Uniform integrability and the Bramson correction for the cascade front

Prove uniform integrability of the derivative martingale associated with the critical tilt of the multitype branching random walk of activation times, and establish the resulting Bramson-type logarithmic delay and decorated-point-process limit for the time-indexed cascade front.

Background

In the dissipative regime, the failed cascade is compared with a multitype Galton–Watson genealogy whose edge activation times produce a branching random walk. The paper proves a ballistic law of large numbers for the deepest failed vertex, with speed c_* determined by a Perron-root variational formula.

The unresolved second-order problem concerns the analogue of the Bramson correction known for branching Brownian motion. The proposed route is to establish uniform integrability and positivity of the derivative martingale at the critical tilt, then transfer the resulting martingale information to a logarithmic front delay and a randomly shifted Gumbel decorated point-process limit. The paper identifies the relevant moment estimate and notes that a complete proof would still require adapting decorated-point-process technology to the genealogical-tree branching random walk.

References

We show that the generation-indexed front admits a genuine BRW comparison, prove that the front depth grows ballistically with an explicit asymptotic speed $c_$ determined by the Perron root of a tilted mean matrix (a linear first-order theorem, unconditional in the dissipative regime), and reduce the second-order behaviour --- the conjectured Bramson-type logarithmic delay $c_ t - \frac{3}{2c_*}\log t$ --- to the construction of a derivative martingale, which we carry out at the linearised level and whose uniform integrability is the one missing analytic input.

— 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 2.1, Section 2.2; see also the abstract and Programme 2.3