907 research outputs found

    Spatial Mixing of Coloring Random Graphs

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    We study the strong spatial mixing (decay of correlation) property of proper qq-colorings of random graph G(n,d/n)G(n, d/n) with a fixed dd. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as G(n,d/n)G(n, d/n), an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for qαd+βq\ge\alpha d+\beta with α>2\alpha>2 and sufficiently large β=O(1)\beta=O(1), with high probability proper qq-colorings of random graph G(n,d/n)G(n, d/n) exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree

    Induced Ramsey-type theorems

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    We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.Comment: 30 page

    Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average

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    In this work we show that for every d<d < \infty and the Ising model defined on G(n,d/n)G(n,d/n), there exists a βd>0\beta_d > 0, such that for all β<βd\beta < \beta_d with probability going to 1 as nn \to \infty, the mixing time of the dynamics on G(n,d/n)G(n,d/n) is polynomial in nn. Our results are the first polynomial time mixing results proven for a natural model on G(n,d/n)G(n,d/n) for d>1d > 1 where the parameters of the model do not depend on nn. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than n \polylog(n). Our proof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex vv of the graph has a neighborhood N(v)N(v) of radius O(logn)O(\log n) in which the induced sub-graph is a tree union at most O(logn)O(\log n) edges and where for each simple path in N(v)N(v) the sum of the vertex degrees along the path is O(logn)O(\log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n,d/n)G(n,d/n) it applies for all external fields and β<βd\beta < \beta_d, where dtanh(βd)=1d \tanh(\beta_d) = 1 is the critical point for decay of correlation for the Ising model on G(n,d/n)G(n,d/n).Comment: Corrected proof of Lemma 2.

    A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

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    Let R\R be the set of all finite graphs GG with the Ramsey property that every coloring of the edges of GG by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let G(n,p)G(n,p) be the random graph on nn vertices with edge probability pp. We prove that there exists a function c^=c^(n)\hat c=\hat c(n) with 000 0, as nn tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
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