Schramm's formula for multiple loop-erased random walks

We revisit the computation of the discrete version of Schramm's formula for the loop-erased random walk derived by Kenyon. The explicit formula in terms of the Green function relies on the use of a complex connection on a graph, for which a line bundle Laplacian is defined. We give explicit results in the scaling limit for the upper half-plane, the cylinder and the Moebius strip. Schramm's formula is then extended to multiple loop-erased random walks.Comment: 59 pages, 19 figures. v2: reformulation of Section 2.3, minor correction

Sandpile probabilities on triangular and hexagonal lattices

We consider the Abelian sandpile model on triangular and hexagonal lattices. We compute several height probabilities on the full plane and on half-planes, and discuss some properties of the universality of the model.Comment: 26 pages, 12 figures. v2 and v3: minor correction

Multipoint correlators in the Abelian sandpile model

We revisit the calculation of height correlations in the two-dimensional Abelian sandpile model by taking advantage of a technique developed recently by Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian, ubiquitous in the context of cycle-rooted spanning forests, with a complex connection. In the case at hand, the connection is constant and localized along a semi-infinite defect line (zipper). In the appropriate limit of a trivial connection, it allows one to count spanning forests whose components contain prescribed sites, which are of direct relevance for height correlations in the sandpile model. Using this technique, we first rederive known 1- and 2-site lattice correlators on the plane and upper half-plane, more efficiently than what has been done so far. We also compute explicitly the (new) next-to-leading order in the distances ($r^{-4}$ for 1-site on the upper half-plane, $r^{-6}$ for 2-site on the plane). We extend these results by computing new correlators involving one arbitrary height and a few heights 1 on the plane and upper half-plane, for the open and closed boundary conditions. We examine our lattice results from the conformal point of view, and confirm the full consistency with the specific features currently conjectured to be present in the associated logarithmic conformal field theory.Comment: 60 pages, 21 figures. v2: reformulation of the grove theorem, minor correction

Applications of the line bundle Laplacian to spanning forests and related lattice models

Over the last thirty years, conformal field theories (CFTs) have proved to be a powerful tool in physics to study two-dimensional lattice models at their critical point. In the scaling limit, discrete statistical correlators are expected to converge to those of a quantum field theory. In the simplest cases, these correlators are rational functions of the distances separating the insertion points. However, correlators with logarithms may also appear in so-called logarithmic conformal field theories (LCFTs), which possess a richer and more complicated structure than their rational counterparts. One way to gain insight into LCFTs is by computing exact correlators in lattice models believed to be described by such theories. Among them is the Abelian sandpile model (ASM), which was introduced as a toy model for dynamical systems that naturally evolve toward their critical state. Its most natural observables are the height variables, which are controlled by a nonlocal probability measure. Computing height correlations in the ASM amounts to enumerating spanning trees with certain nonlocal properties, which is quite cumbersome via standard methods of graph theory. The first objective of this thesis consists therefore in using a recent and more efficient technique to produce explicit results, which is based on a connection encoded in the so-called line bundle Laplacian. On the mathematical side, Schramm-Loewner evolutions (SLEs) were developed in the early 2000s as a more rigorous approach to study scaling limits of random paths and interfaces appearing in two- dimensional critical lattice models. Since then, SLEs have been proved to describe the continuum limit of a few well-known models, such as percolation, the Ising model, and the loop-erased random walk. One of the simplest explicit computations in the SLE framework is Schramm’s formula, which gives the probability that a random curve between two boundary points of a continuous domain goes left of a marked interior point. The second objective of this thesis is to establish a discrete equivalent of this formula for a specific model, namely the loop-erased random walk (LERW), via its relation with spanning trees. The main tool is the line bundle Laplacian, which is also used to extend Schramm’s formula to multiple LERWs. Explicit results are then compared to SLE and CFT predictions.(SC - Sciences) -- UCL, 201

Etude préliminaire sur la localisation du nickel dans une garniérite naturelle

This paper gives the results of a preliminary study carried out on a sample of garnierite which is a mixture of a 7 Å (serpentine-like) and 10 Å (talc-like) minerals. The use of several techniques of investigation, e. g. electron microscope fitted wit XDS (dispersion energy spectrometer) and other volumetric methods, allowed to show that an important part of the nickel is present as an amorphous nickel compound (probably as nickel hydroxide) associated with the 10 Å mineral but not involved in its octahedral layer. Furthermore, it is shown that most of the lattice nickel lies in the octahedral layer of the 10 Å phase, and not homogeneously distributed in the octahedral layer of both the 7 Å and the 10 Å mineral.Dans cet article, nous donnons les résultats d'une étude préliminaire menée sur un échantillon de garniérite constitué d'un mélange de minéraux 1 : 1 et 2 : 1. L'utilisation conjointe de plusieurs méthodes d'investigation, dont notamment la microscopie électronique couplée à un spectromètre à dispersion d'énergie et certaines méthodes volumétriques, a permis de montrer qu'une proportion importante de nickel se trouve sous forme d'un composé amorphe, probablement de l'hydroxyde associé à la phase 10 Å, mais ne faisant pas partie de la couche octaédrique de la phase minérale. En outre, l'analyse par microsonde à haute résolution indique que le nickel réticulaire est localisé principalement dans la phase 10 Å, et non réparti de manière homogène entre les phases à 7 Å et à 10 Å.Poncelet Georges, Jacobs Pierre, Delannay Francis, Genet Michel, Gérard Paul, Herbillon Adrien-Jules. Étude préliminaire sur la localisation du nickel dans une garniérite naturelle. In: Bulletin de Minéralogie, volume 102, 4, 1979. pp. 379-385

Integrated Digital Image Correlation considering gray level and blur variations: Application to distortion measurements of IR camera

International audienceThe acquisition of images with different modalities may involve different alterations with respect to an ideal model. Inhomogeneous brightness and contrast, blur due to non-ideal focusing, distortions are common. It is proposed herein to account for such effects for instance by registering a calibration target image with an actual optical image to measure lens distortions. An Integrated Digital Image Correlation (I-DIC) algorithm is proposed to account for the above artifacts and the algorithm is detailed. The resolution and uncertainty of the technique are first investigated on synthetic images, and then applied to the measurement of distortions for infrared (IR) images. The procedure is shown to reduce drastically the residual level assessing the validity of the image formation model, but more importantly allowing for a much improved registration of images

Critical assessment of the impact of process parameters on vertical roughness and hardness of thin walls of AlSi10Mg processed by laser powder bed fusion

This work deals with the impact of prominent process parameters in laser powder bed fusion (L-PBF) technique, also called selective laser melting (SLM), on relevant properties of AlSi10Mg such as surface roughness and hardness. More particularly, the influence of scanning strategies, track energy density (TED), and of the offset between the contours and the bulk hatching were scrutinized. Basic thin structures such as single wall tracks (SWT) and ultra thin walls (UTW) were used to isolate the effect of the TED and of the offset between parallel melt pool tracks on the vertical roughness, respectively. It is shown that the TED reduces the level of roughness mostly due to a geometrical effect. The optimum range of TED is shown to be at the beginning of the keyhole conditions, at the expense of the amount of sub-surface porosities and of hardness. Increasing the offset also allows reducing the roughness. Thicker structures with contours and bulk hatching were also used and brought the same conclusions. In this work, low levels of roughness (about 4 μm) have been obtained on vertical walls without any post-treatment but at the expense of the mechanical properties

Effect of scanning strategies and laser parameters on roughness, porosities and hardness of SLM AlSi10Mg thin-walls

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Effect of scanning strategy and laser parameters on roughness, porosities and hardness of SLM AlSi10Mg thin-walls

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