26,403 research outputs found
Upper tails for triangles
With the number of triangles in the usual (Erd\H{o}s-R\'enyi) random
graph , and , we show (for some )
\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].
This is tight up to the value of .Comment: 10 page
Life and death of a hero - Lessons learned from modeling the dwarf spheroidal Hercules: an incorrect orbit?
Hercules is a dwarf spheroidal satellite of the Milky Way, found at a
distance of about 138 kpc, and showing evidence of tidal disruption. It is very
elongated and exhibits a velocity gradient of 16 +/- 3 km/s/kpc. Using this
data a possible orbit of Hercules has previously been deduced in the
literature. In this study we make use of a novel approach to find a best fit
model that follows the published orbit. Instead of using trial and error, we
use a systematic approach in order to find a model that fits multiple
observables simultaneously. As such, we investigate a much wider parameter
range of initial conditions and ensure we have found the best match possible.
Using a dark matter free progenitor that undergoes tidal disruption, our
best-fit model can simultaneously match the observed luminosity, central
surface brightness, effective radius, velocity dispersion, and velocity
gradient of Hercules. However, we find it is impossible to reproduce the
observed elongation and the position angle of Hercules at the same time in our
models. This failure persists even when we vary the duration of the simulation
significantly, and consider a more cuspy density distribution for the
progenitor. We discuss how this suggests that the published orbit of Hercules
is very likely to be incorrect.Comment: accepted by MNRAS; 19 pages, 19 figures, 2 table
Dispersion of tracer particles in a compressible flow
The turbulent diffusion of Lagrangian tracer particles has been studied in a
flow on the surface of a large tank of water and in computer simulations. The
effect of flow compressibility is captured in images of particle fields. The
velocity field of floating particles has a divergence, whose probability
density function shows exponential tails. Also studied is the motion of pairs
and triplets of particles. The mean square separation is fitted to
the scaling form ~ t^alpha, and in contrast with the
Richardson-Kolmogorov prediction, an extended range with a reduced scaling
exponent of alpha=1.65 pm 0.1 is found. Clustering is also manifest in strongly
deformed triangles spanned within triplets of tracers.Comment: 6 pages, 4 figure
Upper tails and independence polynomials in random graphs
The upper tail problem in the Erd\H{o}s--R\'enyi random graph
asks to estimate the probability that the number of
copies of a graph in exceeds its expectation by a factor .
Chatterjee and Dembo showed that in the sparse regime of as
with for an explicit ,
this problem reduces to a natural variational problem on weighted graphs, which
was thereafter asymptotically solved by two of the authors in the case where
is a clique. Here we extend the latter work to any fixed graph and
determine a function such that, for as above and any fixed
, the upper tail probability is , where is the maximum degree of . As it turns out, the
leading order constant in the large deviation rate function, , is
governed by the independence polynomial of , defined as where is the number of independent sets of size in . For
instance, if is a regular graph on vertices, then is the
minimum between and the unique positive solution of
Two new Probability inequalities and Concentration Results
Concentration results and probabilistic analysis for combinatorial problems
like the TSP, MWST, graph coloring have received much attention, but generally,
for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example).
Here, we prove two probability inequalities which generalize and strengthen
Martingale inequalities. The inequalities provide the tools to deal with more
general heavy-tailed and inhomogeneous distributions for combinatorial
problems. We prove a wide range of applications - in addition to the TSP, MWST,
graph coloring, we also prove more general results than known previously for
concentration in bin-packing, sub-graph counts, Johnson-Lindenstrauss random
projection theorem. It is hoped that the strength of the inequalities will
serve many more purposes.Comment: 3
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