Pré-Publication, Document De Travail Année : 2025

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

Résumé

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 Tilière 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 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.

Fichier principal
Vignette du fichier
2509.06648v1.pdf (537.35 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence

Dates et versions

hal-05247244 , version 1 (09-09-2025)

Licence

Identifiants

Citer

Théo Ballu. The limit shape of the Leaky Abelian Sandpile Model on isoradial graphs. 2025. ⟨hal-05247244⟩
163 Consultations
67 Téléchargements

Altmetric

Partager

  • More