28 research outputs found
Renormalized energy concentration in random matrices
We define a "renormalized energy" as an explicit functional on arbitrary
point configurations of constant average density in the plane and on the real
line. The definition is inspired by ideas of [SS1,SS3]. Roughly speaking, it is
obtained by subtracting two leading terms from the Coulomb potential on a
growing number of charges. The functional is expected to be a good measure of
disorder of a configuration of points. We give certain formulas for its
expectation for general stationary random point processes. For the random
matrix -sine processes on the real line (beta=1,2,4), and Ginibre point
process and zeros of Gaussian analytic functions process in the plane, we
compute the expectation explicitly. Moreover, we prove that for these processes
the variance of the renormalized energy vanishes, which shows concentration
near the expected value. We also prove that the beta=2 sine process minimizes
the renormalized energy in the class of determinantal point processes with
translation invariant correlation kernels.Comment: last version, to appear in Communications in Mathematical Physic
Multispecies virial expansions
We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange–Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs
Soft and hard wall in a stochastic reaction diffusion equation
We consider a stochastically perturbed reaction diffusion equation in a
bounded interval, with boundary conditions imposing the two stable phases at
the endpoints. We investigate the asymptotic behavior of the front separating
the two stable phases, as the intensity of the noise vanishes and the size of
the interval diverges. In particular, we prove that, in a suitable scaling
limit, the front evolves according to a one-dimensional diffusion process with
a non-linear drift accounting for a "soft" repulsion from the boundary. We
finally show how a "hard" repulsion can be obtained by an extra diffusive
scaling.Comment: 33 page
Analyticity of The Ground State Energy For Massless Nelson Models
We show that the ground state energy of the translationally invariant Nelson
model, describing a particle coupled to a relativistic field of massless
bosons, is an analytic function of the coupling constant and the total
momentum. We derive an explicit expression for the ground state energy which is
used to determine the effective mass.Comment: 33 pages, 1 figure, added a section on the calculation of the
effective mas
A closer look at the uncertainty relation of position and momentum
We consider particles prepared by the von Neumann-L\"uders projection. For
those particles the standard deviation of the momentum is discussed. We show
that infinite standard deviations are not exceptions but rather typical. A
necessary and sufficient condition for finite standard deviations is given.
Finally, a new uncertainty relation is derived and it is shown that the latter
cannot be improved.Comment: 3 pages, introduction shortened, content unchange
On the Global Existence of Bohmian Mechanics
We show that the particle motion in Bohmian mechanics, given by the solution
of an ordinary differential equation, exists globally: For a large class of
potentials the singularities of the velocity field and infinity will not be
reached in finite time for typical initial values. A substantial part of the
analysis is based on the probabilistic significance of the quantum flux. We
elucidate the connection between the conditions necessary for global existence
and the self-adjointness of the Schr\"odinger Hamiltonian.Comment: 35 pages, LaTe
Metastability Driven by Soft Quantum Fluctuation Modes
The semiclassical Euclidean path integral method is applied to compute the
low temperature quantum decay rate for a particle placed in the metastable
minimum of a cubic potential in a {\it finite} time theory. The classical path,
which makes a saddle for the action, is derived in terms of Jacobian elliptic
functions whose periodicity establishes the one-to-one correspondence between
energy of the classical motion and temperature (inverse imaginary time) of the
system. The quantum fluctuation contribution has been computed through the
theory of the functional determinants for periodic boundary conditions. The
decay rate shows a peculiar temperature dependence mainly due to the softening
of the low lying quantum fluctuation eigenvalues. The latter are determined by
solving the Lam\`{e} equation which governs the fluctuation spectrum around the
time dependent classical bounce.Comment: Journal of Low Temperature Physics (2008) Publisher: Springer
Netherland