The top row shows the true external fields,

, versus those predicted from Eq. (18a), and the bottom row shows the true pairwise interaction,

, versus those predicted from Eq. (18b). Values of

ranging from 5 to 10 are shown, with different colors corresponding to different

. For each value of

, data is shown for 45 realization of the true distribution. Insets show the

of the mean external fields (top) and mean pairwise interactions (bottom). The three columns correspond exactly to the columns in . (A, B) (

). There is a very good match between the true and predicted values of both external fields and pairwise interactions. (C, D) (

). Even though

has increased, the match is still good. (E, F) (

). The true and predicted external fields and pairwise interactions do not match as well as the cases shown in (A, B, C, D). There is also now a stronger

in the mean external fields compared to (A) and (B). The

of the pairwise interactions in (F) is weaker than that of the external fields, but still notably stronger than the ones in (B) and (D). This indicates that the perturbative expansion is starting to break down.

## PubMed Commons