35 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

    The missing log in large deviations for triangle counts

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    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

    On the lower tail variational problem for random graphs

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    We study the lower tail large deviation problem for subgraph counts in a random graph. Let XHX_H denote the number of copies of HH in an Erd\H{o}s-R\'enyi random graph G(n,p)\mathcal{G}(n,p). We are interested in estimating the lower tail probability P(XH(1δ)EXH)\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H) for fixed 0<δ<10 < \delta < 1. 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 pnαHp \ge n^{-\alpha_H} (and conjecturally for a larger range of pp). 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 HH, and 0<δ<δH0 < \delta < \delta_H for some δH>0\delta_H > 0, as p0p \to 0 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 HH and δ\delta close to 1.Comment: 15 pages, 5 figures, 1 tabl

    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
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