Gaussian fluctuation limit near the P_1-P_3 boundary

Establish that for z_0=\beta_0+i\tau_0\in P_{1,3}, the renormalized limit (1/2-z^2)^{1/2}W_\infty(z), as z approaches z_0 from P_1, converges in distribution to a conditionally Gaussian random variable with variance proportional to W_\infty(2\beta_0), and establish the corresponding triple-point statement with exponent 1/4 and Z_\infty.

Background

For parameters on P_{1,3}, the limit W_\infty(z) is not directly defined because the additive martingale explodes, although a polynomially renormalized finite-time martingale has a conditionally Gaussian limit. The conjecture proposes the analogous asymptotic behavior of W_\infty(z) when approached from within P_1, including a modified exponent and derivative-martingale normalization at the triple point.

References

We expect a similar behavior for $W_\infty(z)$ when $z$ approaches $P_{1,3}$, as detailed in the following conjecture.

— Fluctuations of additive martingale limits of branching Brownian motion  (2609.10530 - Chen et al., 9 Sep 2026) in Conjecture 3, Section 3, “Related literature and further questions,” paragraph “Complex additive martingales”