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Search Results: 1 - 10 of 951 matches for " Laure Dumaz "
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Large deviations and path properties of the true self-repelling motion
Laure Dumaz
Mathematics , 2011,
Abstract: We derive some large deviation bounds for events related to the "true self-repelling motion", a one-dimensional self-interacting process introduced by Toth and Werner, that has very different path properties than usual diffusion processes. We then use these estimates to study certain of these path properties such as its law of iterated logarithms for both small and large times.
A clever (self-repelling) burglar
Laure Dumaz
Mathematics , 2012, DOI: 10.1214/EJP.v17-1758
Abstract: We derive the following property of the "true self-repelling motion", a continuous real-valued self-interacting process (X_t, t \ge 0) introduced by Balint Toth and Wendelin Werner. Conditionally on its occupation time measure at time one (which is the information about how much time it has spent where before time one), the law of X_1 is uniform in a certain admissible interval. This contrasts with the corresponding conditional distribution for Brownian motion that had been studied by Warren and Yor.
From Sine kernel to Poisson statistics
Romain Allez,Laure Dumaz
Mathematics , 2014, DOI: 10.1214/EJP.v19-3742
Abstract: We study the Sine$_\beta$ process introduced in [B. Valk\'o and B. Vir\'ag. Invent. math. (2009)] when the inverse temperature $\beta$ tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of $\beta$-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine$_\beta$ point process converges weakly to a Poisson point process on $\mathbb{R}$. Thus, the Sine$_\beta$ point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $\beta=\infty$) and the Poisson process.
Random matrices in non-confining potentials
Romain Allez,Laure Dumaz
Mathematics , 2014, DOI: 10.1007/s10955-015-1258-1
Abstract: We consider invariant matrix processes diffusing in non-confining cubic potentials of the form $V_a(x)= x^3/3 - a x, a\in \mathbb{R}$. We construct the trajectories of such processes for all time by restarting them whenever an explosion occurs, from a new (well chosen) initial condition, insuring continuity of the eigenvectors and of the non exploding eigenvalues. We characterize the dynamics of the spectrum in the limit of large dimension and analyze the stationary state of this evolution explicitly. We exhibit a sharp phase transition for the limiting spectral density $\rho_a$ at a critical value $a=a^*$. If $a\geq a^*$, then the potential $V_a$ presents a well near $x=\sqrt{a}$ deep enough to confine all the particles inside, and the spectral density $\rho_a$ is supported on a compact interval. If $a
Tracy-Widom at high temperature
Romain Allez,Laure Dumaz
Statistics , 2013, DOI: 10.1007/s10955-014-1058-z
Abstract: We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature $\beta$ tends to $0$. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy-Widom $\beta$ law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index $k$. Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in [L. Dumaz and B. Vir\'ag. Ann. Inst. H. Poincar\'e Probab. Statist. 49, 4, 915-933, (2013)]. We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when $\beta\to 0$. As an application, we investigate the maximal eigenvalues statistics of $\beta_N$-ensembles when the repulsion parameter $\beta_N\to 0$ when $N\to +\infty$. We study the double scaling limit $N\to +\infty, \beta_N \to 0$ and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of [A. Edelman and B. D. Sutton. J. Stat. Phys. 127, 6, 1121-1165 (2007)] and [J. A. Ram\'irez, B. Rider and B. Vir\'ag. J. Amer. Math. Soc. 24, 919-944 (2011)] from our later study of the stochastic Airy operator.
Marginal densities of the "true" self-repelling motion
Laure Dumaz,Bálint Tóth
Mathematics , 2012,
Abstract: Let X(t) be the true self-repelling motion (TSRM) constructed by B.T. and Wendelin Werner in 1998, L(t,x) its occupation time density (local time) and H(t):=L(t,X(t)) the height of the local time profile at the actual position of the motion. The joint distribution of (X(t),H(t)) was identified by B.T. in 1995 in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of X(t) and H(t). The distribution of X(t) has a particularly surprising shape: It has a sharp local minimum with discontinuous derivative at 0. As a consequence we also obtain a precise version of the large deviation estimate of arXiv:1105.2948v3.
Near-critical spanning forests and renormalization
Stéphane Benoist,Laure Dumaz,Wendelin Werner
Mathematics , 2015,
Abstract: We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their scaling limit as slight perturbations of an uniformly chosen spanning tree. We show how to relate this scaling limit to a stationary distribution of a natural Markov process on a state of abstract graphs with non-constant edge-weights. This simple Markov process can be viewed as a renormalization flow, so that in this two-dimensional case, one can give a rigorous meaning to the fact that there is a unique fixed point (ie. stationary distribution) in two dimensions for this renormalization flow, and that when starting from any periodic two-dimensional lattice, the renormalization flow converges to this fixed point (ie. the Markov process converges in law to its stationary distribution). While the results of this paper are dealing with the planar case and build on the convergence in distribution of branches of the UST to SLE$_2$ as well as on the predicted convergence of the suitably renormalized length of the loop-erased random walk to the "natural parametrization" of the SLE$_2$, this Markov process setup is in fact not restricted to two dimensions.
The right tail exponent of the Tracy-Widom-beta distribution
Laure Dumaz,Bálint Virág
Mathematics , 2011, DOI: 10.1214/11-AIHP475
Abstract: The Tracy-Widom beta distribution is the large dimensional limit of the top eigenvalue of beta random matrix ensembles. We use the stochastic Airy operator representation to show that as a tends to infinity the tail of the Tracy Widom distribution satisfies P(TW_beta > a) = a^(-3/4 beta+o(1)) exp(-2/3 beta a^(3/2)).
Existence of solutions for non-necessarily cooperative systems involving Schr?dinger operators
Laure Cardoulis
International Journal of Mathematics and Mathematical Sciences , 2001, DOI: 10.1155/s016117120100401x
Abstract: We study the existence of a solution for a non-necessarilycooperative system of n equations involving Schrödinger operators defined on ℝN and we study also a limit case (the Fredholm Alternative (FA)). We derive results for semilinear systems.
La Funció Pública Europea - La función pública europea - The Civil Service in Europe
Laure Levi
Revista Catalana de Dret Públic , 2012,
Abstract: La funció pública europea està formada per funcionaris europeus sotmesos a un Estatut i per agents contractuals de dret públic. L’Estatut, principal fonament de la funció pública europea, té la finalitat de regular les relacions jurídiques entre les institucions i els seus funcionaris, establint entre aquests drets i obligacions recíproques. Funcionaris i agents són, per tant, titulars de drets previstos a l'Estatut i al dret primari (com els drets de defensa, el dret a la motivació, la llibertat d'expressió, la llibertat sindical, el respecte de la vida privada i familiar). Però també són titulars de deures i obligacions la violació dels quals pot donar lloc a mesures disciplinàries i a la ruptura de la relació laboral. Les institucions europees esperen que el seu personal respecti les altes normes professionals i ètiques. L’Estatut estableix i organitza els drets i obligacions dels funcionaris. També organitza el règim disciplinari, és a dir, el règim que s'aplica quan un funcionari – o antic funcionari – ignora les seves obligacions professionals, ja s’hagi comès la falta de manera voluntària o per negligència. La función pública europea está formada por funcionarios europeos sometidos a un estatuto y por agentes contractuales de derecho público. El Estatuto, principal fundamento de la función pública europea, tiene como objetivo regular las relaciones jurídicas entre las instituciones y sus funcionarios, estableciendo entre estos derechos y obligaciones recíprocas. Los funcionarios y los agentes son, por lo tanto, titulares de derechos previstos en el estatuto y en el derecho primario (como los derechos de defensa, el derecho a la motivación, la libertad de expresión, la libertad sindical, el respeto de la vida privada y familiar). Pero también son titulares de deberes y obligaciones cuya violación puede dar lugar a medidas disciplinarias y a la ruptura de la relación laboral. Las instituciones europeas esperan que su personal respete las altas normas profesionales y éticas. El estatuto establece y organiza los derechos y obligaciones de los funcionarios. También organiza el régimen disciplinario, es decir, el régimen que se aplica cuando un funcionario —o antiguo funcionario— ignora sus obligaciones profesionales, ya se haya cometido la falta voluntariamente o por negligencia. The European civil service is made up, on the one hand, of European civil servants who are governed by a statute, and agents working under public law employment contract. The statute, which is the main foundation for the European civil service, is aimed at regulating the lega
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