Sandpile solitons via smoothing of superharmonic functions

Abstract

Let F : Z 2 → Z be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that under certain conditions there exists the pointwise minimal superharmonic function which coincides with F "at infinity".We develop such a theory to prove existence of so-called solitons (or strings) in a certain sandpile model, studied by S. Caracciolo, G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the phenomena of the identity in the sandpile group for a square where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we send waves (that is why we call them solitons), and can interact, forming triads and nodes.

Self-organized criticality and pattern emergence through the lens of tropical geometry

“…Tropical curves consist of edges, such that to each direction of the edges there corresponds a line-shaped pattern (a string) such as the one encountered in Figure 2; these patterns can be computed (18). In simulations, we have observed that these strings act like the renormalization group and, thus, ensure the proportional growth of the quadratic patches in Figure 2.…”

“…(See (16) and references therein.) Recently, the patches and a linear pattern in this fractal picture were explained in (17)(18)(19) using discrete superharmonic functions and Apollonian circle packing.…”