Engulfing MDLA threshold and limiting-shape conjecture

Consider an engulfing version of multi-particle diffusion-limited aggregation (MDLA) with particle density $\lambda$ and internal random-walk rate $\gamma$, where $\gamma=\infty$ means that all excess particles attach instantaneously. Engulfing MDLA threshold and shape conjecture. For $\gamma=\infty$, the engulfing MDLA grows if and only if $\lambda>1$. For every sufficiently large $\lambda$ and every $\gamma>0$, the model grows linearly, with its limiting shape converging to a ball as $\gamma\to\infty$. The statement concerns an extension of the Pool/MDLA dynamics motivated by electrochemical deposition; the asserted threshold and limiting-shape behavior remain open in the supplied text.