AI 中文总结
本研究精确预测保守随机置换级联的拓扑结构，闭式计算二维级联 excursion 集的欧拉特征，关联其与多重分形谱，为强非高斯场形态学提供基准。
AI 中文摘要
乘性级联是生成兼具真正非高斯性与尺度不变性的随机场的知名方法，其单点统计与标度统计，即多重分形谱τ(q)、D_q和f(α)已被精确求解，但其几何与拓扑测度尚未明确。本研究表明，保守（微正则）随机置换级联的拓扑结构可被精确预测：利用级联b-ary树的超度量结构及无放回的兄弟分裂，通过转移算子，我们闭式计算了二维级联j阶 excursion 集的数字（立方）欧拉特征（即亏格）；该结果无需蒙特卡洛模拟，与模拟实现的欧拉密度最大残差约为10⁻⁴，属于原子测度的受控离散化误差。我们将亏格标度与多重分形谱解析关联，证明对于权重的几何阶梯，亏格在多重分形宽度参数下严格自相似。该构造采用Greiner等人（Phys. Rev. E 58, 554, 1998）的树累积量机制，专门针对拓扑泛函；保守分裂带有微正则指纹Cov(ln w_a, ln w_b)/Var(ln w)=-1/(n-1)，这使其与正则级联及对数正态级联相区分。该级联是瑞利-莱维飞行的乘性对应物，为强非高斯场的形态学提供了受控基准。
英文摘要
The multiplicative cascade is a well-known method for generating random fields that are both genuinely non-Gaussian and scale invariant. Its one-point and scaling statistics, namely the multifractal spectra $τ(q)$, $D_q$, and $f(α)$, are known exactly; however, its geometric and topological measures are not. In this work, we show that the topological structures of a conservative (microcanonical) random permutation cascade can be precisely predicted. We compute the digital (cubical) Euler characteristic, or genus, of the level-$j$ excursion set of the two-dimensional cascade in closed form, using a transfer operator on the cascade's $b$-ary tree closed by its ultrametric structure and a conservative without-replacement sibling split. The result requires no Monte Carlo and matches simulated realizations with a maximum residual of $\sim\!10^{-4}$ in the Euler density, a controlled discretization artifact of the atomic measure. We tie the genus scaling analytically to the multifractal spectrum, and show that for a geometric ladder of weights the genus is exactly self-similar under the multifractal-width dial. The construction uses the tree-sum cumulant machinery of Greiner et al. (Phys. Rev. E 58, 554, 1998), specialized to topological functionals; the conservative split carries the microcanonical fingerprint $\mathrm{Cov}(\ln w_a,\ln w_b)/\mathrm{Var}(\ln w)=-1/(n-1)$ that distinguishes it from canonical and lognormal cascades. The cascade is the multiplicative counterpart of the Rayleigh--Lévy flight and a controlled benchmark for the morphology of strongly non-Gaussian fields.
Comments7 pages, 4 figures
