2,225 research outputs found

### Estimating and understanding exponential random graph models

We introduce a method for the theoretical analysis of exponential random
graph models. The method is based on a large-deviations approximation to the
normalizing constant shown to be consistent using theory developed by
Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017]. The theory
explains a host of difficulties encountered by applied workers: many distinct
models have essentially the same MLE, rendering the problems ``practically''
ill-posed. We give the first rigorous proofs of ``degeneracy'' observed in
these models. Here, almost all graphs have essentially no edges or are
essentially complete. We supplement recent work of Bhamidi, Bresler and Sly
[2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS)
(2008) 803-812 IEEE] showing that for many models, the extra sufficient
statistics are useless: most realizations look like the results of a simple
Erd\H{o}s-R\'{e}nyi model. We also find classes of models where the limiting
graphs differ from Erd\H{o}s-R\'{e}nyi graphs. A limitation of our approach,
inherited from the limitation of graph limit theory, is that it works only for
dense graphs.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1155 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Carries, shuffling, and symmetric functions

The "carries" when n random numbers are added base b form a Markov chain with
an "amazing" transition matrix determined by Holte. This same Markov chain
occurs in following the number of descents or rising sequences when n cards are
repeatedly riffle shuffled. We give generating and symmetric function proofs
and determine the rate of convergence of this Markov chain to stationarity.
Similar results are given for type B shuffles. We also develop connections with
Gaussian autoregressive processes and the Veronese mapping of commutative
algebra.Comment: 23 page

### Fluctuations of the Bose-Einstein condensate

This article gives a rigorous analysis of the fluctuations of the
Bose-Einstein condensate for a system of non-interacting bosons in an arbitrary
potential, assuming that the system is governed by the canonical ensemble. As a
result of the analysis, we are able to tell the order of fluctuations of the
condensate fraction as well as its limiting distribution upon proper centering
and scaling. This yields interesting results. For example, for a system of $n$
bosons in a 3D harmonic trap near the transition temperature, the order of
fluctuations of the condensate fraction is $n^{-1/2}$ and the limiting
distribution is normal, whereas for the 3D uniform Bose gas, the order of
fluctuations is $n^{-1/3}$ and the limiting distribution is an explicit
non-normal distribution. For a 2D harmonic trap, the order of fluctuations is
$n^{-1/2}(\log n)^{1/2}$, which is larger than $n^{-1/2}$ but the limiting
distribution is still normal. All of these results come as easy consequences of
a general theorem.Comment: 26 pages. Minor changes in new versio

### On barycentric subdivision, with simulations

Consider the barycentric subdivision which cuts a given triangle along its
medians to produce six new triangles. Uniformly choosing one of them and
iterating this procedure gives rise to a Markov chain. We show that almost
surely, the triangles forming this chain become flatter and flatter in the
sense that their isoperimetric values goes to infinity with time. Nevertheless,
if the triangles are renormalized through a similitude to have their longest
edge equal to [0,1]\subset\CC (with 0 also adjacent to the shortest edge),
their aspect does not converge and we identify the limit set of the opposite
vertex with the segment [0,1/2]. In addition we prove that the largest angle
converges to $\pi$ in probability. Our approach is probabilistic and these
results are deduced from the investigation of a limit iterated random function
Markov chain living on the segment [0,1/2]. The stationary distribution of this
limit chain is particularly important in our study. In an appendix we present
related numerical simulations (not included in the version submitted for
publication)

### Graph limits and exchangeable random graphs

We develop a clear connection between deFinetti's theorem for exchangeable
arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph
limits (work of Lovasz and many coauthors). Along the way, we translate the
graph theory into more classical probability.Comment: 26 page

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