Local limits of galton-watson trees conditioned on the number of protected nodes

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes

Discrete harmonic functions on an orthant in Zd

International audienceWe give a positive answer to a conjecture on the uniqueness of harmonic functions in the quarter plane stated by K. Raschel. More precisely we prove the existence and uniqueness of a positive discrete harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed at the boundary of an orthant in Zd. Our methodsallow on the other hand to generalize from the quarter plane to orthants in higher dimensions and to treat the spatially inhomogeneous walks

Intrinsic defects and mid-gap states in quasi-one-dimensional Indium Telluride

Recently, intriguing physical properties have been unraveled in anisotropic semiconductors, in which the in-plane electronic band structure anisotropy often originates from the low crystallographic symmetry. The atomic chain is the ultimate limit in material downscaling for electronics, a frontier for establishing an entirely new field of one-dimensional quantum materials. Electronic and structural properties of chain-like InTe are essential for better understanding of device applications such as thermoelectrics. Here, we use scanning tunneling microscopy/spectroscopy (STM/STS) measurements and density functional theory (DFT) calculations to directly image the in-plane structural anisotropy in tetragonal Indium Telluride (InTe). As results, we report the direct observation of one-dimensional In1+ chains in InTe. We demonstrate that InTe exhibits a band gap of about 0.40 +-0.02 eV located at the M point of the Brillouin zone. Additionally, line defects are observed in our sample, were attributed to In1+ chain vacancy along the c-axis, a general feature in many other TlSe-like compounds. Our STS and DFT results prove that the presence of In1+ induces localized gap state, located near the valence band maximum (VBM). This acceptor state is responsible for the high intrinsic p-type doping of InTe that we also confirm using angle-resolved photoemission spectroscopy.Comment: n

Quantum Confinement and Electronic Structure at the Surface of van der Waals Ferroelectric {\alpha}-In2_{2}Se3_{3}

Two-dimensional (2D) ferroelectric (FE) materials are promising compounds for next-generation nonvolatile memories, due to their low energy consumption and high endurance. Among them, {\alpha}-In$_{2}$Se$_{3}$ has drawn particular attention due to its in- and out-of-plane ferroelectricity, whose robustness has been demonstrated down to the monolayer limit. This is a relatively uncommon behavior since most bulk FE materials lose their ferroelectric character at the 2D limit due to depolarization field. Using angle resolved photoemission spectroscopy (ARPES), we unveil another unusual 2D phenomena appearing in 2H \alpha-In$_{2}$Se$_{3}$ single crystals, the occurrence of a highly metallic two-dimensional electron gas (2DEG) at the surface of vacuum-cleaved crystals. This 2DEG exhibits two confined states which correspond to an electron density of approximatively 10$^{13}$ electrons/cm$^{3}$, also confirmed by thermoelectric measurements. Combination of ARPES and density functional theory (DFT) calculations reveals a direct band gap of energy equal to 1.3 +/- 0.1 eV, with the bottom of the conduction band localized at the center of the Brillouin zone, just below the Fermi level. Such strong n-type doping further supports the quantum confinement of electrons and the formation of the 2DEG.Comment: 20 pages, 12 figure

Ramdom Walk and Galton-Watson trees

Dans cette thèse nous nous sommes intéressés de trois types de problèmes : 1 -Existence et unicité d’une fonction harmonique strictement positive associée à une marche aléatoire inhomogène confinée dans un orthant. 2 -Etude de la convergence en loi des arbres de Galton Watson critiques conditionnés à avoir un nombre assez grand de noeuds protégés. 3 -Etude de la convergence en loi des arbres de Galton Watson conditionnés à avoir une génération anormalement grande.In this thesis we are interested in three types of problems: 1-Existence and uniqueness of a positive harmonic function associated with an inhomogeneous random walk confined in an orthant. 2-Study of convergence in distribution of critical Galton Watson trees conditioned to have a large enoughnumber of protected nodes. 3-Study of the convergence in distribution of Galton Watson trees conditioned to have a large generation

LOCAL LIMITS OF GALTON-WATSON TREES CONDITIONED ON LARGE WIDTH

International audienceWe study the local convergence of critical Galton-Watson trees under various conditionings. We give a sufficient condition, which serves to cover all the previous cases, for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. We also propose an other proof to give the limit in distribution of a critical Galton-Watson tree, with bounded support, conditioned on having a large width

Marches aléatoires et arbres de Galton-Watson

In this thesis we are interested in three types of problems: 1-Existence and uniqueness of a positive harmonic function associated with an inhomogeneous random walk confined in an orthant. 2-Study of convergence in distribution of critical Galton Watson trees conditioned to have a large enoughnumber of protected nodes. 3-Study of the convergence in distribution of Galton Watson trees conditioned to have a large generation.Dans cette thèse nous nous sommes intéressés de trois types de problèmes : 1 -Existence et unicité d’une fonction harmonique strictement positive associée à une marche aléatoire inhomogène confinée dans un orthant. 2 -Etude de la convergence en loi des arbres de Galton Watson critiques conditionnés à avoir un nombre assez grand de noeuds protégés. 3 -Etude de la convergence en loi des arbres de Galton Watson conditionnés à avoir une génération anormalement grande

Very fat geometric galton-watson trees

Let τn be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n ∈ N *) is a deterministic positive sequence. We study the local limit of these trees τn as n → ∞ and observe three distinct regimes: if (an, n ∈ N *) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n ∈ N *) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring

Very fat geometric galton-watson trees

International audienceLet τn be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n ∈ N *) is a deterministic positive sequence. We study the local limit of these trees τn as n → ∞ and observe three distinct regimes: if (an, n ∈ N *) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n ∈ N *) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring