The $q<1$ Random-Cluster Model on Wired Trees: Uniqueness and Negative Dependence

What the math gives to ML

The paper gives a constructive tree recursion for a random-cluster model with q<1, where the nontrivial fixed point appears exactly above p_c=q/(\Delta+q-2), together with conditional negative association of connectivity indicators on distinct wired branches. The transferable asset is a principled source of negatively dependent stochastic gates: unlike independent dropout or ordinary mixture routing, activating one branch suppresses correlations with increasing observables on other branches. A practical adaptation is a tree-structured stochastic routing layer whose branch masks are sampled from a finite wired random-cluster distribution, with q controlling anti-correlation and p controlling activity; the fixed-point recursion supplies a cheap calibration and phase-transition diagnostic.