Effect of UGW disorder on the logarithmic front correction

Determine whether disorder in the unimodular Galton–Watson environment changes the coefficient of the logarithmic correction to the front position, and characterize any difference between quenched and annealed corrections.

Background

The paper distinguishes the front behavior in a fixed tree environment (quenched) from behavior averaged over the unimodular Galton–Watson tree (annealed). It notes that the pressure function already permits different quenched and annealed speeds away from boundary-homogeneous trees.

The unresolved question is whether the tree disorder also modifies the second-order logarithmic correction, potentially replacing the usual 3/2 coefficient by another constant as in branching random walk in random environment. The paper identifies this as the principal genuinely new problem beyond adapting existing branching-random-walk methods.

References

The corresponding question for the logarithmic correction --- whether disorder in the UGW environment shifts the $\frac{3}{2}$ to a different constant, as happens for BRW in random environment --- is open and is the part of this programme that is not mere transcription.

— 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 Remark 2.5, Section 2.3.2