Definition of an essentially RPA field

We will say that the field $a$ is of an ``essentially RPA'' type if:

1. The phase factors are statistically independent and uniformly distributed variables up to $O({\epsilon}^2)$ corrections, i.e.

   $${\cal P}^{(N)} \{s, \xi \} = {1 \over (2 \pi)^{N} } {\cal P}^{(N,a)} \{s \} \; [1 + O({\epsilon}^2)],$$ (17)

   
   
   where

   $${\cal P}^{(N,a)} \{s \} = \left( \prod_{ l {\cal 2 B}_N } \oint_{{\cal S}^{1} } \vert d \xi_l\vert \; \right) {\cal P}^{(N)} \{s, \xi \},$$ (18)

   
   
   is the $N$-mode amplitude PDF.
2. The amplitude variables are almost independent is a sense that for each $M \ll N$ modes the $M$-mode amplitude PDF is equal to the product of the one-mode PDF's up to $O(M/N)$ and $o({\epsilon}^2)$ corrections,

   $${\cal P}^{(M)}_{j_1, j_2, \dots , j_M} = P^{(a)}_{j_1} P^{(a)}_{j_2} \dots P^{(a)}_{j_M} \; [1 + O(M/N) + O({\epsilon}^2)].$$ (19)