Limit theory for the sensitive continuous-type regime

Develop the critical and geometric limit theory for the interacting diffusion cascade outside the saturated regime, where failure strength is a continuous type and the mean offspring operator acts on the failure-strength state space; in particular, establish the total-progeny \(3/2\) law with its correct constant and characterize the boundary dimension using the continuous-type Perron projection and spatial central-limit theory.

Background

Outside saturation, the failure strength of a vertex is inherited by its children and induces correlations among sibling transmissions. The natural branching object is therefore a continuous-type process on hazard domain × failure strength, governed by a positive integral mean operator rather than the finite matrix used in the saturated theory.

The paper provides finite-matrix upper and lower certificates for the operator's criticality, but does not establish the corresponding limit theorems. In particular, the authors identify the total-progeny exponent, its constant, and the Hausdorff dimension of the failed boundary as unresolved continuous-type questions requiring general-state-space branching methods, Perron projections, and spatial fluctuation results.

References

What remains open is the limit theory: the $3/2$ exponent should persist (count variance is still finite, Lemma~\ref{lem:var}), but the constant and the boundary dimension require the spatial CLT and Perron projection on a continuous type.

— Threshold cascades of interacting diffusions on unimodular random trees: a conditional branching-process universality theorem, the annealed pressure, and the dimension of the failed boundary  (2608.15987 - Kumar, 17 Aug 2026) in Discussion, Section 8.3, item 2; Remark 7.2

The continuous-type CRT scaling limit is then reduced to a multitype invariance principle with a Polish type space, which we state as a conjecture with the required hypotheses verified modulo one tightness estimate.

— Two problems for threshold cascades of interacting diffusions on unimodular random trees: front propagation with a Bramson correction, and the continuous-type limit theory  (2608.21125 - Kumar et al., 21 Aug 2026) in Conjecture 3.5, Section 3.3; see also Programme 3.6 and the abstract