Emergence of interfaces from sandpile models

28.06.2021, 14:00
Forschungsseminar Wahrscheinlichkeitstheorie

Wioletta Ruszel (Utrecht)

Interfaces separating two phases (e.g. water and ice) are created in phase coexistence situations such as at 0 degree Celsius. Random interface models are stochastic models which aim at explaining the macroscopic shape of an interface given the microscopic interaction of its particles (e.g. molecules). Prominent random interface models (in continuum space) are the Gaussian free field or fractional Gaussian fields. In this talk we would like to explain how general Gaussian interface models emerge from divisible sandpiles.

A divisible sandpile models is defined as follows. Given a graph G, assign a (real-valued) height s(x) to each vertex x of G. A positive value s(x)>0 is interpreted as a mass and a negative one as a hole. At every time step do the following: if the mass s(x)>1, then keep mass 1 and redistribute the excess among the neighbours. Under some condition, the sandpile configuration will stabilize, meaning that all the heights will be lower or equal to 1. The odometer function u(x) collects the amount of mass emitted from x during stabilization. It turns out that, depending on the initial configuration and redistribution rule, the odometer interface (u(x))_{x\in G} will scale to a Gaussian field.The results presented in this talk are in collaboration with A. Cipriani (TU Delft), L. Chiarini (TU Delft/IMPA), J. de Graaff (TU Delft), R. Hazra (ISI Kolkata) and M. Jara (IMPA)

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