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- ItemThick points for Gaussian free fields with different cut-offs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Cipriani, Alessandra; Hazra, Rajat SubhraMassive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal properties of the thick points of their cut-offs. Under some sufficient conditions for a centered Gaussian process with logarithmic variance we study the set of thick points and derive their Hausdorff dimension. We prove that various cut-offs for Gaussian free fields satisfy these assumptions. We also give sufficient conditions for comparing thick points of different cut-offs.
- ItemExtremes of the supercritical Gaussian free field(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Chiarini, Alberto; Cipriani, Alessandra; Hazra, Rajat SubhraWe show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).
- ItemThe divisible sandpile with heavy-tailed variables(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Cipriani, Alessandra; Hazra, Rajat Subhra; Ruszel, Wioletta M.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 alpha-stable random distribution.