22 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
Emergence of interlacements from the finite volume Bose soup
We show that, conditioned on the (empirical) particle density exceeding the
critical value, the finite volume Bose loop soup converges to the superposition
of the Bosonic loop soup (on the whole space) and the Poisson point process of
random interlacements. The intensity of the latter is given by the excess
density above the critical point. We consider both the free case and the mean
field case.Comment: 27 pages, fixed typo
A note on the intersections of two random walks in two dimensions
In this note we prove a large deviation result for the intersection of the
ranges of two independent random walks in dimension two. This complements the
study of Phetpradap from 2011, where the intersection in dimension three and
above was studied.Comment: 10 page
Space-time random walk loop measures
In this work, we introduce and investigate two novel classes of loop measures, space–time Markovian loop measures and Bosonic loop measures, respectively. We consider loop soups with 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 [8], [10]. Our space–time setting allows obtaining quantum correlation functions as torus limits of space–time correlation functions
Gibbs measures for the repulsive Bose gas
We prove the existence of Gibbs measures for the Feynman representation of
the Bose gas with non-negative interaction in the grand-canonical ensemble. Our
results are valid for all negative chemical potentials as well as slightly
positive chemical potentials. We consider both the Gibbs property of marked
points as well as a Markov--Gibbs property of paths.Comment: 32 pages, 4 figure
Large deviations of the range of the planar random walk on the scale of the mean
We show an upper large deviation bound on the scale of the mean for a
symmetric random walk in the plane with finite sixth moment. This result
complements the study of Van den Berg, Bolthausen and Den Hollander, where the
continuum case of the Wiener Sausage is studied, and in Phetpradap, in which
one is restricted to dimension three and higher.Comment: 30 page
Geometric properties of random walk loop soups
In this thesis the author examines geometric properties of (Poisson) loop soups generated from loop measures with varying weights. The framework incorporates the Markovian loop measure, see [LJ11], as well as the Bosonic loop measure, see [AV20]. The author characterises certain geometric features of the loop soup, such as its percolative properties and correlation structure
An Advanced Tree Algorithm with Interference Cancellation in Uplink and Downlink
In this paper, we propose Advanced Tree-algorithm with Interference
Cancellation (ATIC), a variant of binary tree-algorithm with successive
interference cancellation (SICTA) introduced by Yu and Giannakis. ATIC assumes
that Interference Cancellation (IC) can be performed both by the access point
(AP), as in SICTA, but also by the users. Specifically, after every collision
slot, the AP broadcasts the observed collision as feedback. Users who
participated in the collision then attempt to perform IC by subtracting their
transmissions from the collision signal. This way, the users can resolve
collisions of degree 2 and, using a simple distributed arbitration algorithm
based on user IDs, ensure that the next slot will contain just a single
transmission. We show that ATIC reaches the asymptotic throughput of 0.924 as
the number of initially collided users tends to infinity and reduces the number
of collisions and packet delay. We also compare ATIC with other tree algorithms
and indicate the extra feedback resources it requires.Comment: This paper will be presented at the ASILOMAR Conference on Signals,
Systems, and Computer
Analysis of d-ary Tree Algorithms with Successive Interference Cancellation
In this article, we calculate the mean throughput, number of collisions,
successes, and idle slots for random tree algorithms with successive
interference cancellation. Except for the case of the throughput for the binary
tree, all the results are new. We furthermore disprove the claim that only the
binary tree maximises throughput. Our method works with many observables and
can be used as a blueprint for further analysis.Comment: 30 pages, 2 figures, comments welcom
Formation of infinite loops for an interacting bosonic loop soup
We compute the limiting measure for the Feynman loop representation of the
Bose gas for a non mean-field energy. As predicted in previous works, for high
densities the limiting measure gives positive weight to random interlacements,
indicating the quantum Bose--Einstein condensation. We prove that in many cases
there is a shift in the critical density compared to the free/mean-field case,
and that in these cases the density of the random interlacements has a
jump-discontinuity at the critical point.Comment: 36 pages, 6 figure