连续着色问题中动态转移的优化
Optimization of the dynamic transition in the continuous coloring problem
摘要：
随机约束满足问题可以表现出这样一个阶段，即每个变量$α$的约束数量一方面使系统在理论上可解，但也使寻找解变得困难，这意味着诸如Monte-Carlo方法的常见算法无法找到解。这种困难的发生与动态相变的出现密切相关，在动态相变中，问题的相空间分解成指数数量的簇。这个动态相变的确切位置对于人们选择代表给定问题的哈密顿量的细节来说并不普遍。在本文中，我们开发了一些理论工具，以找到一个系统的方法来建立一个哈密顿量，最大限度地提高动态$α_{\rm d}$阈值。为了说明我们的技术，我们将集中在连续着色的问题上，其中试图在网络的每个节点上设置角度$x_i \in [0;2π]$，使得没有相邻节点比某个阈值角度$θ$更接近，即$\cos(x_i - x_j) \leq \cosθ$。这个问题既可以看作是离散图着色问题的连续版本，也可以看作是Mari-Krzakala-Kurchan（MKK）模型的一维版本。这个模型的相关性源于这样一个事实，即稀疏随机图上的连续约束满足问题在统计物理学中仍然很大程度上未被探索。我们表明，足够小的角度$θ$这个模型提出了一个随机的一阶过渡和计算的动力学，凝聚和Kesten-Stigum过渡，我们还比较了分析预测与蒙特卡罗模拟值的$θ= 2π/q$，$q \in \mathbb{N}$。选择这样的值的$q$，使我们能够很容易地比较我们的结果与著名的离散着色问题。
英文摘要：
Random constraint satisfaction problems can exhibit a phase where the number of constraints per variable $α$ makes the system solvable in theory on the one hand, but also makes the search for a solution hard, meaning that common algorithms such as Monte-Carlo method fail to find a solution. The onset of this hardness is deeply linked to the appearance of a dynamical phase transition where the phase space of the problem breaks into an exponential number of clusters. The exact position of this dynamical phase transition is not universal with respect to the details of the Hamiltonian one chooses to represent a given problem. In this paper, we develop some theoretical tools in order to find a systematic way to build a Hamiltonian that maximizes the dynamic $α_{\rm d}$ threshold. To illustrate our techniques, we will concentrate on the problem of continuous coloring, where one tries to set an angle $x_i \in [0;2π]$ on each node of a network in such a way that no adjacent nodes are closer than some threshold angle $θ$, that is $\cos(x_i - x_j) \leq \cosθ$. This problem can be both seen as a continuous version of the discrete graph coloring problem or as a one-dimensional version of the the Mari-Krzakala-Kurchan (MKK) model. The relevance of this model stems from the fact that continuous constraint satisfaction problems on sparse random graphs remain largely unexplored in statistical physics. We show that for sufficiently small angle $θ$ this model presents a random first order transition and compute the dynamical, condensation and Kesten-Stigum transitions; we also compare the analytical predictions with Monte Carlo simulations for values of $θ= 2π/q$, $q \in \mathbb{N}$. Choosing such values of $q$ allows us to easily compare our results with the renowned problem of discrete coloring.
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