2,225 research outputs found

    Estimating and understanding exponential random graph models

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    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

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    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

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    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 nn bosons in a 3D harmonic trap near the transition temperature, the order of fluctuations of the condensate fraction is n−1/2n^{-1/2} and the limiting distribution is normal, whereas for the 3D uniform Bose gas, the order of fluctuations is n−1/3n^{-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)1/2n^{-1/2}(\log n)^{1/2}, which is larger than n−1/2n^{-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

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    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 π\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

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    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