26,403 research outputs found

    Upper tails for triangles

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    With ξ\xi the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph G(m,p)G(m,p), p>1/mp>1/m and η>0\eta>0, we show (for some Cη>0C_{\eta}>0) \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 CηC_{\eta}.Comment: 10 page

    Life and death of a hero - Lessons learned from modeling the dwarf spheroidal Hercules: an incorrect orbit?

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    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

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    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

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    The upper tail problem in the Erd\H{o}s--R\'enyi random graph GGn,pG\sim\mathcal{G}_{n,p} asks to estimate the probability that the number of copies of a graph HH in GG exceeds its expectation by a factor 1+δ1+\delta. Chatterjee and Dembo showed that in the sparse regime of p0p\to 0 as nn\to\infty with pnαp \geq n^{-\alpha} for an explicit α=αH>0\alpha=\alpha_H>0, 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 HH is a clique. Here we extend the latter work to any fixed graph HH and determine a function cH(δ)c_H(\delta) such that, for pp as above and any fixed δ>0\delta>0, the upper tail probability is exp[(cH(δ)+o(1))n2pΔlog(1/p)]\exp[-(c_H(\delta)+o(1))n^2 p^\Delta \log(1/p)], where Δ\Delta is the maximum degree of HH. As it turns out, the leading order constant in the large deviation rate function, cH(δ)c_H(\delta), is governed by the independence polynomial of HH, defined as PH(x)=iH(k)xkP_H(x)=\sum i_H(k) x^k where iH(k)i_H(k) is the number of independent sets of size kk in HH. For instance, if HH is a regular graph on mm vertices, then cH(δ)c_H(\delta) is the minimum between 12δ2/m\frac12 \delta^{2/m} and the unique positive solution of PH(x)=1+δP_H(x) = 1+\delta

    Two new Probability inequalities and Concentration Results

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    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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