In this paper we propose a polynomial-time deterministic algorithm for approximately counting the k-colourings of the random graph G(n, d/n), for constant d>0. In particular, our algorithm computes in polynomial time a $(1\pm n^{-\Omega(1)})$ -approximation of the so-called ‘free energy’ of the k-colourings of G(n, d/n), for $k\geq (1+\varepsilon) d$ with probability $1-o(1)$ over the graph instances.Our algorithm uses spatial correlation decay to compute numerically estimates of marginals of the Gibbs distribution. Spatial correlation decay has been used in different counting schemes for deterministic counting. So far algorithms have exploited a certain kind of set-to-point correlation decay, e.g. the so-called Gibbs uniqueness. Here we deviate from this setting and exploit a point-to-point correlation decay. The spatial mixing requirement is that for a pair of vertices the correlation between their corresponding configurations becomes weaker with their distance.Furthermore, our approach generalizes in that it allows us to compute the Gibbs marginals for small sets of nearby vertices. Also, we establish a connection between the fluctuations of the number of colourings of G(n, d/n) and the fluctuations of the number of short cycles and edges in the graph.

中文翻译：

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使用子图序列对图着色进行确定性计数

在本文中，我们提出了多项式时间确定性的近似计算的算法ķ- 随机图的着色G(日，日/日)，对于常数d>0。特别是，我们的算法在多项式时间内计算$(1\pm n^{-\Omega(1)})$- 所谓的“自由能”的近似值ķ- 着色G(日，日/日）， 为了$k\geq (1+\varepsilon) d$有概率$1-o(1)$在图实例上。我们的算法使用空间相关衰减计算吉布斯分布边缘的数值估计。空间相关衰减已用于确定性计数的不同计数方案中。到目前为止，算法已经利用了某种定点相关衰减，例如所谓的吉布斯唯一性. 在这里，我们偏离了这个设置并利用了点对点相关衰减。空间混合要求是，对于一对顶点，它们的相应配置之间的相关性随着它们的距离而变弱。此外，我们的方法概括为它允许我们计算附近顶点的小集合的吉布斯边缘。此外，我们建立了着色数量波动之间的联系G(日，日/日) 以及图中短圈数和边数的波动。