4,554 research outputs found
Space-time random walk loop measures
In this work, we investigate a novel setting of Markovian loop measures and
introduce a new class of loop measures called Bosonic loop measures. Namely, we
consider loop soups with varying intensity (chemical potential in
physics terms), and secondly, we study Markovian loop measures on graphs with
an additional "time" dimension leading to so-called space-time random walks and
their loop measures and Poisson point loop processes. Interesting phenomena
appear when the additional coordinate of the space-time process is on a
discrete torus with non-symmetric jump rates. The projection of these
space-time random walk loop measures onto the space dimensions is loop measures
on the spatial graph, and in the scaling limit of the discrete torus, these
loop measures converge to the so-called [Bosonic loop measures]. This provides
a natural probabilistic definition of [Bosonic loop measures]. These novel loop
measures have similarities with the standard Markovian loop measures only that
they give weights to loops of certain lengths, namely any length which is
multiple of a given length which serves as an additional
parameter. We complement our study with generalised versions of Dynkin's
isomorphism theorem (including a version for the whole complex field) as well
as Symanzik's moment formulae for complex Gaussian measures. Due to the lacking
symmetry of our space-time random walks, the distributions of the occupation
time fields are given in terms of complex Gaussian measures over complex-valued
random fields ([B92,BIS09]. Our space-time setting allows obtaining quantum
correlation functions as torus limits of space-time correlation functions.Comment: 3 figure
The Widom-Rowlinson Model on the Delaunay Graph
We establish phase transitions for continuum Delaunay multi-type particle
systems (continuum Potts or Widom-Rowlinson models) with a repulsive
interaction between particles of different types. Our interaction potential
depends solely on the length of the Delaunay edges. We show that a phase
transition occurs for sufficiently large activities and for sufficiently large
potential parameter proving an old conjecture of Lebowitz and Lieb extended to
the Delaunay structure. Our approach involves a Delaunay random-cluster
representation analogous to the Fortuin-Kasteleyn representation of the Potts
model. The phase transition manifests itself in the mixed site-bond percolation
of the corresponding random-cluster model. Our proofs rely mainly on geometric
properties of Delaunay tessellations in and on recent studies
[DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is
a uniform bound on the number of connected components in the Delaunay graph
which provides a novel approach to Delaunay Widom Rowlinson models based on
purely geometric arguments. The interaction potential ensures that shorter
Delaunay edges are more likely to be open and thus offsets the possibility of
having an unbounded number of connected components.Comment: 36 pages, 11 figure
Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks
We study large deviations principles for random processes on the
lattice with finite time horizon under a symmetrised
measure where all initial and terminal points are uniformly given by a random
permutation. That is, given a permutation of elements and a
vector of initial points we let the random processes
terminate in the points and then sum over
all possible permutations and initial points, weighted with an initial
distribution. There is a two-level random mechanism and we prove two-level
large deviations principles for the mean of empirical path measures, for the
mean of paths and for the mean of occupation local times under this symmetrised
measure. The symmetrised measure cannot be written as any product of single
random process distributions. We show a couple of important applications of
these results in quantum statistical mechanics using the Feynman-Kac formulae
representing traces of certain trace class operators. In particular we prove a
non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein
statistics and mean field interactions.
A special case of our large deviations principle for the mean of occupation
local times of simple random walks has the Donsker-Varadhan rate function
as the rate function for the limit but for finite time . We give an interpretation in quantum statistical mechanics for this
surprising result
Phase transitions in Delaunay Potts models
We establish phase transitions for classes of continuum Delaunay multi-type
particle systems (continuum Potts models) with infinite range repulsive
interaction between particles of different type. In one class of the Delaunay
Potts models studied the repulsive interaction is a triangle (multi-body)
interaction whereas in the second class the interaction is between pairs
(edges) of the Delaunay graph. The result for the edge model is an extension of
finite range results in \cite{BBD04} for the Delaunay graph and in \cite{GH96}
for continuum Potts models to an infinite range repulsion decaying with the
edge length. This is a proof of an old conjecture of Lebowitz and Lieb. The
repulsive triangle interactions have infinite range as well and depend on the
underlying geometry and thus are a first step towards studying phase
transitions for geometry-dependent multi-body systems. Our approach involves a
Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn
representation of the Potts model. The phase transitions manifest themselves in
the percolation of the corresponding random-cluster model. Our proofs rely on
recent studies \cite{DDG12} of Gibbs measures for geometry-dependent
interactions
Dissipative periodic and chaotic patterns to the KdV--Burgers and Gardner equations
We investigate the KdV-Burgers and Gardner equations with dissipation and
external perturbation terms by the approach of dynamical systems and
Shil'nikov's analysis. The stability of the equilibrium point is considered,
and Hopf bifurcations are investigated after a certain scaling that reduces the
parameter space of a three-mode dynamical system which now depends only on two
parameters. The Hopf curve divides the two-dimensional space into two regions.
On the left region the equilibrium point is stable leading to dissapative
periodic orbits. While changing the bifurcation parameter given by the velocity
of the traveling waves, the equilibrium point becomes unstable and a unique
stable limit cycle bifurcates from the origin. This limit cycle is the result
of a supercritical Hopf bifurcation which is proved using the Lyapunov
coefficient together with the Routh-Hurwitz criterion. On the right side of the
Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using
Shil'nikov's theorem which requires the construction of a homoclinic orbit,
while for the Gardner equation the supercritical Hopf bifurcation leads only to
a stable periodic orbit.Comment: 13 pages, 12 figure
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