13,572 research outputs found
Functional summary statistics for point processes on the sphere with an application to determinantal point processes
We study point processes on , the -dimensional unit sphere
, considering both the isotropic and the anisotropic case, and
focusing mostly on the spherical case . The first part studies reduced
Palm distributions and functional summary statistics, including nearest
neighbour functions, empty space functions, and Ripley's and inhomogeneous
-functions. The second part partly discusses the appealing properties of
determinantal point process (DPP) models on the sphere and partly considers the
application of functional summary statistics to DPPs. In fact DPPs exhibit
repulsiveness, but we also use them together with certain dependent thinnings
when constructing point process models on the sphere with aggregation on the
large scale and regularity on the small scale. We conclude with a discussion on
future work on statistics for spatial point processes on the sphere
Random point sets and their diffraction
The diffraction of various random subsets of the integer lattice
, such as the coin tossing and related systems, are well
understood. Here, we go one important step beyond and consider random point
sets in . We present several systems with an effective
stochastic interaction that still allow for explicit calculations of the
autocorrelation and the diffraction measure. We concentrate on one-dimensional
examples for illustrative purposes, and briefly indicate possible
generalisations to higher dimensions.
In particular, we discuss the stationary Poisson process in
and the renewal process on the line. The latter permits a unified approach to a
rather large class of one-dimensional structures, including random tilings.
Moreover, we present some stationary point processes that are derived from the
classical random matrix ensembles as introduced in the pioneering work of Dyson
and Ginibre. Their re-consideration from the diffraction point of view improves
the intuition on systems with randomness and mixed spectra.Comment: 9 pages, 2 figures; talk presented at ICQ 11 (Sapporo
A dynamical classification of the range of pair interactions
We formalize a classification of pair interactions based on the convergence
properties of the {\it forces} acting on particles as a function of system
size. We do so by considering the behavior of the probability distribution
function (PDF) P(F) of the force field F in a particle distribution in the
limit that the size of the system is taken to infinity at constant particle
density, i.e., in the "usual" thermodynamic limit. For a pair interaction
potential V(r) with V(r) \rightarrow \infty) \sim 1/r^a defining a {\it
bounded} pair force, we show that P(F) converges continuously to a well-defined
and rapidly decreasing PDF if and only if the {\it pair force} is absolutely
integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to
this case as {\it dynamically short-range}, because the dominant contribution
to the force on a typical particle in this limit arises from particles in a
finite neighborhood around it. For the {\it dynamically long-range} case, i.e.,
a \leq d-1, on the other hand, the dominant contribution to the force comes
from the mean field due to the bulk, which becomes undefined in this limit. We
discuss also how, for a \leq d-1 (and notably, for the case of gravity, a=d-2)
P(F) may, in some cases, be defined in a weaker sense. This involves a
regularization of the force summation which is generalization of the procedure
employed to define gravitational forces in an infinite static homogeneous
universe. We explain that the relevant classification in this context is,
however, that which divides pair forces with a > d-2 (or a < d-2), for which
the PDF of the {\it difference in forces} is defined (or not defined) in the
infinite system limit, without any regularization. In the former case dynamics
can, as for the (marginal) case of gravity, be defined consistently in an
infinite uniform system.Comment: 12 pages, 1 figure; significantly shortened and focussed, additional
references, version to appear in J. Stat. Phy
Piecewise Constant Martingales and Lazy Clocks
This paper discusses the possibility to find and construct \textit{piecewise
constant martingales}, that is, martingales with piecewise constant sample
paths evolving in a connected subset of . After a brief review of
standard possible techniques, we propose a construction based on the sampling
of latent martingales with \textit{lazy clocks} . These
are time-change processes staying in arrears of the true time but that
can synchronize at random times to the real clock. This specific choice makes
the resulting time-changed process a martingale
(called a \textit{lazy martingale}) without any assumptions on , and
in most cases, the lazy clock is adapted to the filtration of the lazy
martingale . This would not be the case if the stochastic clock
could be ahead of the real clock, as typically the case using standard
time-change processes. The proposed approach yields an easy way to construct
analytically tractable lazy martingales evolving on (intervals of)
.Comment: 17 pages, 8 figure
- …