Universality in critical exponents for toppling waves of the BTW sandpile model on two-dimensional lattices

Universality and scaling for systems driven to criticality by a tuning parameter has been well studied. However, there are very few corresponding studies for the models of self-organized criticality, e.g., the Bak, Tang, and Wiesenfeld (BTW) sandpile model. It is well known that every avalanche of the BTW sandpile model may be represented as a sequence of waves and the asymptotic probability distributions of all waves and last waves have critical exponents, 1 and 11/8, respectively. By an inversion symmetry, Hu, Ivashkevich, Lin, and Priezzhev showed that in the BTW sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power-law relationship with critical exponent 5/8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1 (Phys. Rev. Lett. 85 (2000) 4048). Such predictions have been confirmed by extensive numerical simulations of the BTW sandpile model on square lattices. Very recently, we used Monte Carlo simulations to find that the waves of the BTW model on square, honeycomb, triangular, and random lattices have the same set of critical exponents.