Rényi complexity in mean-field disordered systems

-Referee 1
We thank the referee for their positive opinion in favor of publication, while noting that the motivation section still has room for improvement.

As Referee 1 pointed out, the Rényi entropy and the free energy in the Monasson method are directly linked (or mathematically equivalent), which is quite evident. It follows that the Rényi entropy is essentially a generalized annealed Franz-Parisi potential.

This connection is interesting for two main reasons. First, Rényi entropy is generally easier to compute or measure compared to Shannon entropy due to its simpler functional form. This simplicity is one of the main reasons why Rényi entropy has been widely used in various areas of physics, such as quantum many-body systems, and continues to attract growing attention.

Second, this strong link between Rényi entropy, developed in information theory, and methodologies in disordered systems, could broaden perspectives and open new pathways for understanding both fields (see also strengths raised by Referee 2). A historical example of such a cross-disciplinary impact is the adoption of large deviation theory from probability theory into statistical physics. (We agree this comparison may sound slightly exaggerated, but we hope the message is clear.) Initially, physicists had been using similar mathematical tools —like free energies and generating functions— without realizing their connection to large deviation theory. Once this link was established, it led to a surge of new research, as we have witnessed.

Similarly, we believe that the connection between Rényi entropy and methods in disordered systems holds the potential to broaden the horizons of both fields, going beyond simple mathematical equivalence.

Such implications are already evident in our paper in two key areas. First, the Rényi entropy is a non-increasing function of the Rényi index $m$, a well-known property in information theory. This property directly translates to the generalized Franz-Parisi potential, where the free energy difference should decrease with increasing
$m$ in any model in any spacial dimension. This result is valuable not only for theoretical understanding but also as a guide for numerical simulations. In previous studies (e.g., 10.1103/PhysRevE.88.022313, 10.1073/pnas.1407934111, 10.1103/PhysRevE.89.022309), the annealed Franz-Parisi potential ($m=2$) was often used as a proxy for the quenched Franz-Parisi potential to estimate the (Shannon) configurational entropy. However, to our knowledge, discussions on how closely the annealed version approximates the quenched one were not provided. Based on the general properties of Rényi entropy, we can now immediately conclude that the annealed Franz-Parisi energy difference provides a lower bound for the quenched one (and thus the configurational entropy). Moreover, in the mean-field models we studied, both vanish at the same temperature.

A more non-trivial example where the connection between Rényi entropy and disordered systems is beneficial lies in the inequalities presented in Eq. (20). These inequalities, derived using techniques from information theory, include an upper bound for the Rényi entropy when
$m>1$. We found that this bound is saturated under specific conditions, namely in the RSB$_b$ regime. In fact, information theory deals extensively with such inequalities involving entropy. We anticipate that more connections of this kind could be further exploited.

So far, we have mainly discussed the benefits for disordered systems, drawing insights from Rényi entropy. However, we believe the reverse direction is also worth exploring. In equilibrium statistical mechanics, the Rényi index $m$ corresponds to the number of replicas, providing an interpretation of Rényi entropy within the framework of replica theory. This may open the door for further transfer of ideas from disordered physics to information theory, especially considering the growing overlap between these fields nowadays.

In the previous manuscript, while the motivation in terms of a (mean-field) guideline for numerical measurements was emphasized, the interdisciplinary aspect of the Rényi entropy —linking information theory and techniques in disordered systems— was not clearly stated. In the current manuscript, following the referee's suggestions, we explicitly highlight this interdisciplinary motivation in the introduction and conclusions parts and provide additional explanations in the text.

Following the referee's last comment, we note that currently, we are performing numerical simulations to compute the Rényi complexity for a model glass-forming liquid in finite dimensions based on the Kurchan-Levine approach. This involves some technical challenges, such as identifying similar patches, which are not directly related to the essential features of the Rényi complexity presented in this manuscript. These simulations require additional efforts, and therefore, we plan to present them in a separate publication.



Referee 2

We thank the referee for the positive evaluation of our manuscript and for suggesting two modifications that improve the clarity of the paper.


Requested change 1
The Edwards-Anderson order parameter qEA is used in Eq. (12) before being defined. In the rest of the paper, the authors define it as the overlap at which the annealed Franz-Parisi potential develops a second minimum below the (m-dependent) “Mode-Coupling temperature”. A few lines around Eq. (12), possibly explaining how this definition is consistent with the historical one given by Edwards and Anderson, could help the reader. Notice that, according to the authors’ definition, qEA=qEA(m,T), such that the identification with the physical overlap is non-trivial.

Answer1
We thank the referee for this comment. Indeed, in the previous manuscript, we directly referred to the location of the local minimum as the Edwards-Anderson parameter. However, this requires a more nuanced explanation, as the former involves the overlap parameter arising in replica computations, while the latter characterizes the freezing of degrees of freedom in the physical system under consideration.
The correspondence between these two quantities is rooted in physical considerations and the ansatz of replica symmetry (and its possible breaking), which makes the identification non-trivial.

In the revised manuscript, following the referee's suggestion, we have rewritten the discussion around Eq.(12) to properly reflect this distinction and provide the necessary clarification.


Requested change 2

The behavior of the p-spin model is slightly different than the one of the RFEM, where the Rényi complexity is well defined for all T. Indeed, as clear from Fig. 6, there are 3 non trivial temperatures for each index m: from lower temperatures, TK where the complexity starts being non-zero, Tc(m) where it becomes RS, and a generalized mode-coupling transition at a temperature that I will call Td(m), as it seems to depend on m. To clarify this fact, I suggest the authors to define explicitly this temperature in the text, possibly as Td(m), reserving TMC=lim m→1 Td(m).

Answer 2
Following the referee's suggestion we have explicitly defined the temperature Td(m) in section A1, and provided its expression in Eq. (A.8).
In addition, we have incorporated this $m$-dependent dynamic transition temperature into the phase diagram in Fig.7.