Natural Measures on Polyominoes Induced by the Abelian Sandpile Model

Abstract

We introduce a natural Boltzmann measure over polyominoes induced by boundary avalanches in the Abelian Sandpile Model. Through the study of a suitable associated process, we give an argument suggesting that the probability distribution of the avalnche sizes has a power-law decay with exponent 3/2, in contrast with the present understanding of bulk avalanches in the model (which has some exponent between 1 and 5/4), and to the ordinary generating function of polyominoes (which is conjectured to have a logarithmic singularity, i.e. exponent 1). We provide some numerical evidence for our claims, and evaluate some other statistical observables on our process, most notably the density of triple points.

Figures (4)

• Figure 1: An example of the correspondence between the BT boundary avalanche process and the permutation boundary avalanche processes for the possible choices of $\sigma$, for the small graph with $V=3$ and $B=2$ depicted on the top-left corner. Top: the list of the 7 spanning forests, and the corresponding list of $(|T_1|,|T_2|)$. Bottom: the 7 recurrent configurations, and the associated lists of $(|P_1|,|P_2|)$ for the 2 permutations of the boundary edges. The three unordered lists are the same (namely, $(3,0)$, $(2,1)$, $(1,2)$ and $(0,3)$ are repeated 2,2,1,2 times, respectively), this being the consequence, for this graph, of the statement that the permutation boundary avalanche process and the BT boundary avalanche process on the uniform measure over recurrent configurations are the same probabilistic process.
• Figure 2: Averages of the $k$-th largest polyomino in a process (except the giant one), multiplied by $k^2$, and rescaled so that the first value is 1, on the data presented in Figure \ref{['distriProce']}. A far-fetching conjecture based on the formula (\ref{['eq.3876478']}) would suggest that this function is 1, up to values of $k \ll L_x$.
• Figure 3: Left: plot of the ordered list of $10^4$ avalanche sizes, for a folded-cylinder geometry on the square lattice, of size $101 \times 158$. We adopt a log-log plot, with a superposed red line of slope $-1/2$, which highlights the validity of the ansatz in equation (\ref{['eq.distroansatz1']}) in this case. Right: plot of the ordered list of $L_x \times 10^2=10100$ avalanche sizes, for $100$ realisations of the permutation process.
• Figure 4: Some examples of realisations of the permutation boundary avalanche process on the square lattice in a folded-cylinder geometry with $L_x=101$ (only the most relevant part of the cylinder is shown). The hue value of the colour describes the position of the corresponding boundary edge in the permutation $\sigma$.