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

2022/10/03 by Robin Kaiser, Kaiser, Robin, Ecaterina Sava‐Huss +1
Mathematics · Physics and Astronomy · #05C81 #60J10 #60J45 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2210.00810

openalex publication_date 2022/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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