Extend maximum-particle difference estimates to all window widths

Prove that the estimates for u_M(t,√2t+z−K)−u_M(t,√2t+z+K) hold for every K>0, rather than only for sufficiently large K, in both the supercritical and critical regimes.

Background

The paper uses upper and lower estimates for the difference between two distribution functions of the BBM maximum, separated by a spatial window of width 2K. These estimates are needed to control contributions from different regions of the initial point process and to derive tightness properties.

The authors establish the estimates only after choosing K sufficiently large. They explicitly state that they expect the estimates to hold for every positive K but were unable to prove that stronger assertion. Thus, the unresolved issue concerns sharpening the technical bounds, not the principal fixed-point characterization proved in the paper.

References

We expect quant-bound-for-super-critical-for-difference, quant-bound-for-difference-critical to hold for all K>0 rather than only for K large enough. We were able to verify quant-bound-for-super-critical-for-difference, quant-bound-for-difference-critical only for large enough K>0, which will be sufficient for our purposes.

quant-bound-for-super-critical-for-difference:

C1e−2zte−z2/2t≤uM(t,2t+z−K)−uM(t,2t+z+K)≤C2e−2zte−z2/2t.C_1\frac{e^{-\sqrt 2 z}}{\sqrt t}e^{-z^2/2t} \le u_M(t, \sqrt{2}t + z - K ) - u_M(t, \sqrt{2}t + z + K )\le C_2\frac{e^{-\sqrt 2 z}}{\sqrt t}e^{-z^2/2t}.

— Locally finite fixed points of branching Brownian motion  (2608.23202 - Chen et al., 24 Aug 2026) in Lemma 2.8 / Lemma “Lemma-on-difference-U_M-for-critical,” Section 2, subsection “Some estimates on solution to F-KPP equation”