12,879 research outputs found
The missing log in large deviations for triangle counts
This paper solves the problem of sharp large deviation estimates for the
upper tail of the number of triangles in an Erdos-Renyi random graph, by
establishing a logarithmic factor in the exponent that was missing till now. It
is possible that the method of proof may extend to general subgraph counts.Comment: 15 pages. Title changed. To appear in Random Structures Algorithm
Detailed comparison of Milky Way models based on stellar population synthesis and SDSS star counts at the north Galactic pole
We test the ability of the TRILEGAL and Besancon models to reproduce the CMD
of SDSS data at the north Galactic pole (NGP). We show that a Hess diagram
analysis of colour-magnitude diagrams is much more powerful than luminosity
functions (LFs) in determining the Milky Way structure. We derive a
best-fitting TRILEGAL model to simulate the NGP field in the (g-r, g) CMD of
SDSS filters via Hess diagrams. For the Besancon model, we simulate the LFs and
Hess diagrams in all SDSS filters. We use a chi2 analysis and determine the
median of the relative deviations in the Hess diagrams to quantify the quality
of the fits by the TRILEGAL models and the Besancon model in comparison and
compare this with the Just-Jahreiss model. The input isochrones in the
colour-absolute magnitude diagrams of the thick disc and halo are tested via
the observed fiducial isochrones of globular clusters (GCs). We find that the
default parameter set lacking a thick disc component gives the best
representation of the LF in TRILEGAL. The Hess diagram reveals that a
metal-poor thick disc is needed. In the Hess diagram, the median relative
deviation of the TRILEGAL model and the SDSS data amounts to 25 percent,
whereas for the Just-Jahreiss model the deviation is only 5.6 percent. The
isochrone analysis shows that the representation of the MS of (at least
metal-poor) stellar populations in the SDSS system is reliable. In contrast,
the RGBs fail to match the observed fiducial sequences of GCs. The Besancon
model shows a similar median relative deviation of 26 percent in (g-r, g). In
the u band, the deviations are larger. There are significant offsets between
the isochrone set used in the Besancon model and the observed fiducial
isochrones. In contrast to Hess diagrams, LFs are insensitive to the detailed
structure of the Milky Way components due to the extended spatial distribution
along the line of sight.Comment: 21 pages, 17 figures and 5 tables. Accepted by publication of A&
On the lower tail variational problem for random graphs
We study the lower tail large deviation problem for subgraph counts in a
random graph. Let denote the number of copies of in an
Erd\H{o}s-R\'enyi random graph . We are interested in
estimating the lower tail probability for fixed .
Thanks to the results of Chatterjee, Dembo, and Varadhan, this large
deviation problem has been reduced to a natural variational problem over
graphons, at least for (and conjecturally for a larger
range of ). We study this variational problem and provide a partial
characterization of the so-called "replica symmetric" phase. Informally, our
main result says that for every , and for some
, as slowly, the main contribution to the lower tail
probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted
edge density. On the other hand, this is false for non-bipartite and
close to 1.Comment: 15 pages, 5 figures, 1 tabl
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