591 research outputs found
The right tail exponent of the Tracy-Widom-beta distribution
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))
Tracy-Widom at high temperature
We investigate the marginal distribution of the bottom eigenvalues of the
stochastic Airy operator when the inverse temperature tends to . We
prove that the minimal eigenvalue, whose fluctuations are governed by the
Tracy-Widom 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 . 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 . As an application, we investigate the
maximal eigenvalues statistics of -ensembles when the repulsion
parameter when . We study the double scaling limit
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.Comment: 5 figure
Random matrices in non-confining potentials
We consider invariant matrix processes diffusing in non-confining cubic
potentials of the form . 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 at a critical value
. If , then the potential presents a well near
deep enough to confine all the particles inside, and the spectral
density is supported on a compact interval. If however, the
steady state is in fact dynamical with a macroscopic stationary flux of
particles flowing across the system. In this regime, the eigenvalues allocate
according to a stationary density profile with full support in
, flanked with heavy tails such that as
. Our method applies to other non-confining potentials and we
further investigate a family of quartic potentials, which were already studied
in Br\'ezin et al. to count planar diagrams.Comment: 32 pages, 7 figure
Marginal densities of the "true" self-repelling motion
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.Comment: 20 pages, 7 figure
A clever (self-repelling) burglar
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.Comment: 21 pages, 8 figure
Anderson localization for the -d Schr\"odinger operator with white noise potential
We consider the random Schr\"odinger operator on obtained by
perturbing the Laplacian with a white noise. We prove that Anderson
localization holds for this operator: almost surely the spectral measure is
pure point and the eigenfunctions are exponentially localized. We give two
separate proofs of this result. We also present a detailed construction of the
operator and relate it to the parabolic Anderson model. Finally, we discuss the
case where the noise is smoothed out.Comment: 42 page
From Sine kernel to Poisson statistics
[no abstract available
Optimization of electricity / hydrogen cogeneration from generation IV nuclear energy systems
One of the great motivations of studying and developing Generation IV (Gen IV) reactors of VHTR (Very High Temperature Reactor) design concept is their capacity to efficiently produce both electricity and H2 (hydrogen). This study aims at developing an optimization methodology for cogeneration systems of H2 and electricity, from Gen IV nuclear reactors, with respect to energy constraints, economics and conjuncture in term of demand. It lies within a scope of a collaboration between the Laboratoire de GĂ©nie Chimique (Toulouse, France) and the Commissariat Ă lâEnergie Atomique (CEA, Cadarache, France) in order to compare various cogeneration systems from both energy and economics viewpoint.
This paper presents the results of an optimization study based on the âminimal destruction of exergyâ or âexergy lossâ concept. This criterion, used within the framework of a mono-objective genetic algorithm optimizer, was applied successfully to electric and heat production from Gen IV systems
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