Asymptotic Height Distribution in High-Dimensional Sandpiles
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Asymptotic Height Distribution in High-Dimensional Sandpiles Antal A. Járai1 · Minwei Sun1 Received: 21 May 2019 / Revised: 26 October 2019 © The Author(s) 2019
Abstract We give an asymptotic formula for the single-site height distribution of Abelian sandpiles on Zd as d → ∞, in terms of Poisson(1) probabilities. We provide error estimates. Keywords Abelian sandpile · Uniform spanning forest · Wilson’s method · Loop-erased random walk Mathematics Subject Classification (2010) Primary 60K35 · Secondary 82C20
1 Introduction We consider the Abelian sandpile model on the nearest neighbour lattice Zd ; see Sect. 1.1 for definitions and background. Let P denote the weak limit of the stationary distributions P L in finite boxes [−L, L]d ∩ Zd . Let η denote a sample configuration from the measure P. Let pd (i) = P[η(o) = i], i = 0, . . . , 2d − 1, denote the height probabilities at the origin in d dimensions. The following theorem is our main result that states the asymptotic form of these probabilities as d → ∞. Theorem 1.1 (i) For 0 ≤ i ≤ d 1/2 , we have pd (i) =
i e−1 1j! j=0
B
2d − j
+O
i i i 1 −1 1 + O = . e d2 2d j! d2
(1.1)
j=0
Minwei Sun [email protected] Antal A. Járai [email protected]
1
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
123
Journal of Theoretical Probability
(ii) If d 1/2 < i ≤ 2d − 1, we have pd (i) = pd (d 1/2 ) + O(d −3/2 ). In particular, pd (i) ∼ (2d)−1 , if i, d → ∞. The appearance of the Poisson(1) distribution in the above formula is closely related to the result of Aldous [1] that the degree distribution of the origin in the uniform spanning forest in Zd tends to 1 plus a Poisson(1) random variable as d → ∞. Indeed our proof of (1.1) is achieved by showing that in the uniform spanning forest of Zd , the number of neighbours w of the origin o, such that the unique path from w to infinity passes through o is asymptotically the same as the degree of o minus 1, that is, Poisson(1). In [11], we compared the formula (1.1) to numerical simulations in d = 32 on a finite box with L = 128, and there is excellent agreement with the asymptotics already for these values. Other graphs where information on the height distribution is available are as follows. Dhar and Majumdar [7] studied the Abelian sandpile model on the Bethe lattice, and the exact expressions for various distribution functions including the height distribution at a vertex were obtained using combinatorial methods. For the single-site height distribution they obtained (see [7, Eqn. (8.2)]) i d +1 1 (d − 1)d− j+1 . pBethe,d (i) = 2 j (d − 1) d d j=0
If one lets the degree d → ∞ in this formula, one obtains the form in the right hand side of (1.1) for any fixed i (with 2d replaced by d). Exact expressions for the distribution of height probabilities were derived by Papoyan and Shcherbakov [20] on the Husimi lattice of triangles with an arbitrary coordination number q. However, on d-dimensional cubic lattices of d ≥ 2, exact results for the height probability are
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