Anonymous Graph Exploration with Binoculars

International audienceWe investigate the exploration of networks by a mobile agent. It is long known that, without global information about the graph, it is not possible to make the agent halts after the exploration except if the graph is a tree. We therefore endow the agent with binoculars, a sensing device that can show the local structure of the environment at a constant distance of the agent current location.We show that, with binoculars, it is possible to explore and halt in a large class of non-tree networks. We give a complete characterization of the class of networks that can be explored using binoculars using standard notions of discrete topology. This class is much larger than the class of trees: it contains in particular chordal graphs, plane triangulations and triangulations of the projective plane. Our characterization is constructive, we present an Exploration algorithm that is universal; this algorithm explores any network explorable with binoculars, and never halts in non-explorable networks

Vers la théorie de l'indice de Conley sur les complexes cubiques

Robin Forman a défini pour la première fois le champ vectoriel combinatoire sur les complexes simpliciaux en 1998. Kaczynski, Mrozek et Wanner ont proposé d'enrichir cette notion en flux combinatoire en 2016, ce qui permet de développer la théorie d'indice de Conley sur les complexes simpliciaux. Dans cette thèse, une preuve du chaos dans le modèle combinatoire de Lorenz par conjugaison est présentée. L'approche par flux combinatoire vers l'indice de Conley est adaptée aux complexes cubiques et la théorie est développée dans une certaine mesure. Un analogue cubique de l'application semi-continue supérieurement à valeurs contractibles est également établi.Abstract: Combinatorial vector fields on simplicial complexes have been first defined by Robin Forman in 1998. The enrichment of this notion to combinatorial flow is proposed by Kaczynski, Mrozek, and Wanner in 2016, which allows the development of Conley index theory on simplicial complexes. In this thesis, a proof for chaos in combinatorial Lorenz model by means of conjugacy is presented. The combinatorial flow approach toward Conley index is adapted to cubical complexes and theory is developed to some extent. A cubical analogue of contractible valued upper semi continuous map is also established

Random sampling of lattice configurations using local Markov chains

Algorithms based on Markov chains are ubiquitous across scientific disciplines, as they provide a method for extracting statistical information about large, complicated systems. Although these algorithms may be applied to arbitrary graphs, many physical applications are more naturally studied under the restriction to regular lattices. We study several local Markov chains on lattices, exploring how small changes to some parameters can greatly influence efficiency of the algorithms. We begin by examining a natural Markov Chain that arises in the context of "monotonic surfaces", where some point on a surface is sightly raised or lowered each step, but with a greater rate of raising than lowering. We show that this chain is rapidly mixing (converges quickly to the equilibrium) using a coupling argument; the novelty of our proof is that it requires defining an exponentially increasing distance function on pairs of surfaces, allowing us to derive near optimal results in many settings. Next, we present new methods for lower bounding the time local chains may take to converge to equilibrium. For many models that we study, there seems to be a phase transition as a parameter is changed, so that the chain is rapidly mixing above a critical point and slow mixing below it. Unfortunately, it is not always possible to make this intuition rigorous. We present the first proofs of slow mixing for three sampling problems motivated by statistical physics and nanotechnology: independent sets on the triangular lattice (the hard-core lattice gas model), weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model), and non-saturated Ising (tile-based self-assembly). Previous proofs of slow mixing for other models have been based on contour arguments that allow us prove that a bottleneck in the state space constricts the mixing. The standard contour arguments do not seem to apply to these problems, so we modify this approach by introducing the notion of "fat contours" that can have nontrivial area. We use these to prove that the local chains defined for these models are slow mixing. Finally, we study another important issue that arises in the context of phase transitions in physical systems, namely how the boundary of a lattice can affect the efficiency of the Markov chain. We examine a local chain on the perfect and near-perfect matchings of the square-octagon lattice, and show for one boundary condition the chain will mix in polynomial time, while for another it will mix exponentially slowly. Strikingly, the two boundary conditions only differ at four vertices. These are the first rigorous proofs of such a phenomenon on lattice graphs.Ph.D.Committee Chair: Randall, Dana; Committee Member: Heitsch, Christine; Committee Member: Mihail, Milena; Committee Member: Trotter, Tom; Committee Member: Vigoda, Eri