Scaling Limits in Divisible Sandpiles: A Fourier Multiplier Approach
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Scaling Limits in Divisible Sandpiles: A Fourier Multiplier Approach Alessandra Cipriani1 · Jan de Graaff1 · Wioletta M. Ruszel1 Received: 23 December 2018 / Revised: 20 July 2019 © The Author(s) 2019
Abstract In this paper we investigate scaling limits of the odometer in divisible sandpiles on d-dimensional tori following up the works of Chiarini et al. (Odometer of longrange sandpiles in the torus: mean behaviour and scaling limits, 2018), Cipriani et al. (Probab Theory Relat Fields 172:829–868, 2017; Stoch Process Appl 128(9):3054– 3081, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalized Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form (−Δ)−s/2 W for s > 2 and W a spatial white noise on the d-dimensional unit torus. Keywords Divisible sandpile · Fourier analysis · Generalized Gaussian field · Abstract Wiener space Mathematics Subject Classification (2010) 31B30 · 60J45 · 60G15 · 82C22 · 60G22
1 Introduction and Main Results Gaussian random fields arise naturally in the study of many statistical physical models. In particular fractional Gaussian fields F G Fs (D) := (−Δ)−s/2 W , where W denotes
The first author acknowledges the support of the Grant 613.009.102 of the Netherlands Organisation for Scientific Research (NWO). The first and third author would like to thank Leandro Chiarini and Rajat Subhra Hazra for helpful discussions.
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Wioletta M. Ruszel [email protected] Alessandra Cipriani [email protected] Jan de Graaff [email protected]
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TU Delft (DIAM), Building 28, van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands
123
Journal of Theoretical Probability
a spatial white noise, s ∈ R and D ⊂ Rd , typically arise in the context of random phenomena with long-range dependence and are closely related to renormalization. Examples of fractional Gaussian fields include the Gaussian free field and the continuum bi-Laplacian model. We refer the reader to Lodhia et al. [14] and references therein for a complete survey on fractional Gaussian fields. In this paper we study a class of divisible sandpile models and show that the scaling limit of its odometer functions converges to a Gaussian limiting field indexed by a Fourier multiplier. The divisible sandpile was introduced by Levine and Peres [11,12], and it is defined as follows: A divisible sandpile configuration on the discrete torus Zdn of side-length n is a function s : Zdn → R, where s(x) indicates a mass of particles or a hole at site x. Note that here, unlike the classical Abelian sandpile model [2,8], s(x) is a real-valued number. Given (σ (x))x∈Zdn a sequence of centered (possibly correlated) multivariate Gaussian random variables, we choose s to be equal to s(x) = 1 + σ (x) −
1 σ (z). nd d
(1.1)
z∈Zn
If a vertex x ∈ Zdn is unstable, i.e., s(x) > 1, it topples by keeping mass 1 for itself and distributing the exces
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