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Limits of determinantal processes near a tacnode

Alexei Borodin, Maurice Duits (2011)

Annales de l'I.H.P. Probabilités et statistiques

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process....

Linear convergence in the approximation of rank-one convex envelopes

Sören Bartels (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A linearly convergent iterative algorithm that approximates the rank-1 convex envelope f r c of a given function f : ℝ n × m → ℝ , i.e. the largest function below f which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.

Linear convergence in the approximation of rank-one convex envelopes

Sören Bartels (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A linearly convergent iterative algorithm that approximates the rank-1 convex envelope  f r c of a given function f : ℝ n × m → ℝ , i.e. the largest function below f which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.

Line-energy Ginzburg-Landau models : zero-energy states

Pierre-Emmanuel Jabin, Felix Otto, BenoÎt Perthame (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a class of two-dimensional Ginzburg-Landau problems which are characterized by energy density concentrations on a one-dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero-energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero-energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful...

Lipschitz percolation.

Dirr, Nicolas, Dondl, Patrick W., Grimmett, Geoffrey R., Holroyd, Alexander E., Scheutzow, Michael (2010)

Electronic Communications in Probability [electronic only]

Local minimizers with vortex filaments for a Gross-Pitaevsky functional

Robert L. Jerrard (2007)

ESAIM: Control, Optimisation and Calculus of Variations

This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filaments in a rotationally forced Bose-Einstein condensate. This functional is derived as a Γ-limit of scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence...

Local order at arbitrary distances in finite-dimensional spin-glass models

Pierluigi Contucci, Francesco Unguendoli (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

For a finite dimensional spin-glass model we prove low temperature local order i.e. the property of concentration of the overlap distribution close to the value 1. The theorem hold for both local observables and for products of observables at arbitrary mutual distance: when the Hamiltonian includes the Edwards-Anderson interaction we prove bond local order, when it includes the random-field interaction we prove site local order.

Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov (2014)

Annales de l'I.H.P. Probabilités et statistiques

Random interlacements at level u is a one parameter family of connected random subsets of ℤ d , d ≥ 3 (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level u , exhibits a non-trivial percolation phase transition in u (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements...

Localisation for non-monotone Schrödinger operators

Alexander Elgart, Mira Shamis, Sasha Sodin (2014)

Journal of the European Mathematical Society

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schrödinger operators with non-monotone random potentials, on the d -dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis...

Localisation pour des opérateurs de Schrödinger aléatoires dans L 2 ( ℝ d ) : un modèle semi-classique

Frédéric Klopp (1995)

Annales de l'institut Fourier

Dans L 2 ( ℝ d ) , nous démontrons un résultat de localisation exponentielle pour un opérateur de Schrödinger semi-classique à potentiel périodique perturbé par de petites perturbations aléatoires indépendantes identiquement distribuées placées au fond de chaque puits. Pour ce faire, on montre que notre opérateur, restreint à un intervalle d’énergie convenable, est unitairement équivalent à une matrice aléatoire infinie dont on contrôle bien les coefficients. Puis, pour ce type de matrices, on prouve un résultat...

Localization for Schrödinger operators with Poisson random potential

Abel Klein, Peter Hislop, François Germinet (2007)

Journal of the European Mathematical Society

We prove exponential and dynamical localization for the Schr¨odinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.

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