AI 中文总结
本研究针对植入图问题，证明了不同临界情形下低阶不可区分性结合树宽条件（或无需该条件）可推出带噪声版本的渐近不可区分性，给出了低阶启发式的严格推论及证明方法。
AI 中文摘要
低阶启发式已成为预测平均情形植入-零假设问题中计算阈值的广泛使用框架。然而，近期一系列反例表明，低阶不可区分性通常并不能排除高效的抗噪声区分器；参见Buhai等人（2025）和Mao（2026）的研究。受这些进展推动，Hsieh等人（2026）启动了对低阶启发式严格推论的研究。在本工作中，我们针对植入图问题延续这一研究方向。\n 设$Q_n=G(n,c/n)$，$P_n$通过将确定性图$Γ_n$的均匀随机副本植入$Q_n$的独立样本中得到。在$c>1$的超临界情形下，我们证明若$P_n$与$Q_n$是度-$D_n$不可区分的，且$\text{tw}(Γ_n)=o(D_n/\text{log }n)$，则$P_n$的带噪声版本与$Q_n$渐近不可区分。此处$\text{tw}(Γ_n)$表示$Γ_n$的树宽，是衡量图可被分解为树状片段效率的指标。在$0<c≤1$的临界和次临界情形下，只要$D_n=ω(\text{log }n)$，无需任何树宽假设即可得到相同结论。\n 我们的证明包含两个主要部分。第一，我们揭示了傅里叶展开中的子图计数因子与自同构因子，和同构三元组计数之间的对应关系。第二，我们将分解树切割为子树，将每个大的傅里叶支撑拆分为仅在少数界面顶点处相交的低阶片段，并利用噪声吸收重新组装这些片段的代价。在临界及以下情形，低阶假设排除了短环，而噪声会破坏剩余的长环。
英文摘要
The low-degree heuristic has become a widely used framework for predicting computational thresholds in average-case planted-versus-null problems. However, a recent sequence of counterexamples shows that low-degree indistinguishability does not, in general, rule out efficient noise-tolerant distinguishers; see Buhai et al. (2025) and Mao (2026). Motivated by these developments, Hsieh et al. (2026) initiated the study of rigorous consequences of the low-degree heuristic. In this work, we continue this program for planted-graph problems. Let $Q_n=G(n,c/n)$, and let $P_n$ be obtained by planting a uniformly random copy of a deterministic graph $Γ_n$ into an independent sample from $Q_n$. In the supercritical regime $c>1$, we show that if $P_n$ is degree-$D_n$ indistinguishable from $Q_n$ and $\operatorname{tw}(Γ_n)=o(D_n/\log n)$, then a noisy version of $P_n$ is asymptotically indistinguishable from $Q_n$. Here $\operatorname{tw}(Γ_n)$ denotes the treewidth of $Γ_n$, a measure of how efficiently the graph can be decomposed into tree-like pieces. In the critical and subcritical regimes $0<c\leq 1$, the same conclusion holds whenever $D_n=ω(\log n)$, without any treewidth assumption. Our proof has two main ingredients. First, we uncover a correspondence between the subgraph-count and automorphism factors in the Fourier expansion and counts of isomorphism triples. Second, we cut the decomposition tree into subtrees, breaking each large Fourier support into low-degree pieces that meet at only a few interface vertices, and use noise to absorb the cost of reassembling them. At and below criticality, the low-degree assumption rules out short cycles, while noise destroys the remaining long cycles.
Comments38 pages
