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THE APOLLONIAN STRUCTURE OF INTEGER SUPERHARMONIC MATRICES
, 2013
"... We prove that the set of quadratic growths attainable by integervalued superharmonic functions on the lattice Z² has the structure of an Apollonian circle packing, affirming a conjecture posed by the authors in [7], and completely characterizing the PDE which determines the continuum scaling limit ..."
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We prove that the set of quadratic growths attainable by integervalued superharmonic functions on the lattice Z² has the structure of an Apollonian circle packing, affirming a conjecture posed by the authors in [7], and completely characterizing the PDE which determines the continuum scaling limit of the Abelian sandpile on the lattice Z².
THE DIVISIBLE SANDPILE AT CRITICAL DENSITY
"... Abstract. The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertextransitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤ 1. The process stabilizes almost surely if m < 1 and it almost surely does n ..."
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Abstract. The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertextransitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤ 1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m> 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete biLaplacian Gaussian field. 1.