Equivalence of conditional and external field ensembles in absorbing-state phase transitions

I comment on the relation between two sampling methods for absorbing-state models. It is shown that a certain ensemble without external field conditional to activity coincides with the unconditional ensemble for sufficiently small external field. The actual physical processes involved are identical and the derivation of the identity of the scaling behavior relies on a single (established) scaling law. While the conditional ensemble by construction does not contain information about the system with large external field, it contains all the information about the limit of vanishing external field and about the vicinity of the critical point: finite size scaling as well as critical scaling in the temperature-like variable or in (small) external field.

Figure 1

Cartoon of a continuous record of activity in a system with a small external field $h$ which triggers new avalanches typically with some time of complete quiescence between them. The total time span covered is $T_0=300$. The record could have been produced by compiling the three instances shown in Fig. 2.

Figure 2

Cartoon of three avalanches produced by placing seeds in an otherwise inactive system. The three instances could have been taken from the continuous record in Fig. 1.