平面渗流中谱集与关键集的熵与奇异性

中文摘要 AI 辅助

渗流穿越的谱样本与关键集在均匀乘积测度下具有相同的一维和二维包含概率。我们证明，对于临界方格键渗流和三角格点渗流穿越，两者的香农熵均与总影响量可比，且在每一个保留的非根空间分辨率下具有匹配的估计。三角熵界对一类非齐次近临界乘积测度是一致的。对于临界三角格点方形穿越，这两个定律渐近互异：一个基本三角面在关键集中被禁止，而在大谱样本中以正密度出现。低于显式阈值的每个固定面密度，其非空谱概率与$R^2\alpha_4(R)^2$可比，而两个定律的非空重叠具有衰减指数$11/12$。在所有耦合下，它们的最小期望对称差距离与总影响量可比。当保留概率$\rho_R$满足$\rho_R^3R^2\alpha_4(R)\to\infty$时，几何分离在独立稀释后持续存在；当该量趋于零时，面检测器失效。证明结合了空间编码、臂估计、局部傅里叶消去以及在外条件作用下保持的精确六环强制不等式。

英文摘要

The spectral sample and pivotal set of a percolation crossing have the same one- and two-coordinate inclusion probabilities under the uniform product measure. We prove that both Shannon entropies are comparable to total influence for critical square-lattice bond and triangular-lattice site crossings, with matching estimates at every retained nonroot spatial resolution. The triangular entropy bounds are uniform over a class of inhomogeneous near-critical product measures. For critical triangular-site square crossings, the two laws are asymptotically mutually singular: an elementary triangular face is forbidden in the pivotal set and occurs with positive density in a large spectral sample. Every fixed face density below an explicit threshold has nonempty spectral probability comparable to $R^2α_4(R)^2$, while the nonempty overlap of the two laws has decay exponent $11/12$. Their minimum expected symmetric-difference distance over all couplings is comparable to total influence. Geometric separation persists after independent thinning whenever the retention probability $ρ_R$ satisfies $ρ_R^3R^2α_4(R)\to\infty$; the face detector fails when this quantity tends to zero. The proofs combine spatial encoding, arm estimates, local Fourier cancellation, and an exact six-cycle coercivity inequality preserved under exterior conditioning.

发表机构

• Einstein Institute of Mathematics, The Hebrew University of Jerusalem（希伯来大学爱因斯坦数学研究所）

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