Initial-distribution scaling conjecture for one-dimensional multi-particle DLA

Let $Z_1,Z_2,\dots$ be independent identically distributed random variables with

$$\mathbb E Z_1=\lambda,\qquad \operatorname{Var}(Z_1)=\sigma^2,$$

and initialize the one-dimensional multi-particle Diffusion Limited Aggregation model with $Z_i$ particles at vertex $i$. Let $X_t$ be the size of the aggregate. Initial-distribution scaling conjecture. With high probability, $X_t=\Theta(\sqrt t)$ if $\lambda<1$, $X_t=\Theta(t)$ if $\lambda>1$, and $X_t=\Theta(t^{2/3})$ if $\lambda=1$. Moreover, in the critical case,

$$\left\{t^{-2/3}X_{st}\right\}_{s>0}\overset{d}{\longrightarrow}\left\{\int_0^s Z_x\,dx\right\}_{s>0},\qquad t\to\infty,$$

where $Z_t$ solves

$$dZ_t=(4\sigma^2-4)Z_t^4\,dt+2\sigma Z_t^{5/2}\,dB_t,$$

with $Z_0=\infty$. This conjecture concerns how the initial distribution affects the scaling limit, especially at criticality, where the aggregate is expected to grow faster than the time needed for large intervals to reach stationarity. The supplied text gives no resolution status.