8 - Analytical results

Summary

In this chapter, some important analytical techniques and results are discussed. The first two sections are concerned with mean-field theory, which is routinely applied in SOC, and renormalisation which has had a number of celebrated successes in SOC. As discussed in Sec. 8.3, Dhar (1990a) famously translated the set of rules governing an SOC model into operators, which provides a completely different, namely algebraic perspective. Directed models, discussed in Sec. 8.4, offer a rich basis of exactly solvable models for the analytical methods discussed in this chapter. In the final section, Sec. 8.5, SOC is translated into the language of the theory of interfaces.

It is interesting to review the variety of theoretical languages that SOC models have been cast in. Mean-field theories express SOC models (almost) at the level of updating rules and thus more or less explicitly in terms of a master equation. The same applies for some of the renormalisation group procedures (Vespignani, Zapperi, and Loreto, 1997), although Díaz-Guilera (1992) suggested very early an equation of motion of the local particle density in the form of a Langevin equation. The language of interfaces overlaps with this perspective in the case of Hwa and Kardar's (1989a) surface evolution equations, whereas the absorbing state (AS) approach as well as depinning use a similar formalism but a different physical interpretation – what evolves in the former case is the configuration of the system, while it is the number of charges in the latter.