We study the problem of detecting a planted star in the Erd{ő}s--R{é}nyi random graph $G(n,m)$, formulated as a hypothesis test. We determine the scaling window for critical detection in $m$ in terms of the star size, and characterize the asymptotic total variation distance between the null and alternative hypotheses in this window. In the course of the proofs we show a condensation phase transition in the likelihood ratio that closely resembles that of the random energy model from spin glass theory.

翻译：我们研究了在Erdős–Rényi随机图$G(n,m)$中检测植入星图的问题，该问题被表述为假设检验。我们依据星图大小确定了临界检测在$m$中的缩放窗口，并刻画了该窗口内零假设与备择假设之间渐近总变差距离的性质。在证明过程中，我们揭示了似然比中的凝聚相变现象，该现象与自旋玻璃理论中的随机能量模型相变高度相似。