Scaling limit of the odometer in divisible sandpiles

Cipriani, Alessandra; Hazra, Rajat Subhra; Ruszel, Wioletta M.

doi:10.1007/s00440-017-0821-x

Cited by 10 publications

(18 citation statements)

References 23 publications

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“…Remark 5. Pluggin in the value α = 2 in the above Theorem matches the main result of Cipriani et al (2016), concerned specifically with the Gaussian case.…”

Section: Basic Setup and Main Resultssupporting

confidence: 74%

“…

This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al (2015) and Cipriani et al (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an α-stable random distribution.

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mentioning

confidence: 86%

“…The proof of Theorem 5 relies on two steps. As done in Cipriani et al (2016), the proof is based on determining the scaling limit in a "simpler" case, that is, when the variables σ in Proposition 4 are i.i.d. SαS(1).…”

Section: 2mentioning

confidence: 99%

“…In Cipriani et al (2016) the authors considered a general divisible sandpile model with i.i.d. initial distribution on a discrete torus and proved the conjecture of Levine et al (2015) on the torus T d determining the limiting field.…”

Section: Introductionmentioning

confidence: 99%

“…In this article we are interested in exploring the properties of the divisible sandpile model when the initial mass comes from heavy-tailed distributions. We are interested in extending results from both Cipriani et al (2016) and Levine et al (2015), namely we first study the dichotomy between stabilizing versus exploding configurations and secondly determine the scaling limit of the odometer function for heavy-tailed distributions on the torus. The novelty of the article is to consider the stabilization versus explosion dichotomy for divisible sandpiles for more general initial distributions by removing the finite variance assumption at the critical 1 value E(s) = 1 and to study scaling limits for those generalized random variables.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The divisible sandpile with heavy-tailed variables

Cipriani

Hazra

Ruszel

2018

Stochastic Processes and their Applications

Self Cite

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“…Remark 5. Pluggin in the value α = 2 in the above Theorem matches the main result of Cipriani et al (2016), concerned specifically with the Gaussian case.…”

Section: Basic Setup and Main Resultssupporting

confidence: 74%

“…

…”

mentioning

confidence: 86%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The divisible sandpile with heavy-tailed variables

Cipriani

Hazra

Ruszel

2018

Stochastic Processes and their Applications

Self Cite

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Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous

Schiavo,

Herry,

Kopfer

et al. 2024

Mathematische Nachrichten

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For an arbitrary dimension , we study: the polyharmonic Gaussian field on the discrete torus , that is the random field whose law on given by where is the Lebesgue measure and is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on where is a regularity parameter. As , we prove convergence of the fields to the polyharmonic Gaussian field on the continuous torus , as well as convergence of the random measures to the LQG measure on , for all .

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Stochastic Homogenization of Gaussian Fields on Random Media

Chiarini

Ruszel

2023

Ann. Henri Poincaré

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In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $$\Xi ^{g,\textbf{a}} $$ Ξ g , a and bi-Laplacian fields $$\Xi ^{b,\textbf{a}}$$ Ξ b , a . They can be characterized as follows: for $$f=\delta $$ f = δ the solution u of $$\nabla \cdot \textbf{a} \nabla u =f$$ ∇ · a ∇ u = f , $$\textbf{a}$$ a is a uniformly elliptic random environment, is the covariance of $$\Xi ^{g,\textbf{a}}$$ Ξ g , a . When f is the white noise, the field $$\Xi ^{b,\textbf{a}}$$ Ξ b , a can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain $$D\subset \mathbb {R}^d$$ D ⊂ R d , or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator $$\Delta $$ Δ , we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator $${{\,\mathrm{\bar{\textbf{a}}}\,}}\Delta $$ a ¯ Δ , with constant $${{\,\mathrm{\bar{\textbf{a}}}\,}}$$ a ¯ depending on the law of the environment $$\textbf{a}$$ a . The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014).

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Scaling limit of the odometer in divisible sandpiles

Cited by 10 publications

References 23 publications

The divisible sandpile with heavy-tailed variables

The divisible sandpile with heavy-tailed variables

Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous

Stochastic Homogenization of Gaussian Fields on Random Media

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