Undergrad Theory of fluctuations in disordered systems

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SUMMARY

The discussion centers on the symmetry properties of interaction terms in disordered systems as outlined in the paper by Urbani and Parisi. Specifically, the participants analyze the expression < f(na,nb) f(nc,nd) f(ne,nf) > and the implications of assuming equals . The confusion arises regarding the correct symmetry transformations, particularly the exchanges involving indices b and f, as well as c and d. The references to the "Static replica approach to critical correlations in glassy systems" further emphasize the need for symmetrization in the equations derived.

PREREQUISITES
  • Understanding of statistical mechanics and disordered systems
  • Familiarity with the replica method in theoretical physics
  • Knowledge of correlation functions and their symmetries
  • Ability to interpret complex mathematical expressions in physics literature
NEXT STEPS
  • Study the replica method in detail, focusing on its application to disordered systems
  • Research symmetry properties in correlation functions, particularly in glassy systems
  • Examine the derivation and implications of Eq. (97) in the referenced paper
  • Explore the concept of critical correlations in statistical mechanics
USEFUL FOR

The discussion is beneficial for theoretical physicists, graduate students in statistical mechanics, and researchers focusing on disordered systems and critical phenomena.

giulio_hep
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TL;DR
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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