Current status on Richardson's model (growth model) A very simple stochastic growth model on a lattice is the Richardson's model (Actually originally defined by Murray Eden in the 60s).
Each point of the lattice can be either occupied or vacant, once they are occupied they remain so forever, and vacant points become occupied at a rate equal to the fraction of the occupied neighbours (so a point can become occupied only if at least one of its neighbours is occupied). Eventually all points will be occupied, but the limiting shape of the aggregation of occupied points roughly looks like a circle. There is a link (1) that discuss a bit more about this model and gives a few papers. However none of these papers are recent.
Question:
I would be curious to know if there are still some interesting open problems on this model, and more interestingly, if despite its simplicity this model can accurately describes any real biological phenomena.
(I can easily imagine that this model can be seen as a special case of some epidemic model on a lattice, but this is not really what this post aims at)
Here is a picture (black vacant, white occupied) that I took from Eden's original paper

link: https://services.math.duke.edu/~rtd/survey/survc1.html
original article : Eden, Murray. "A two-dimensional growth process." Proceedings of the fourth Berkeley symposium on mathematical statistics and probability. Vol. 4. University of California Press Berkeley, 1961.
 A: Minor remark: I have always known this model as the Eden model, it seems that most probabilists currently refer to it as such.
To answer your question, yes there are many interesting open questions, mainly regarding the fluctuations of the interface around its limiting shape. The Eden model is the prototypical example of a model conjectured to belong to the KPZ universality class. This immediately yields a wealth of conjectures. For example, after a time of order $t$, the deviations from a limiting shape of radius $t$ (the exact shape itself isn't known I believe) should be of order $t^{1/3}$.
The law of this fluctuation in any fixed direction is further conjectured to converge to the GUE Tracy-Widom distribution as $t \to \infty$. If, instead of starting with one single occupied site, one starts with an occupied half-space, then the law of the interface between occupied and empty sites, suitably recentred and rescaled, is conjectured to converge to the KPZ fixed point. All of these conjectures are expected to be extremely hard since the Eden model has no known integrable structure, so that the exact calculations that allowed to show similar results for some other models aren't available. You'll find more details in the various review articles by people like Quastel, Spohn, Corwin, etc.
