The limit shape of the Leaky Abelian Sandpile Model

The leaky abelian sandpile model (Leaky-ASM) is a growth model in which $n$ grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1-1/d$ of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case $d=1$.
We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed $d>1$, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.
We will also discuss the limit shape in the regime when the dissipation parameter $d$ converges to 1 as $n$ grows, as this is related to the ordinary ASM with a modified initial configuration.