Cette thèse porte sur le modèle de Hofstadter i.e., un électron qui se déplace sur un réseau carré couplé à un champ magnétique homogène et perpendiculaire au réseau. Son spectre en énergie est l'un des célèbres fractals de la physique quantique, connu sous le nom "le papillon de Hofstadter". Cette thèse consiste en deux parties principales: la première est l'étude du lien profond entre le modèle de Hofstadter et la distribution de l’aire algébrique entourée par les marches aléatoires sur un réseau carré bidimensionnel. La seconde partie se concentre sur les caractéristiques spécifiques du papillon de Hofstadter et l'étude de la largeur de bande du spectre. On a trouvé une formule exacte pour la trace de l'Hamiltonien de Hofstadter en termes des coefficients de Kreft, et également pour les moments supérieurs de la largeur de bande.Cette thèse est organisée comme suit. Dans le chapitre 1, on commence par la motivation de notre travail. Une introduction générale du modèle de Hofstadter ainsi que des marches aléatoires sera présentée. Dans le chapitre 2, on va montrer comment utiliser le lien entre les marches aléatoires et le modèle de Hofstadter. Une méthode de calcul de la fonction génératrice de l'aire algébrique entourée par les marches aléatoires planaires sera expliquée en détail. Dans le chapitre 3, on va présenter une autre méthode pour étudier ces questions en utilisant le point de vue "point spectrum traces" et retrouver la trace de Hofstadter complète. De plus, l'avantage de cette construction est qu'elle peut être généralisée au cas de "l'amost Mathieu opérateur". Dans le chapitre 4, on va introduire la méthode développée par D.J.Thouless pour le calcul de la largeur de bande du spectre de Hofstadter. En suivant la même logique, on va montrer comment généraliser la formule de la largeur de bande de Thouless à son n-ième moment, à définir plus précisément ultérieurement. / This thesis is about the Hofstadter model, i.e., a single electron moving on a two-dimensional lattice coupled to a perpendicular homogeneous magnetic field. Its spectrum is one of the famous fractals in quantum mechanics, known as the Hofstadter's butterfly. There are two main subjects in this thesis: the first is the study of the deep connection between the Hofstadter model and the distribution of the algebraic area enclosed by two-dimensional random walks. The second focuses on the distinctive features of the Hofstadter's butterfly and the study of the bandwidth of the spectrum. We found an exact expression for the trace of the Hofstadter Hamiltonian in terms of the Kreft coefficients, and for the higher moments of the bandwidth.This thesis is organized as follows. In chapter 1, we begin with the motivation of our work and a general introduction to the Hofstadter model as well as to random walks will be presented. In chapter 2, we will show how to use the connection between random walks and the Hofstadter model. A method to calculate the generating function of the algebraic area distribution enclosed by planar random walks will be explained in details. In chapter 3, we will present another method to study these issues, by using the point spectrum traces to recover the full Hofstadter trace. Moreover, the advantage of this construction is that it can be generalized to the almost Mathieu operator. In chapter 4, we will introduce the method which was initially developed by D.J.Thouless to calculate the bandwidth of the Hofstadter spectrum. By following the same logic, I will show how to generalize the Thouless bandwidth formula to its n-th moment, to be more precisely defined later.

Cette thèse se situe à l'interface entre combinatoire et probabilités,et contribue à l'étude de différents modèles issus de la mécanique statistique : polymères, marches aléatoires inter-agissantes ou en milieu aléatoire, cartes aléatoires. Le premier modèle que nous étudions est une famille de mesures de probabilités sur les chemins auto-évitants de longueur infinie sur un réseau régulier, construites à partir de marches aléatoires biaisées sur l'arbre des chemins auto-évitants finis. Ces mesures, introduites par Beretti et Sokal, existent pour tout biais strictement supérieur à l'inverse de la constante de connectivité, et leur limite en ce biais critique serait l'un des définitions naturelles de la marche aléatoire uniforme en longueur infinie. Le but de ce travail, en collaboration avec Vincent Beffara, est de comprendre le lien entre cette limite, si elle existe, et d'autres chemins aléatoires notamment la mesure de Kesten (qui est la limite faible de la marche auto-évitante uniforme dans le demi-plan) et les interfaces de percolation de Bernoulli critique; d'une certaine façon le modèle constitue une interpolation entre les deux. Dans une deuxième partie, nous considérons des marches aléatoires en conductances aléatoires sur un arbre quelconque, dans le cas où la loi des conductances est à queue lourde. L’objectif de notre travail, en collaboration avec Andrea Collevecchio et Daniel Kious, est de montrer une transition de phase par rapport au paramètre de la queue; on exprime le paramètre critique comme une fonction explicite de l'arbre sous-jacent. Parallèlement, nous étudions des modèles de marches aléatoires excitées sur des arbres et leurs transitions de phase. En particulier, nous étendons une conjecture de Volkov et généralisons des résultats de Bas devant et Singh. Enfin, une troisième partie en collaboration avec Vincent Beffara et Benjamin Lévêque porte sur les cartes aléatoires en genre supérieur : nous montrons l'existence de limites d'échelle, le long de sous-suites, pour les triangulations simples uniformes sur le tore, étendant à ce cas les résultats d'Adario-Berri et Albenque (sur les triangulations simples de la sphère) et de Bettinelli (sur les quadrangulations du tore). La question de l'unicité de la limite et de son universalité restent ouvertes, mais nous obtenons des résultats partiels dans ce sens. / This thesis is at the interface between combinatorics and probability,and contributes to the study of a few models stemming from statisticalmechanics: polymers, self-interacting random walks and random walks inrandom environment, random maps.bigskipThe first model that we investigate is a one-parameter family ofprobability measures on self-avoiding paths of infinite length on aregular lattice, constructed from biased random walks on the tree offinite self-avoiding paths. These measures, initially introduced byBeretti and Sokal, exist for every bias larger than the inverseconnectivity constant, and their limit at the critical bias would beaamong the natural definitions of the uniform self-avoiding walk ofinfinite length. The aim of our work, in collaboration with VincentBeffara, is to understand the link between this limit, if it indeedexists, and other random infinite paths such as Kesten's measure(which is the weak limit of uniformly random finite self-avoidingwalks in the half-plane) and critical Bernoulli percolationinterfaces; the model can be seen as an interpolation between thesetwo.In a second part, we consider random walks with random conductances ona tree, in the case when the law of the conductances has heavy tail.Our aim, in collabration with Andrea Collevecchio and Daniel Kious, isto show a phase transition in the tail parameter; we express thecritical point as an explicit function of the underlying tree.In parallel, we study excited random walks on trees and their phasetransitions: we extend a conjecture of Volkov's and generalize resultsby Basdevant and Singh.Finally, a third part in collaboration with Vincent Beffara andBenjamin Lévêque contributes to the study of random maps of highergenus: we show the existence of subsequential scaling limits foruniformly random simple triangulations of the torus, extending to thatsetup fromer results by Adario-Berri and Albenque (on simpletriangulations of the sphere) and by Bettinelli (on quadrangulationsof the torus). The question of uniqueness and universality of thelimit remain open, but we obtain partial results in that direction.

This thesis studies two Markov processes describing the evolution of a system of many interacting random components. These processes undergo an absorbing-state phase transition, i.e., as one variates the parameter values, the process exhibits a transition from a convergence regime to one of the absorbing-states to an active regime. In Chapter 2 we study Activated Random Walk, which is an interacting particle system where the particles can be of two types and their number is conserved. Firstly, we provide a new lower bound for the critical density on Z as a function of the jump distribution and of the sleeping rate and we prove that the critical density is not a constant function of the jump distribution. Secondly, we prove that on Zd in the case of biased jump distribution the critical density is strictly less than one, provided that the sleeping rate is small enough. This answers a question that has been asked by Dickman, Rolla, Sidoravicius [9, 28] in the case of biased jump distribution. Our results have been presented in [33]. In Chapter 3 we study a class of probabilistic cellular automata which are related by a natural coupling to a special type of oriented percolation model. Firstly, we consider the process on a finite torus of size n, which is ergodic for any parameter value. By employing dynamic-renormalization techniques, we prove that the average absorption time grows exponentially (resp. logarithmically) with n when the model on Z is in the active (resp. absorbing) regime. This answers a question that has been asked by Toom [37]. Secondly, we study how the neighbourhood of the model affects the critical probability for the process on Z. We provide a lower bound for the critical probability as a function of the neighbourhood and we show that our estimates are sharp by comparing them with our numerical estimates. Our results have been presented in [34, 35].

Cette thèse présente une méthode de segmentation de modèles 3D en parties significatives ou fonctionnelles. La segmentation s’effectue par "transfert" d’une segmentation exemple : la segmentation d’un modèle est calculée en transférant les segments d’une segmentation exemple d’un objet appartenant à la même classe de modèles 3D. Pour ce faire, nous avons adapté et étendu la méthode de segmentation par les marches aléatoires et transformé notre problème en un problème de localisation et mise en correspondance de faces germes. Notre méthode comporte quatre étapes fondamentales : la mise en correspondance entre le modèle exemple et le modèle cible, la localisation automatique de germes sur le modèle cible pour initialiser les régions, le calcul des segments du modèle cible et l’amélioration de leurs frontières. En constatant que les critères de similarité diffèrent selon que les objets sont de type rigide (chaises, avions,…) ou de type articulé (humains, quadrupèdes,…), nous décomposons notre approche en deux. La première dédiée aux objets rigides, où la mise en correspondance est basée sur le calcul des transformations rigides afin d’aligner au mieux les parties significatives des deux objets comparés. La deuxième dédiée aux modèles articulés, où la mise en correspondance des parties fonctionnelles, présentant des variations de poses plus importantes, est basée sur des squelettes calculés via des diagrammes de Reeb. Nous montrons à travers des évaluations qualitatives et quantitatives que notre méthode obtient des résultats meilleurs que les techniques de segmentation individuelle et comparables aux techniques de co-segmentation avec un temps de calcul nettement inférieur. / In this dissertation, we present a new method to segment 3D models into their functional parts. The segmentation is performed by a transfer approach: a semantic-oriented segmentation of an object is calculated using a pre-segmented example model from the same class (chairs, humans, etc.). To this end, we adapted and extended the random walk segmentation method which allowed us to transform our problem into a problem of locating and matching seed faces. Our method consists of four fundamental steps: establishing correspondences between the example and the target model, localizing seeds to initialize regions in the target model, computing the segments and refining their boundaries in the target model. We decomposed our approach in two, taking into account similarity criteria which differ regarding the object type (rigid vs. articulated). The first approach is dedicated to rigid objects (chairs, airplanes, etc.), where the matching is based on rigid transformations to determine the best alignment between the functional parts of the compared objects. The second one focused on articulated objects (humans, quadrupeds, etc.), where coarse topological shape attributes are used in a skeleton-based approach to cover larger pose variations when computing correspondences between functional parts. We show through qualitative and quantitative evaluations that our method improves upon individual segmentation techniques and obtains results that are close to the co-segmentation techniques results with an important calculation time reduction.

No description available.

Demyelination is the process where the insulating layer of axons known as myelin is damaged. This affects the propagation of action potentials along axons which can have deteriorating consequences on the motor activity of an organism. Thus it is important to understand the biophysical effects of demyelination to improve the diagnostics of its diseases. We trained a Convolutional Neural Network (CNN) on Coherent anti-Stokes Raman scattering (CARS) microscope images of mice spinal cord inflicted with the demyelinating disease Experimental Autoimmune Encephalomyelitis (EAE). Our CNN was able to classify the images reliably based on clinical scores assigned to the mice. We then synthesized our own images of the spinal cord regions using a 2D Biased Random Walk. These images are simplified versions of the original CARS images and show homogenously myelinated axons, unlike the heterogeneous nerve fibres found in real spinal cords. The images were fed into the trained CNN as an attempt to develop a clinical connection to the biophysical effects of demyelination. We found that the trained CNN was indeed able to capture structural features related to demyelination which can allow us to constrain demyelination models such that they include the simulated parameters of the synthesized images.

Graphs are a great tool for representing complex physical and social systems, where the interactions among many units, from tens of animal species in a food-web, to millions of users in a social network, give rise to emergent, complex system behaviours. In the field of network science this representation, which is usually called a complex network, can be complicated at will to better represent the real system under study. For instance, interactions may be directed or may differ in their strength or cost, leading to directed weighted networks, but they may also depend on time, like in temporal networks, or nodes (i.e. the units of the system) may interact in different ways, in which case edge-coloured multi-graphs and multi-layer networks represent better the system. Besides this rich repertoire of network structures, we cannot forgot that edges represent interactions and that this interactions are not static, but are, instead, purposely established to reach some function of the system, as for instance, routing people and goods through a transportation network or cognition, through the exchange of neuro-physiological signals in the brain. Building on the foundations of spectral graph theory, of non-linear dimensionality reduction and diffusion maps, and of the recently introduced diffusion distance [Phys. Rev. Lett. 118, 168301 (2017)] we use the simple yet powerful tool of continuous-time Markov chains on networks to model their process-driven geometry and characterise their functional shape. The main results are: (i) the generalisation of the diffusion geometry framework to different types of interconnected systems (from edge-coloured multigraphs to multi-layer networks) and of random walk dynamics [Phys. Rev. E 103, 042301 (2021)] and (ii) the introduction of new descriptors based on the diffusion geometry to quantify and describe the micro- (through the network depth [J. Complex Netw. 8, 4 (2020)]), meso- (functional rich-club) and macro-scale (using statistics of the pairwise distances between the network's nodes [Comm. Phys. 4, 125 (2021)]) of complex networks.

No description available.

No description available.

Quantum walks have been utilized by many quantum algorithms which provide improved performance over their classical counterparts. Quantum search algorithms, the quantum analogues of spatial search algorithms, have been studied on a wide variety of structures. We study quantum walks and searches on the Cayley graphs of finitely-generated free groups. Return properties are analyzed via Green’s functions, and quantum searches are examined. Additionally, the stopping times and success rates of quantum searches on random networks are experimentally estimated. / Mathematics