Universal intervals in the homomorphism order of digraphs

In this thesis we solve some open problems related to the homomorphism order of digraphs. We begin by introducing the basic concepts of graphs and homomorphisms and studying some properties of the homomorphism order of digraphs. Then we present the new results. First, we show that the class of digraphs containing cycles has the fractal property (strengthening the density property) . Then we show a density theorem for the class of proper oriented trees. Here we say that a tree is proper if it is not a path. Such result was claimed in 2005 but none proof have been published ever since. We also show that the class of proper oriented trees, in addition to be dense, has the fractal property. We end by considering the consequences of these results and the remaining open questions in this area.Outgoin

Multicritical continuous random trees

We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root singularity in their generating function. The scaling limit involves continuous trees with branching points of order up to k+1. We derive explicit integral representations for the average profile of this k-th order multicritical continuous random tree, as well as for its history distributions measuring multi-point correlations. The latter distributions involve non-positive universal weights at the branching points together with fractional derivative couplings. We prove universality by rederiving the same results within a purely continuous axiomatic approach based on the resolution of a set of consistency relations for the multi-point correlations. The average profile is shown to obey a fractional differential equation whose solution involves hypergeometric functions and matches the integral formula of the discrete approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps

Characterizing universal intervals in the homomorphism order of digraphs

In this thesis we characterize all intervals in the homomorphism order of digraphs in terms of universality. To do this, we first show that every interval of the class of digraphs containing cycles is universal. Then we focus our interest in the class of oriented trees (digraphs with no cycles). We give a density theorem for the class of oriented paths and a density theorem for the class of oriented trees, and we strengthen these results by characterizing all universal intervals in these classes. We conclude by summarising all statements and characterizing the universal intervals in the class of digraphs. This solves an open problem in the area

Cellular automata and self-organized criticality

Cellular automata provide a fascinating class of dynamical systems capable of diverse complex behavior. These include simplified models for many phenomena seen in nature. Among other things, they provide insight into self-organized criticality, wherein dissipative systems naturally drive themselves to a critical state with important phenomena occurring over a wide range of length and time scales.Comment: 23 pages, 12 figures (most in color); uses sprocl.tex; chapter submitted for "Some new directions in science on computers," G. Bhanot, S. Chen, and P. Seiden, ed

Derivations and Dirichlet forms on fractals

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, and refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.Comment: to appear in the Journal of Functional Analysis 201

Ihara's zeta function for periodic graphs and its approximation in the amenable case

In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised by Grigorchuk and Zuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs.Comment: 21 pages, 4 figure

SLE for theoretical physicists

This article provides an introduction to Schramm(stochastic)-Loewner evolution (SLE) and to its connection with conformal field theory, from the point of view of its application to two-dimensional critical behaviour. The emphasis is on the conceptual ideas rather than rigorous proofs.Comment: 43 pages, to appear in Annals of Physics; v.2: published version with minor correction