Universal and non-universal large deviations in critical systems

In this paper, the authors study large deviations in critical systems. While they
mainly focus on a situation at equilibrium (namely the O(n) model and its variants),
they also comment on some non-equilibrium situations, mainly the Kardar-Parisi-Zhang (KPZ) equation
in 1+1 dimension. One of the motivation of the paper is the power-law correction (with corresponding
exponent \psi) to the leading stretched exponential tail that describes the (typical) fluctuations of the
magnetisation in these systems. The value of this exponent was conjectured to be $\psi = (\delta-1)/2$
where $\delta$ is the isotherm exponent. Furthermore, in Ref. [1] this conjecture was extended to non-equilibrium systems.
The second focus of this paper is to explore the crossover from this typical stretched exponential behavior to the large deviation
regime (which they call Cramer's regime).

Concerning the first point, the authors present different approaches (hierarchical model, mean-field limit,
functional renormalization group, perturbative approach in $d=4-\epsilon$ and numerical simulations) which all
suggest that Eq. (1) with indeed the exponent $\psi$ as predicted (multiplied by $n$ for a $n$-component system), confirming the conjectures stated
in previous works. However they also point out, using previous exact results for the KPZ equation that
the value of the exponent $\psi$ is different. Concerning the large deviations, they actually show that there
are two regimes of large deviations: a first one for moderate values of the parameter $s$ -- which to a large extent
is universal in the renormalization group sense -- and a second one for very large values of $n$ which is non universal.

The paper is well written, physically sound and present interesting and timely results. I believe that they
will be of interest to the statistical physics community working on large deviations and/or critical phenomena. Therefore, I would
like to recommend the publication of the present manuscript in SciPost. I only have minor comments that the authors may address
in a revised version of their manuscript.

1) On line 40-44 the authors discuss an algebraic day but instead
mention a stretched exponential behavior right after.

2) On line 43, when they discuss the Ising model, they should say that in this
case 's' is the global magnetisation of the system.

3) In Eq. (6) the denominator should be \sqrt{2\pi/I''(0)}

4) In Eq. (7), on the left hand side, this should be $s$ and not s.

5) Below Eq. (B6) the text needs to formatted properly.

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The paper studies the probability distribution functions of O(n) models at their critical points. It compares the results obtained with different approaches and discusses the corrections to the leading behavior. The topic is old but relevant. There has recently been notable progress on this problem from the RG theory point of view.
The manuscript is written in a very reader-friendly style, which I perceived as a "mixture" of a short review and a research paper. I like, appreciate, and support this way of writing, but I insist that the revised manuscript makes a more sharp distinction between the previous and the new results. At the present point, it is not quite clear from the reading (at least for me), which results presented here are new and which appeared in literature before ( Refs. 11, 18 in particular, but also earlier papers).
In addition to this, I have the following points and questions:
- The abstract strongly says that the results: "challenge existing assumptions about power-law corrections...". If I am correct, the relevant reference to this sentence appears only in the discussion section. The first part of this discussion of Sec.5 only invoked previous results (out of equilibrium), the second one (relating to MC of 3d Ising) is short and rather soft or speculative. The authors should also say more explicitly and clearly what "assumptions" they mean and discuss the references. It is also unusual that a result very important for the paper is mentioned only in the concluding section (and is also rather speculative). In contrast, the result sections strongly confirm the expected behavior, but only for situations/approximations that have mean-field features (see the point below).
- I got the impression that (except for MC) the calculations of the paper have vanishing anomalous dimensions (due to either approximation or the properties of the model). In particular this seems to imply (quite unsatisfactorily) that the tail is the same for all values of n, which is not true in general. In consequence the impact of n only appears at the level of the correction. The important quantity beta/nu is also always the same at the level of the carried out calculations.
I do not see why the role of \eta should not be discussed, and also computed with functional RG or the perturbative approach (even at low levels of accuracy). I also see it as a bit logically tense: the paper studies correction factors with approximations which do not capture the correct leading behavior (which in my understanding involves \eta and depends on the value of n).
- It might be very interesting to discuss d=2 (and n=1). It seems like none of the present approaches can be applied to this case. Is this right? I understand that the motivation for such study can be limited due to the difficulty of reaching the universal regime in MC (but this difficulty occurs also in d=3).
- What is meant by "CLT breaks down"? A mathematical theorem cannot break down. I understand that the theorem assumptions are not met. But which assumptions? Similarly: what are the assumptions and statement of what the authors call "generalised CLT"?
- There is imbalance in detailing technical points of distinct methods. I propose to move most of Sec.3.4 (the technical part) to the appendix.

See the report -\ eta in particular.

Ask for major revision

Scientifically sound, clearly written, of interest to theoretical community, different analytic and numerical methods, exceptional quality of numerical data.

Lacking some background for the non-expert reader, absence of consequences for experiment

This is an interesting and generally well written paper addressing the statistical properties of spatially averaged variables in the large deviation range of fluctuations, with particular reference to order parameter fluctuations at thermodynamic equilibrium and near a second order phase transition. I think that it could make a valid contribution to Sci Post and I recommend it's publication. Some points to be considered before publication are included below.

1. One thing I have noted is that the exponents used in the analysis, beta/nu, delta, eta depend uniquely on the eigenvalue exponent y_h related to the field variable. I think this is because of the particular choice of order parameter fluctuations at T=T_c and small field. If one studied fluctuations of the enthalpy, H=U-Nhm, even at T=T_c, the critical scaling would involve y_t as well as y_h. Or, for the order parameter, working at zero field, with reduced temperature t would (I think) introduce the second exponent, y_t through the critical exponent nu. Could this be done ? Is it of interest ? Or does it just add complexity without content ?

2. I was a bit confused about some of the universality classes appearing in the analysis. In particular for the hierarchical model, I was expecting a tree calculation to yield mean field critical exponents but the expression seems to predict delta=5. Is that correct ? It is quite close to the D=3 value but I don't see dimensionality in the hierarchy. In addition to referencing [22] it would be worthwhile discussing this in the text.

3. Similarly, it would be useful to be reminded what one can expect in the large n limit. Naively I was again expecting mean field exponents, but we have 2beta/nu=d-1 and delta=2d/(d-2). References would be useful. These expressions predict unexpected results for d=4.

4. I appreciated the discussion of finite size effects and the consequences for entering the universal large deviation regime in d=2 and d=3. Does this mean that experimentally the regime might just be accessible in d=3 ? What about systems like helium-4 where the lambda transition can be approached with exquisite precision and where finite size effects such as the critical Casimir force can be measured ?

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