Theory of fluctuations in disordered systems

[giulio_hep]

TL;DR Summary

In the computation of the dynamic exponents from the
cubic expansion, I'm asking clarifications and a clear explanation about the interaction term and what are the symmetries in the monomials

I'm reading the https://www.phys.uniroma1.it/fisica/sites/default/files/DOTT_FISICA/MENU/03DOTTORANDI/TesiFin26/Urbani.pdf at paragrph 4.6.2 "The interaction term".

They write a right hand side:

< f(n_a,n_b) f(n_c,n_d) f(n_e,n_f) >

and they want to use a symmetry, for example they assume that <n_a³n_dn_f²> is equal to <n_a³n_b²n_c>

It looks like c = d and b = f at first sight, but which is the correct symmetry really? I can't find an explanation in the previous pages: any idea? Thanks

 

[giulio_hep]

Indeed my doubt is somehow reinforced from what I read in "Static replica approach to critical correlations in glassy systems" (same authors, among which again Pierfrancesco Urbani and this year's Nobel Prize, Giorgio Parisi) ref. 12A540-22 paragraph "C. Expression of λ in HNC", page 23, where it is written:

Here we have again to symmetrize Eq. (153) with respect to the exchanges
a ↔ b, c ↔ d, e ↔ f, ab ↔ cd, ab ↔ ef, cd ↔ ef
because these have been used explicitly to derive Eq. (97).

Again, while the exchange of c with d would make sense for symmetry of f(.,.), my intuition was also able to get the exchange of ab with ef, but it is still a mystery for me to understand the exchange of just b with f... or (put together) of df ↔ cb