稀疏随机图中插值方法与标度极限的组合方法
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
摘要：
在Erdös-Rényi图$\mathbb {G}(N,\lfloor cN\rfloor)$和随机$r$ -正则图$\mathbb {G}(N,r)$上建立了几种组合模型自由能极限的存在性.对于各种各样的模型，包括独立集，MAX-CUT，着色和K-SAT，我们证明了在正温度和零温度下的自由能，适当地重新缩放，收敛到一个极限，因为底层图形的大小发散到无穷大。在零温度的情况下，这被解释为相应的组合最优化问题的缩放极限的存在。例如，作为一个特殊的情况下，我们证明了在这些图中的最大独立集的大小，规范化的节点数收敛到一个限制w. h. p.这解决了一个开放的问题，这是由奥尔德斯（一些开放的问题）提出的，他最喜欢的六个开放的问题之一。它也被提到作为一个开放的问题在其他几个地方：猜想2.20在Wormald [在组合数学调查，1999年（坎特伯雷）（1999年）239-298剑桥大学出版社]; Bollobás和Riordan [随机结构算法39（2011年）1-38]; Janson和Mesason [Combin.可能吧Comput. 17（2008）259-264]以及Aldous和Steele [In Probability on Discrete Structures（2004）1-72 Springer]。
英文摘要：
We establish the existence of free energy limits for several combinatorial models on Erdös-Rényi graph $\mathbb {G}(N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb {G}(N,r)$. For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollobás and Riordan [Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].
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