The time-inhomogeneous branching random walk asymptotic conjecture

Let $L_s$ be the offspring point process at time $s$, let $\kappa_s$ be its log-Laplace transform, and let $\kappa_s^*$ be the Fenchel–Legendre transform. Assume that the greatest root $v_s$ of $\kappa_s^*(v)=0$ is well-defined for every $s\in[0,1]$, and define

$$v_{\mathrm{nat}}=\int_0^1v_s\,\mathrm{d}s,$$

with $\theta_s=\partial_v\kappa_s^*(v_s)$ and $\sigma_s^2=1/\partial_v^2\kappa_s^*(v_s)$. Let $N(T)=e^{L(T)}$, and let $\mathcal{X}^{N(T)}_T$ be the generation-$T$ configuration of an $N(T)$-branching random walk started from one particle at the origin, with offspring distributions $(\mathcal{L}_{t/T})$.

Time-inhomogeneous branching random walk asymptotic conjecture. If $L(T)\sim\alpha T^{1/3}$ for $\alpha\in\mathbb{R}$, then

$$\max\big(\mathcal{X}^{N(T)}_T\big)=v_{\mathrm{nat}}T+\left[\int_0^1\frac{\theta_s\sigma_s^2}{\alpha^2}\Psi\left(\frac{\alpha^3\dot\theta_s}{\theta_s^3\sigma_s^2}\right)\,\mathrm{d}s\right]T^{1/3}+o_{\mathbb{P}}(T^{1/3}).$$

If $1\ll L(T)\ll T^{1/3}$, then

$$\max\big(\mathcal{X}^{N(T)}_T\big)=v_{\mathrm{nat}}T-\left(\frac{\pi^2}{2}\int_0^1\theta_s\sigma_s^2\,\mathrm{d}s\right)\frac{T}{L(T)^2}+o_{\mathbb{P}}\left(\frac{T}{L(T)^2}\right).$$

If $T^{1/3}\ll L(T)\ll T$, then

$$\max\big(\mathcal{X}^{N(T)}_T\big)=v_{\mathrm{nat}}T+\left[\int_0^1\frac{(\dot\theta_s)^-}{\theta_s^2}\,\mathrm{d}s\right]L(T)+o_{\mathbb{P}}(L(T)),$$

where $(\cdot)^-$ denotes the negative part; if additionally $\dot\theta_s\geq0$ for all $s\in[0,1]$, then

$$\max\big(\mathcal{X}^{N(T)}_T\big)=v_{\mathrm{nat}}T-\frac{\mathrm{a}_1}{2^{1/3}}\left[\int_0^1\frac{(\dot\theta_s\sigma_s)^{2/3}}{\theta_s}\,\mathrm{d}s\right]T^{1/3}+o_{\mathbb{P}}(T^{1/3}).$$

These formulas are conjectured extensions of the Gaussian $N$-particle branching Brownian motion results to general branching random walks; the regimes $L(T)\ll T^{1/3}$ and $T^{1/3}\ll L(T)\ll T$ are covered, while the linear regime $L(T)\asymp T$ is treated separately.