2020/10/05 by Alevy, Ian, Mkrtchyan, Sevak · 1 citation
#35R35 #60G50 #60K35 #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2010.01946
The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in ℤ2 and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. We compute the limit shape as a function of d in the symmetric case where each topple sends an equal amount of sand to each neighbor. The limit shape converges to a circle as d→ 1 and a diamond as d→∞. We compute the limit shape by comparing the odometer function at a site to the probability that a killed random walk dies at that site. When d→ 1 the Leaky-ASM converges to the abelian sandpile model (ASM) with a modified initial configuration. We also prove the limit shape is a circle when simultaneously with n→∞ we have that d=dn converges to 1 slower than any power of n. To gain information about the ASM faster convergence is necessary.