The limit shape of the Leaky Abelian Sandpile Model on isoradial graphs

Abstract

We study the limit shape of the boundary of the leaky sandpile model on isoradial graphs. These graphs are equipped with conductances and masses introduced by Boutillier, de Tili\`ere and Raschel, which are defined with the help of the geometry of the graph and Jacobi elliptic functions. Building on the link between the shape of the boundary of the sandpile and the Green function of the graph, together with the asymptotics of the Green function obtained by the previously mentioned authors, we prove an explicit formula for the limit shape, both in $\mathbb{R}^d$ where the graph can be seen as a monotone surface under our main geometric assumption, and on the plane where it naturally lies. We then study the limit regime where the elliptic modulus, the parameter of elliptic functions, tends to $0$, which comes down to approaching the massless case, and obtain a circle as a universal limit shape.