55 research outputs found

    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

    The asymptotic induced matching number of hypergraphs: balanced binary strings

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    We compute the asymptotic induced matching number of the kk-partite kk-uniform hypergraphs whose edges are the kk-bit strings of Hamming weight k/2k/2, for any large enough even number kk. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science. Phrased in the language of tensors, as a direct consequence of our result, we determine the asymptotic subrank of any tensor with support given by the aforementioned hypergraphs. In the context of quantum information theory, our result amounts to an asymptotically optimal kk-party stochastic local operations and classical communication (slocc) protocol for the problem of distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement

    A proof of the Upper Matching Conjecture for large graphs

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    We prove that the `Upper Matching Conjecture' of Friedland, Krop, and Markstr\"om and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every dd and every large enough nn divisible by 2d2d, a union of n/(2d)n/(2d) copies of the complete dd-regular bipartite graph maximizes the number of independent sets and matchings of size kk for each kk over all dd-regular graphs on nn vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth

    Tight bounds on the coefficients of partition functions via stability

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    We show how to use the recently-developed occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs. As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström, and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of q-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least 5

    A reverse Sidorenko inequality

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    Let HH be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph GG without isolated vertices, the weighted number of graph homomorphisms hom(G,H)\hom(G, H) satisfies the inequality hom(G,H)uvE(G)hom(Kdu,dv,H)1/(dudv), \hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)}, where dud_u denotes the degree of vertex uu in GG. In particular, one has hom(G,H)1/E(G)hom(Kd,d,H)1/d2 \hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2} for every dd-regular triangle-free GG. The triangle-free hypothesis on GG is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of GG is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings. For graph colorings, corresponding to H=KqH = K_q, we show that the triangle-free hypothesis on GG may be dropped; this is also valid if some of the vertices of KqK_q are looped. A corollary is that among dd-regular graphs, G=Kd,dG = K_{d,d} maximizes the quantity cq(G)1/V(G)c_q(G)^{1/|V(G)|} for every qq and dd, where cq(G)c_q(G) counts proper qq-colorings of GG. Finally, we show that if the edge-weight matrix of HH is positive semidefinite, then hom(G,H)vV(G)hom(Kdv+1,H)1/(dv+1). \hom(G, H) \le \prod_{v \in V(G)} \hom(K_{d_v+1}, H )^{1/(d_v+1)}. This implies that among dd-regular graphs, G=Kd+1G = K_{d+1} maximizes hom(G,H)1/V(G)\hom(G, H)^{1/|V(G)|}. For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn.Comment: 30 page

    Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs

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    For a graph GG, let Z(G,λ)Z(G,\lambda) be the partition function of the monomer-dimer system defined by kmk(G)λk\sum_k m_k(G)\lambda^k, where mk(G)m_k(G) is the number of matchings of size kk in GG. We consider graphs of bounded degree and develop a sublinear-time algorithm for estimating logZ(G,λ)\log Z(G,\lambda) at an arbitrary value λ>0\lambda>0 within additive error ϵn\epsilon n with high probability. The query complexity of our algorithm does not depend on the size of GG and is polynomial in 1/ϵ1/\epsilon, and we also provide a lower bound quadratic in 1/ϵ1/\epsilon for this problem. This is the first analysis of a sublinear-time approximation algorithm for a # P-complete problem. Our approach is based on the correlation decay of the Gibbs distribution associated with Z(G,λ)Z(G,\lambda). We show that our algorithm approximates the probability for a vertex to be covered by a matching, sampled according to this Gibbs distribution, in a near-optimal sublinear time. We extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability, where again the query complexity is polynomial in 1/ϵ1/\epsilon and the lower bound is quadratic in 1/ϵ1/\epsilon. Our algorithms are simple to implement and of practical use when dealing with massive datasets. Our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity
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