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Annealed averages in spin and matrix models

by Laura Foini, Jorge Kurchan

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Submission summary

Authors (as registered SciPost users): Laura Foini
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-04-23 13:57
Submitted by: Foini, Laura
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


A disordered system is denominated 'annealed' when the interactions themselves may evolve and adjust their values to lower the free energy. The opposite ('quenched') situation when disorder is fixed, is the one relevant for physical spin-glasses, and has received vastly more attention. Other problems however are more natural in the annealed situation: in this work we discuss examples where annealed averages are interesting, in the context of matrix models. We first discuss how in practice, when system and disorder adapt together, annealed systems develop 'planted' solutions spontaneously, as the ones found in the study of inference problems. In the second part, we study the probability distribution of elements of a matrix derived from a rotationally invariant (not necessarily Gaussian) ensemble, a problem that maps into the annealed average of a spin glass model.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-6-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2104.04363v2, delivered 2021-06-26, doi: 10.21468/SciPost.Report.3126


1- Interesting phenomenology is found that is useful both for RMT specialists and statistical physicists.
2- The results are interesting and have a nice phenomenology.
3-I also think that the paper opens up quite a few perspectives for future study.


1- The paper has a tendency to vagueness at some points and the overall presentation could be improved.
2-Exotic choice of notation


In this paper the authors study an annealed version of the spherical SK model and show how it can be used to establish results on the statistics of the elements and sub matrices of random matrices. The paper is interesting and fairly well written and I found it very interesting although a bit difficult to follow at some of the exposition. There are points where the paper changes direction quickly, and the precise model under discussion could be better emphasized in some passages.
I think that the discussion of the phenomenon of eigenvalue detachment is a very interesting and intriguing phenomenon and could be related to some recent results in random matrix theory (RMT). The paper is suitable for publication after some improvements are made mainly at the level of presentation.

Requested changes

1-In the introduction the use of the dagger to note the dual vector to s is misleading as it suggest that the spin vector is complex.

2-The notation P(A_ij) is misleading it should be P_ij(a) as the probability distribution depends on the the pair ij.

3-The notation Ds is a bit odd - why not simply d{\bf s} as the integrals are finite dimensional and D is often used to functional integrals.

4-The phase transition in the spherical SK model does not always occur, it does for the Wigner semi-circle law but for density of states which do not vanish at the edge of the support of the density of states there should be no transition. It is basically the same transition as Bose Einstein condensation. I wonder what happens to the phenomenology of eigenvalue detachment in this case, does the whole scenario remain the same ?

5-In the annealed case the discussion of the formation of a molecule seems to be closely related to Baik-Ben Arous-Peche (BBP) transition for spiked random matrices J. Baik, G. Ben Arous, and S. Peche, Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, Ann. Probab. 33, 1643 (2005), this describes how a rank one perturbation to a random matrix can shift the largest eigenvalue. It might be work discussing this given the papers overall link with RMT.

6-I found the section 2.8 a bit out of place and expected 2.9 to be a continuation of this discussion, it would be better to put this in the introduction or the conclusion and it doesn’t really merit its own section.

7-The model 35 for p=2 is the standard SK model no ? It might be helpful to say this. In figure 6 it would be useful to recall that it is p=2 Ising. In general it might be useful to be more precise about which p=2 is being discussed.

8-Before equation 49 it would be clearer to call 49 the generating function of the probability distribution in equation 46.

The statement at the bottom of page 16 “Let us emphasize that this is a large N result, valid for r much smaller than N” is a bit vague. Should r be of order 1 or does N^1/2 work ?

  • validity: high
  • significance: high
  • originality: high
  • clarity: ok
  • formatting: reasonable
  • grammar: good

Author:  Laura Foini  on 2021-10-26  [id 1882]

(in reply to Report 2 on 2021-06-26)

We thank the referee for the appreciation to our work and the suggestions to improve the presentation.
Please find attached a pdf with the reply to the comments and the questions raised by the referee.
Kind regards,

Laura Foini



Anonymous Report 1 on 2021-6-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2104.04363v2, delivered 2021-06-02, doi: 10.21468/SciPost.Report.3013


see attached pdf report


see attached pdf report


see attached pdf report

Requested changes

see attached pdf report


  • validity: high
  • significance: high
  • originality: -
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Laura Foini  on 2021-10-26  [id 1881]

(in reply to Report 1 on 2021-06-02)
answer to question

We thank the referee for the appreciation to our work and the suggestions to improve the presentation.
In the attached file we reply to the comments and to the questions that the referee pointed out.
Kind regards,

Laura Foini



Anonymous on 2021-11-12  [id 1939]

(in reply to Laura Foini on 2021-10-26 [id 1881])

The authors have addressed most of the issues raised in my report. I would only like them to add one or more relevant references where the issue is discussed which they claim they cannot go into in detail here (the validity of the saddle point approach applied to integration over order(N) variables).

Author:  Laura Foini  on 2021-12-09  [id 2019]

(in reply to Anonymous Comment on 2021-11-12 [id 1939])

In standard random matrix ensembles the energy goes as N^2 times an action order one while the minimization is done over N eigenvalues. This implies that the large parameter in the action is much larger than the number of variables over which the saddle point is performed.
In the revised version we have written explicitly the factor N^2 in Eq. 2 and subsequent.
A complementary approach is to transform the integral over the discrete sets of eigenvalues as a functional integral over the density of eigenvalues and the saddle point is taken over such measure.
A detailed discussion of the terms at different order in N which appear in this procedure (and the fact that the entropic term order N is subleading) is discussed in Livan, Vivo, Novaes, Introduction to random matrices: theory and practice, Springer (2018), that we already cite in our work.
Mathematicians have made rigorous these results at least for a certain class of potentials, an example is discussed in the book E.B. Saff and V. Totik, Logarithmic Potentials with External Field, Springer-Verlag, 1997. We hope that this will reply to the concerns of the referee.

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