aller kombinatorischen Einbettungen planarer Graphen und zum anderen eine Klasse diskreter Funktionen mit bestimmten kombinatorischen Restriktionen (generalized parking functions). Für das Studium dieser Klassen entwickeln und verwenden wir zählkombinatorische Methoden die auf erzeugenden Funktionen basieren. Dies erlaubt uns, Konzentrationsresultate für die Gradsequenzen verschiedener Typen zufälliger kombinatorischer Einbettungen planarer Graphen zu erzielen. Darüber hinaus erhalten wir Konzentrationsresultate für den charakteristischen Parameter, den Defekt, zufälliger Instanzen der untersuchten diskreten Funktionen. Im Bereich der randomisierten Suchheuristiken präsentieren und erweitern wir verschiedene wahrscheinlichkeitstheoretische Methoden der Analyse. Ein besonderer Fokus liegt dabei auf der Analyse der Drift stochastischer Prozesse. Wir wenden diese Methoden in der Laufzeitanalyse evolutionärer Algorithmen an. Insbesondere zeigen wir, wie mit Hilfe von Driftanalyse die erwarteten Laufzeiten evolutionärer Algorithmen auf verschiedenen klassischen Problemen der kombinatorischen Optimierung auf einheitliche Weise abgeschätzt werden können

Subgraph statistics in subcritical graph classes

Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs.Postprint (author's final draft

Conditionnement de grands arbres aléatoires et configurations planes non-croisées

Les limites d échelle de grands arbres aléatoires jouent un rôle central dans cette thèse.Nous nous intéressons plus spécifiquement au comportement asymptotique de plusieurs fonctions codant des arbres de Galton-Watson conditionnés. Nous envisageons plusieurs types de conditionnements faisant intervenir différentes quantités telles que le nombre total de sommets ou le nombre total de feuilles, avec des lois de reproductions différentes.Lorsque la loi de reproduction est critique et appartient au domaine d attraction d uneloi stable, un phénomène d universalité se produit : ces arbres ressemblent à un même arbre aléatoire continu, l arbre de Lévy stable. En revanche, lorsque la criticalité est brisée, la communauté de physique théorique a remarqué que des phénomènes de condensation peuvent survenir, ce qui signifie qu avec grande probabilité, un sommet de l arbre a un degré macroscopique comparable à la taille totale de l arbre. Une partie de cette thèse consiste à mieux comprendre ce phénomène de condensation. Finalement, nous étudions des configurations non croisées aléatoires, obtenues à partir d un polygône régulier en traçant des diagonales qui ne s intersectent pas intérieurement, et remarquons qu elles sont étroitement reliées à des arbres de Galton-Watson conditionnés à avoir un nombre de feuilles fixé. En particulier, ce lien jette un nouveau pont entre les dissections uniformes et les arbres de Galton-Watson, ce qui permet d obtenir d intéressantes conséquences de nature combinatoire.Scaling limits of large random trees play an important role in this thesis. We are more precisely interested in the asymptotic behavior of several functions coding conditioned Galton-Watson trees. We consider several types of conditioning, involving different quantities such as the total number of vertices or leaves, as well as several types of offspring distributions. When the offspring distribution is critical and belongs to the domainof attraction of a stable law, a universality phenomenon occurs: these trees look like the samecontinuous random tree, the so-called stable Lévy tree. However, when the offspring distributionis not critical, the theoretical physics community has noticed that condensation phenomenamay occur, meaning that with high probability there exists a unique vertex with macroscopicdegree comparable to the total size of the tree. The goal of one of our contributions is to graspa better understanding of this phenomenon. Last but not least, we study random non-crossingconfigurations consisting of diagonals of regular polygons, and notice that they are intimatelyrelated to Galton-Watson trees conditioned on having a fixed number of leaves. In particular,this link sheds new light on uniform dissections and allows us to obtain some interesting resultsof a combinatorial flavor.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF