Exact topology of conservative multiplicative cascades: An ultrametric transfer-operator genus

Abstract

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.