随机约束满足问题的有偏测度：较大的交互作用范围和渐近展开
Biased measures for random Constraint Satisfaction Problems: larger interaction range and asymptotic expansion
摘要：
我们研究了一个典型的随机约束满足问题，双着色的$k$ -均匀随机超图，当它的解决方案是加权非均匀的，软之间的相互作用属于不同的超边的聚类过渡。我们表明，过渡的阈值$α_{\rm d}(k)$可以进一步增加相对于一个受限制的相互作用内的hyperedges，并执行渐近扩展的$α_{\rm d}(k)$在大$k$限制。我们发现，$α_{\rm d}(k) = \frac{2^{k-1}}{k}(\ln k + \ln \ln k + γ_{\rm d} + o(1))$，其中常数$γ_{\rm d}$是严格大于一致措施的解决方案。
英文摘要：
We investigate the clustering transition undergone by an exemplary random constraint satisfaction problem, the bicoloring of $k$-uniform random hypergraphs, when its solutions are weighted non-uniformly, with a soft interaction between variables belonging to distinct hyperedges. We show that the threshold $α_{\rm d}(k)$ for the transition can be further increased with respect to a restricted interaction within the hyperedges, and perform an asymptotic expansion of $α_{\rm d}(k)$ in the large $k$ limit. We find that $α_{\rm d}(k) = \frac{2^{k-1}}{k}(\ln k + \ln \ln k + γ_{\rm d} + o(1))$, where the constant $γ_{\rm d}$ is strictly larger than for the uniform measure over solutions.
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