Results 1 
6 of
6
Infinite systems of noncolliding generalized meanders and RiemannLiouville differintegrals
, 2005
"... Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν+1)dimensional Bessel process with ν> −1, in which the inhomogeneity is indexed by κ ∈ [0, 2(ν + 1)). We introduce the noncolliding particle systems of the generalized meanders and prove that they are the Pfaffian p ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν+1)dimensional Bessel process with ν> −1, in which the inhomogeneity is indexed by κ ∈ [0, 2(ν + 1)). We introduce the noncolliding particle systems of the generalized meanders and prove that they are the Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the RiemannLiouville differintegrals of functions comprising the Bessel functions Jν used in the fractional calculus, where orders of differintegration are determined by ν − κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.
No Multiple Collisions for Mutually Repelling Brownian Particles
, 2005
"... Summary. Although Brownian particles with small mutual electrostatic repulsion may collide, multiple collisions at positive time are always forbidden. 1 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Summary. Although Brownian particles with small mutual electrostatic repulsion may collide, multiple collisions at positive time are always forbidden. 1
Interacting Brownian particles with strong repulsion
"... We consider a nite system of interacting Brownian particles with strong repulsion on the line or on the circle. The involved equations may be interpreted as the multidimensional extension of some singular stochastic differential equations on a halfline or on a compact interval. As the number o ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We consider a nite system of interacting Brownian particles with strong repulsion on the line or on the circle. The involved equations may be interpreted as the multidimensional extension of some singular stochastic differential equations on a halfline or on a compact interval. As the number of particles increases without limit, we study the behaviour of the associated empirical measure process. This process converges to a deterministic measurevalued process which is the classical solution of a nonlinear integropartial dierential equation.
Bosonization, vicinal surfaces, and hydrodynamic fluctuation theory, preprint in preparation
, 1998
"... Abstract. Through a Euclidean path integral we establish that the density fluctuations of a Fermi fluid in one dimension are related to vicinal surfaces and to the stochastic dynamics of particles interacting through long range forces with inverse distance decay. In the surface picture one easily ob ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. Through a Euclidean path integral we establish that the density fluctuations of a Fermi fluid in one dimension are related to vicinal surfaces and to the stochastic dynamics of particles interacting through long range forces with inverse distance decay. In the surface picture one easily obtains the Haldane relation and identifies the scaling exponents governing the low energy, Luttinger liquid behavior. For the stochastic particle model we develop a hydrodynamic fluctuation theory, through which in some cases the large distance Gaussian fluctuations are proved nonperturbatively. As pointed out by Haldane some time ago [1,2], spinless fermions in one dimension interacting through a short range potential have universal ground state correlations. The universal properties are computed on the basis of the Tomanaga–Luttinger hamiltonian [3–5] where low energy characteristics turn out to be labelled by two free parameters, traditionally
BROWNIAN PARTICLES WITH ELECTROSTATIC REPULSION ON THE CIRCLE: DYSON’S MODEL FOR UNITARY RANDOM MATRICES REVISITED
, 2000
"... Abstract. The Brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices N N is interpreted as a system of N interacting Brownian particles on the circle with electrostatic interparticles repulsion. The aim of this paper is to dene the nite particle system in a ge ..."
Abstract
 Add to MetaCart
Abstract. The Brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices N N is interpreted as a system of N interacting Brownian particles on the circle with electrostatic interparticles repulsion. The aim of this paper is to dene the nite particle system in a general setting including collisions between particles. Then, we study the behaviour of this system when the number of particles N goes to innity (through the empirical measure process). We prove that a limiting measurevalued process exists and is the unique solution of a deterministic secondorder PDE. The uniform law on [−;] is the only limiting distribution of t when t goes to innity and t has an analytical density.