Searcharxiv⌕ Search

arXiv subjects

Stephane Ouvry

Publications and source records attributed to Stephane Ouvry.

32 records · Page 2Linked to original sources
arXiv2012

Arithmetic area for m planar Brownian paths

We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE information, valid in the 1-path case, on the 0-winding sectors arithmetic area.

math-ph↗
arXiv2011

Algebraic and arithmetic area for $m$ planar Brownian paths

The leading and next to leading terms of the average arithmetic area $< S(m)>$ enclosed by $m\to\infty$ independent closed Brownian planar paths, with a given length $t$ and starting from and ending at the same point, is calculated. The leading term is found to be $< S(m) > \sim {πt\over 2}\ln m$ and the $0$-winding sector arithmetic area inside the $m$ paths is subleading in the asymptotic regime. A closed form expression for the algebraic area distribution is also obtained and discussed.

math-ph↗
arXiv2010

Statistical Curse of the Second Half Rank

In competitions involving many participants running many races the final rank is determined by the score of each participant, obtained by adding its ranks in each individual race. The "Statistical Curse of the Second Half Rank" is the observation that if the score of a participant is even modestly worse than the middle score, then its final rank will be much worse (that is, much further away from the middle rank) than might have been expected. We give an explanation of this effect for the case of a large number of races using the Central Limit Theorem. We present exact quantitative results in this limit and demonstrate that the score probability distribution will be gaussian with scores packing near the center. We also derive the final rank probability distribution for the case of two races and we present some exact formulae verified by numerical simulations for the case of three races. The variant in which the worst result of each boat is dropped from its final score is also analyzed and solved for the case of two races.

cond-mat.stat-mech↗
arXiv2008

On the Lieb-Liniger model in the infinite coupling constant limit

We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model.

cond-mat.stat-mech↗
arXiv2008

Numerical studies of planar closed random walks

Lattice numerical simulations for planar closed random walks and their winding sectors are presented. The frontiers of the random walks and of their winding sectors have a Hausdorff dimension $d_H=4/3$. However, when properly defined by taking into account the inner 0-winding sectors, the frontiers of the random walks have a Hausdorff dimension $d_H\approx 1.77$.

cond-mat.stat-mech↗
arXiv2007

Integer Partitions and Exclusion Statistics

We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $ p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.

cond-mat.stat-mech↗
arXiv2005

Dimensional reduction on a sphere

The question of the dimensional reduction of two-dimensional (2d) quantum models on a sphere to one-dimensional (1d) models on a circle is adressed. A possible application is to look at a relation between the 2d anyon model and the 1d Calogero-Sutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength ωor on a circle of radius R - both parameters ωand R have to be viewed as long distance regulators - the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the Calogero-Moser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the Calogero-Sutherland model on a circle of the same radius. First, the free one-body case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the N-body interacting anyon model is considered : it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus lead to define a new spherical anyon-like model deduced from the Aharonov-Bohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode.

cond-mat.mes-hall↗
arXiv2005

Random Aharonov-Bohm vortices and some funny families of integrals

A review of the random magnetic impurity model, introduced in the context of the integer Quantum Hall effect, is presented. It models an electron moving in a plane and coupled to random Aharonov-Bohm vortices carrying a fraction of the quantum of flux. Recent results on its perturbative expansion are given. In particular, some funny families of integrals show up to be related to the Riemann $ζ(3)$ and $ζ(2)$.

cond-mat.mes-hall↗
arXiv2002

The Lowest Landau Level Anyon Equation of State in the Anti-screening Regime

The thermodynamics of the anyon model projected on the lowest Landau level (LLL) of an external magnetic field is addressed in the anti-screening regime, where the flux tubes carried by the anyons are parallel to the magnetic field. It is claimed that the LLL-anyon equation of state, which is known in the screening regime, can be analytically continued in the statistical parameter across the Fermi point to the antiscreening regime up to the vicinity (whose width tends to zero when the magnetic field becomes infinite) of the Bose point. There, an unphysical discontinuity arises due to the dropping of the non-LLL eigenstates which join the LLL, making the LLL approximation no longer valid. However, taking into account the effect of the non-LLL states at the Bose point would only smoothen the discontinuity and not alter the physics which is captured by the LLL projection: Close to the Bose point, the critical filling factor either goes to infinity (usual bosons) in the screening situation, or to 1/2 in the anti-screening situation, the difference between the flux tubes orientation being relevant even when they carry an infinitesimal fraction of the flux quantum. An exclusion statistics interpretation is adduced, which explains this situation in semiclassical terms. It is further shown how the exact solutions of the 3-anyon problem support this scenario as far as the third cluster coefficient is concerned.

cond-mat.mes-hall↗
arXiv2001

Discrete Thermodynamic Bethe Ansatz

We propose discrete TBA equations for models with discrete spectrum. We illustrate our construction on the Calogero-Moser model and determine the discrete 2-body TBA function which yields the exact N-body Calogero-Moser thermodynamics. We apply this algorithm to the Lieb-Liniger model in a harmonic well, a model which is relevant for the microscopic description of harmonically trapped Bose-Einstein condensates in one dimension. We find that the discrete TBA reproduces correctly the N-body groundstate energy of the Lieb-Liniger model in a harmonic well at first order in perturbation theory, but corrections do appear at second order.

cond-mat.stat-mech↗
arXiv1998

Persistent Currents and Magnetization in two-dimensional Magnetic Quantum Systems

Persistent currents and magnetization are considered for a two-dimensional electron (or gas of electrons) coupled to various magnetic fields. Thermodynamic formulae for the magnetization and the persistent current are established and the ``classical'' relationship between current and magnetization is shown to hold for systems invariant both by translation and rotation. Applications are given, including the point vortex superposed to an homogeneous magnetic field, the quantum Hall geometry (an electric field and an homogeneous magnetic field) and the random magnetic impurity problem (a random distribution of point vortices).

cond-mat↗
arXiv1993

Topological 2-Dimensional Quantum Mechanics

We define a Chern- Simons Lagrangian for a system of planar particles topologically interacting at a distance. The anyon model appears as a particular case where all the particles are identical. We propose exact N-body eigenstates, set up a perturbative algorithm, discuss the case where some particles are fixed on a lattice, and also consider curved manifolds. PACS numbers: 05.30.-d, 11.10.-z

hep-th↗