Poghosyan–Poghosyan–Priezzhev–Ruelle conjecture on threshold and stationary sandpile densities

Let $\zeta_{\tau}(h)$ denote the threshold density of the fixed-energy sandpile with constant initial condition $h$, and let $\zeta_s$ denote the stationary density of Dhar's abelian sandpile. Poghosyan–Poghosyan–Priezzhev–Ruelle conjecture.

$$\zeta_{\tau}(h) \to \zeta_s \quad\text{as}\quad h \to -\infty.$$

Although threshold and stationary distributions are not equal in the limit as the initial condition tends to $-\infty$, the conjecture asserts that their total sand densities converge to the same value. The paper proves this convergence in distribution for the total amount of sand, thereby confirming the conjecture.