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Scaling limits in divisible sandpiles: a Fourier multiplier approach. (English) Zbl 1452.31012

Summary: In this paper we investigate scaling limits of the odometer in divisible sandpiles on \(d\)-dimensional tori following up the works of L. Chiarini, M. Jara and W. M. Ruszel [“Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits”, Preprint, arXiv:1808.06078], the first author et al. [Probab. Theory Relat. Fields 172, No. 3–4, 829–868 (2018; Zbl 1403.31001); Stochastic Processes Appl. 128, No. 9, 3054–3081 (2018; Zbl 1405.60143)]. 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 \((-\varDelta )^{-s/2} W\) for \(s>2\) and \(W\) a spatial white noise on the \(d\)-dimensional unit torus.

MSC:

31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
60J45 Probabilistic potential theory
60G15 Gaussian processes
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