arXiv subjects

Wioletta M. Ruszel

Publications and source records attributed to Wioletta M. Ruszel.

24 records · Page 2Linked to original sources

arXiv2019

Scaling limits in divisible sandpiles: a Fourier multiplier approach

In this paper we complete the investigation of scaling limits of the odometer in divisible sandpiles on $d$-dimensional tori generalising the works Chiarini et al. (2018), Cipriani et al. (2017, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalised 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 $(-Δ)^{-(1+s)} W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.

math.PR↗

arXiv2019

Odometers of Divisible Sandpile Models: Scaling Limits, iDLA and Obstacle Problems. A Survey

The divisible sandpile model is a fixed-energy continuous counterpart of the Abelian sandpile model. We start with a random initial configuration and redistribute mass deterministically. Under certain conditions the sandpile will stabilize. The associated odometer function describes the amount of mass emitted from each vertex during stabilization. In this survey we describe recent scaling limit results of the odometer function depending on different initial configurations and redistribution rules. Moreover we review connections to the obstacle problem from potential theory, including the connection between odometers and limiting shapes of growth models such as iDLA. Finally we state some open problems.

math.PR↗

arXiv2018

Contour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields

We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-α}}$ with $α\in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich-Spencer contours for $α\neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for $α=0$ and conjectured by Cassandro et al for the region they could treat, $α\in (0,α_{+})$ for $α_+=\log(3)/\log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg1$, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any $α\in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*\cdot(1+|x|)^{-γ}$ and $γ>\max\{1-α, 1-α^* \}$ where $α^*\approx 0.2714$, the transition still persists.

math-ph↗

arXiv2018

Absence of Dobrushin states for $2d$ long-range Ising models

We consider the two-dimensional Ising model with long-range pair interactions of the form $J_{xy}\sim|x-y|^{-α}$ with $α>2$, mostly when $J_{xy} \geq 0$. We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed $\pm$-boundary conditions) do not exist. We discuss possible extensions of this result in the direction of the Aizenman-Higuchi theorem, or concerning fluctuations of interfaces. We also mention the existence of rigid interfaces in two long-range anisotropic contexts.

math.PR↗

arXiv2017

Scaling limit of the odometer in divisible sandpiles

In a recent work Levine et al. (2015) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

math.PR↗

arXiv2016

The divisible sandpile with heavy-tailed variables

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.

math.PR↗