A glimpse of the conformal structure of random planar maps

We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm--Loewner evolution) process of parameter $\kappa = 6$ and to combine the locality property of the SLE_{6} together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary. Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension $\frac{1}{3}$ almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF).Comment: To appear in Commun. Math. Phy

d-반순서의 경쟁그래프의 연구

학위논문 (박사)-- 서울대학교 대학원 : 사범대학 수학교육과, 2018. 2. 김서령.The \emph{competition graph} $C(D)$ of a digraph $D$ is defined to be a graph whose vertex set is the same as $D$ and which has an edge joining two distinct vertices $x$ and $y$ if and only if there are arcs $(x,z)$ and $(y,z)$ for some vertex $z$ in $D$. Competition graphs have been extensively studied for more than four decades. Cohen~\cite{cohen1968interval, cohen1977food, cohen1978food} empirically observed that most competition graphs of acyclic digraphs representing food webs are interval graphs. Roberts~\cite{roberts1978food} asked whether or not Cohen's observation was just an artifact of the construction, and then concluded that it was not by showing that if $G$ is an arbitrary graph, then $G$ together with additional isolated vertices as many as the number of edges of $G$ is the competition graph of some acyclic digraph. Then he asked for a characterization of acyclic digraphs whose competition graphs are interval graphs. Since then, the problem has remained elusive and it has been one of the basic open problems in the study of competition graphs. There have been a lot of efforts to settle the problem and some progress has been made. While Cho and Kim~\cite{cho2005class} tried to answer his question, they could show that the competition graphs of doubly partial orders are interval graphs. They also showed that an interval graph together with sufficiently many isolated vertices is the competition graph of a doubly partial order. In this thesis, we study the competition graphs of $d$-partial orders some of which generalize the results on the competition graphs of doubly partial orders. For a positive integer $d$, a digraph $D$ is called a \emph{$d$-partial order} if V(D) \subset \RR^d and there is an arc from a vertex $\mathbf{x}$ to a vertex $\mathbf{y}$ if and only if $\mathbf{x}$ is componentwise greater than $\mathbf{y}$. A doubly partial order is a $2$-partial order. We show that every graph $G$ is the competition graph of a $d$-partial order for some nonnegative integer $d$, call the smallest such $d$ the \emph{partial order competition dimension} of $G$, and denote it by $\dim_\text{poc}(G)$. This notion extends the statement that the competition graph of a doubly partial order is interval and the statement that any interval graph can be the competition graph of a doubly partial order as long as sufficiently many isolated vertices are added, which were proven by Cho and Kim~\cite{cho2005class}. Then we study the partial order competition dimensions of some interesting families of graphs. We also study the $m$-step competition graphs and the competition hypergraph of $d$-partial orders.1 Introduction 1 1.1 Basic notions in graph theory 1 1.2 Competition graphs 6 1.2.1 A brief history of competition graphs 6 1.2.2 Competition numbers 7 1.2.3 Interval competition graphs 10 1.3 Variants of competition graphs 14 1.3.1 m-step competition graphs 15 1.3.2 Competition hypergraphs 16 1.4 A preview of the thesis 18 2 On the competition graphs of d-partial orders 1 20 2.1 The notion of d-partial order 20 2.2 The competition graphs of d-partial orders 21 2.2.1 The regular (d − 1)-dimensional simplex △ d−1 (p) 22 2.2.2 A bijection from H d + to a set of regular (d − 1)-simplices 23 2.2.3 A characterization of the competition graphs of d-partial orders 25 2.2.4 Intersection graphs and competition graphs of d-partial orders 27 2.3 The partial order competition dimension of a graph 29 3 On the partial order competition dimensions of chordal graphs 2 38 3.1 Basic properties on the competition graphs of 3-partial orders 39 3.2 The partial order competition dimensions of diamond-free chordal graphs 42 3.3 Chordal graphs having partial order competition dimension greater than three 46 4 The partial order competition dimensions of bipartite graphs 3 53 4.1 Order types of two points in R 3 53 4.2 An upper bound for the the partial order competition dimension of a graph 57 4.3 Partial order competition dimensions of bipartite graphs 64 5 On the m-step competition graphs of d-partial orders 4 69 5.1 A characterization of the m-step competition graphs of dpartial orders 69 5.2 Partial order m-step competition dimensions of graphs 71 5.3 dim poc (Gm) in the aspect of dim poc (G) 76 5.4 Partial order competition exponents of graphs 79 6 On the competition hypergraphs of d-partial orders 5 81 6.1 A characterization of the competition hypergraphs of d-partial orders 81 6.2 The partial order competition hyper-dimension of a hypergraph 82 6.3 Interval competition hypergraphs 88 Abstract (in Korean) 99Docto

Towards conformal invariance of 2D lattice models

Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm-Loewner Evolution.Comment: ICM 2006 paper with a few typos correcte

Mating of trees for random planar maps and Liouville quantum gravity: a survey

We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between random planar maps and LQG. Applications discussed in this article include scaling limit results for various functionals of decorated random planar maps, estimates for graph distances and random walk on (not necessarily uniform) random planar maps, computations of the Hausdorff dimensions of sets associated with SLE, scaling limit results for random planar maps conformally embedded in the plane, and special symmetries for $\sqrt{8/3}$-LQG which allow one to prove its equivalence with the Brownian map.Comment: 68 pages, 12 figure

Fabrication and experimental evaluation of common domes having waffle-like stiffening. part i- program development

Determination of minimum weight shape and stiffening configuration for doubly curved shells subjected to external buckling pressure

On the helly property of some intersection graphs

An EPG graph G is an edge-intersection graph of paths on a grid. In this doctoral thesis we will mainly explore the EPG graphs, in particular B1-EPG graphs. However, other classes of intersection graphs will be studied such as VPG, EPT and VPT graph classes, in addition to the parameters Helly number and strong Helly number to EPG and VPG graphs. We will present the proof of NP-completeness to Helly-B1-EPG graph recognition problem. We investigate the parameters Helly number and the strong Helly number in both graph classes, EPG and VPG in order to determine lower bounds and upper bounds for this parameters. We completely solve the problem of determining the Helly and strong Helly numbers, for Bk-EPG, and Bk-VPG graphs, for each value k. Next, we present the result that every Chordal B1-EPG graph is simultaneously in the VPT and EPT graph classes. In particular, we describe structures that occur in B1-EPG graphs that do not support a Helly-B1-EPG representation and thus we define some sets of subgraphs that delimit Helly subfamilies. In addition, features of some non-trivial graph families that are properly contained in Helly-B1 EPG are also presented.EPG é um grafo de aresta-interseção de caminhos sobre uma grade. Nesta tese de doutorado exploraremos principalmente os grafos EPG, em particular os grafos B1-EPG. Entretanto, outras classes de grafos de interseção serão estu dadas, como as classes de grafos VPG, EPT e VPT, além dos parâmetros número de Helly e número de Helly forte nos grafos EPG e VPG. Apresentaremos uma prova de NP-completude para o problema de reconhecimento de grafos B1-EPG Helly. Investigamos os parâmetros número de Helly e o número de Helly forte nessas duas classes de grafos, EPG e VPG, a fim de determinar limites inferiores e superi ores para esses parâmetros. Resolvemos completamente o problema de determinar o número de Helly e o número de Helly forte para os grafos Bk-EPG e Bk-VPG, para cada valor k. Em seguida, apresentamos o resultado de que todo grafo B1-EPG Chordal está simultaneamente nas classes de grafos VPT e EPT. Em particular, descrevemos estruturas que ocorrem em grafos B1-EPG que não suportam uma representação B1-EPG-Helly e assim definimos alguns conjuntos de subgrafos que delimitam sub famílias Helly. Além disso, também são apresentadas características de algumas famílias de grafos não triviais que estão propriamente contidas em B1-EPG-Hell

Consistent random vertex-orderings of graphs

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties $\mathcal{P}$ the only such random orderings are uniform, and give some examples of non-uniform orderings when they exist