Random rotor walks and i.i.d. sandpiles on Sierpinski graphs

TL;DR

This paper investigates the behavior of rotor walks and sandpile models on the Sierpinski gasket graph, establishing recurrence and non-stabilization conditions for certain configurations.

Contribution

It proves recurrence of rotor walks with random initial rotors and provides necessary conditions for the stabilization of i.i.d. sandpiles on the Sierpinski gasket.

Findings

Rotor walk with random initial configuration is recurrent on SG.

i.i.d. sandpiles with expected chips ≥ 3 do not stabilize almost surely.

Divisible sandpiles at critical density 1 do not stabilize almost surely.

Abstract

We prove that, on the infinite Sierpinski gasket graph SG, rotor walk with random initial configuration of rotors is recurrent. We also give a necessary condition for an i.i.d. sandpile to stabilize. In particular, we prove that an i.i.d. sandpile with expected number of chips per site greater or equal to three does not stabilize almost surely. Furthermore, the proof also applies to divisible sandpiles and shows that divisible sandpile at critical density one does not stabilize almost surely on SG.