I Theory of fluctuations in disordered systems

giulio_hep
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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(na,nb) f(nc,nd) f(ne,nf) >

and they want to use a symmetry, for example they assume that <na3ndnf2> is equal to <na3nb2nc>

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
 
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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
 
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