N-Site approximations and CAM analysis for a stochastic sandpile

TL;DR

This paper develops n-site cluster approximations with a height restriction for a one-dimensional stochastic sandpile, providing estimates for critical density and critical exponents that align well with simulation results.

Contribution

It introduces a simplified n-site approximation method with height restriction for stochastic sandpiles, enabling accurate estimation of critical parameters.

Findings

Critical particle density estimated as 0.930(1).

Order parameter exponent estimated as 0.41(1).

Relaxation time exponent approximately 2.5.

Abstract

I develop n-site cluster approximations for a stochastic sandpile in one dimension. A height restriction is imposed to limit the number of states: each site can harbor at most two particles (height z_i \leq 2). (This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded.) On the basis of results for n \leq 11 sites, I estimate the critical particle density as zeta_c = 0.930(1), in good agreement with simulations. A coherent anomaly analysis yields estimates for the order parameter exponent [beta = 0.41(1)] and the relaxation time exponent (nu_|| \simeq 2.5).