  In the present note we analyze the one-dimensional multi-particle diffusion
limited aggregation (MDLA) model: the initial number of particles at each
positive integer site has Poisson distribution with mean $\mu$, independently
of all other sites. Particles perform independent continuous-time simple
symmetric random walks until they come to the site neighbouring the sticky
aggregate, which initially consists only of the origin. If a particle tries to
jump on the aggregate, the size of the aggregate increases by one, i.e., its
rightmost point moves to the right by one unit. All particles which are present
at the site neighbouring the aggregate at the moment when the aggregate
advances, are immediately deleted. The $d-$dimensional MDLA model, $d \geq 1$,
was introduced in 1980 by Rosenstock and Marquardt, and studied numerically by
Voss (1984). The one dimensional model exhibits a phase transition for the rate
of growth of the aggregate: it was proven by Kesten and Sidoravicius (2008)
that if $\mu<1$ then the size $R(t)$ of the aggregate grows like $\sqrt{t}$ and
Sly (2016+) proved that if $\mu>1$ then $R(t)$ grows linearly. In this note we
give heuristic predictions about the constant $c(\mu)$ for which $R(t)\approx
c(\mu)\sqrt{t}$ in the subcritical case $\mu<1$, $R(t)\approx
c(1+\varepsilon)t$ in the barely supercritical case $\mu=1+\varepsilon$ and
$R(t) \approx c(1) t^{2/3}$ in the critical case $\mu=1$. We compare our
predictions with new computer simulation results of the 1-dimensional
multi-particle DLA model.
