Direct Evidence for Conformal Invariance of Avalanche Frontier in Sandpile Models

Abstract

Appreciation of Stochastic Loewner evolution (SLE${}_{\kappa }$), as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model (ASM) are numerically shown to be conformally invariant and can be described by SLE with diffusivity $\kappa =2$. This value is the same as value obtained for loop erased random walks (LERW). The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.

6 pages, 4 figures

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Cited by in corpus (12)

• Recent advances in percolation theory and its applications
• Holographic applications of logarithmic conformal field theories
• Avalanche frontiers in dissipative abelian sandpile model as off-critical SLE(2)
• Abelian Sandpile Model on the Honeycomb Lattice
• Three Dimensional Ising Model, Percolation Theory and Conformal Invariance
• Observation of SLE$(\kappa ,\rho )$ on the Critical Statistical Models
• Logarithmic conformal invariance in the Abelian sandpile model
• Statistical Investigation of Avalanches of Three Dimensional Small-World Networks and their Boundary and Bulk Cross-Sections
• A Heisenberg double addition to the logarithmic Kazhdan--Lusztig duality
• Self-Repelling Bi-Exploration Process
• Numerical Determination of Boundary Condition Changing Operators
• Sandpiles Subjected to Sinusoidal Drive