Functional Renormalization Group Approach for Signal Detection

The authors present a review on a renormalization group approach to signal
detection.I believe that this work is valuable on a conceptual level and the
subject is interesting, however I see some issues with the presented material
that I feel should be covered better or if there are no answers to them it should
be clearly said that this is the case.

Since the paper is complex and long I shall discuss it by logical parts

A) In the first part of the paper authors make a nice introduction to the topic
covering first 2 sections.

B) In the 3rd section the authors attempt to make a link with the field theory

C) In the 4th and 5th sections authors present explicit calculations using renormalization group
procedures.

I have minor questions related with the introductiory parts:

A1) It is not clear from the text what is shown in Fig.4. upper panel As I
understand the MP distribution is obtained from multiplying 2 fully random
matrices. How does the figure illustrate deviation from universality
when the difference could also be understood as a MP distribution with a
different cutoff.

A2) A picture is introduced by which dimensionality is illustrated to be a deciding factor in the
relevance of eigenvectors because it induces different spectra in momentum space. I think this is
a dangerous analogy because the dimensionality is not all that comes into play. Which eigendirections
survive in the IR limit depends on the field theory as well, so I do not unerstand what such an
analogy gains us.

B1) My biggest problem with the paper is the 3rd section. I find it confusing and vague and I recommend
rewriting it in a simpler and more transparent way, sacrificing some of the material for clarity.
There seem to be a lot of ideas there, however as far as I see the only important point is left unanswered
and this is: how does looking at field theoretical models with RG help us detect signals
in continuous spectra? My point is this, when we are doing renormalization group on a field
theoretical model such as the phi^4 theory, we are at the end interested in low energy excitations
of the problem. Why are these excitations relevnat for the signal detection? If we look in comparison
the nicely introduced Wigner semicircle spectrum and some peaks imbedded in it, nothing guarantees
that what happens in the low energy limit of the theory is relevant to detecting whatever peaks
we want to detect. So what I recommend to the authors is offering as simple as possible answer
to this question as the main point point of Sec 3.

B2) If the description of such signals as said in my previous point is not what authors have in mind
they should be clear about what they are hoping to achieve with their RG approach. What kind of a signal
is a signal that they are intersted in and that is analogous to the IR degrees of freedom that
have survived after the integration over small wavevector modes

B3) As far as I see in later the authors introduce a phi^4 like field theory with in general
a nonlocal kernel of interaction as we see in Eq. 3.4. which they then discuss near the Gaussian
limit and as the field theory teaches us find that in some relevant cases the Gaussian theory is unstable
and that higher couplings above quadratic need to be taken into the account. What I find interesting
is that their theory is massive. Why is this the case or in other words why is this case relevant for
their consideration?

B4) Their interaction kernel was introduced as nonlocal. If this is the case why then do they
only consider the standard derivative expansion of the gradient term in p^2n? Why is their field theory
not e.g. with long range interactions? In this case they would have e.g. a fractional gradient
term.

B5) Also why do they only consider the simplest situation of the scalar phi^4 theory, let me elaborate? In
3.3 they discuss the symmetries of the model. Why would they not have some more complicated order parameter
field given the symmetries?

In section 4 they introduce the Wetterich-Morris formalism and they do two kinds of calculations: a)
the calculation of flows at local potential level and b) in field expansion either around 0 or the minimum.
To me this looks completely like a standard calculation that was done a lot of times before in virtually
all the reviews on NPRG. What I think they do different although I do not think it is said in a clear way
is impose an arbitrary distribution of q modes, which I presume, gives them a difference from the completely standard
calculation.

They motivate this calculation by stating (*) that:

"A motivation justifying the non-perturbative formalism use was the surprising
observation that, for most common noise, models power counting shows that the
first perturbations to the Gaussian model - the quartic and sixtic terms - are always
relevant at the tail of the spectrum, (see empirical statement (1))."

The gist of the standard calculation on the other hand is that if you tweak the initial condition
above T_c then you flow into the disordered phase and if you tweak them below T_c you flow into
the ordered phase.

C1) The difference between the standard calculation and their calculation should be stressed better
because it took me some time to understand that they are indeed not doing a standard phi^4
calculation.

C2) If they claim that given their distribution of momenta the picture of the flow is
pretty much as in the standard case (which I think they do since evidence to this effect is
given later in sec 5), I do not understand what they mean by their statement (*) above introduced
as motivation. Do they mean that if they tweak the couplings right that the dimensionless couplings are
going to grow? If this is so this statement has nothing to do with nonperturbative physics it just has
to do with the flow into the ordered phase. Even if you flow into the disordered phase you are going
to obtain some finite dimensionfull values for the couplings when the flow stops. Authors please
clarify this!

C3) The authors seem to claim that there is no fixed point of their flow, however Fig 21 testifies that
there might be one. This is very peculiar. The terminal part of the flow should be dominated by the shape
momentum distribution near q=0. Why do the authors not discuss the asymptotic equations separately?
Considering these equations might give an analytic answer whether there is a nontrivial fixed point or not
in their case.

At the beginning of Sec 5.2 they state (**):

"In section 4, we illustrated the dependence of the canonical dimensions on the
scale, for an MP distribution, and emphasized two points. The first point is that
at a large scale only two couplings are relevant, the quartic and the sixtic, the
latter tending to be asymptotically marginal."

This statement is misleading and incomplete. As said before, in all the cases either the flow to
disordered or the ordered phase if we look at the full function U(phi), it is going to be a
nontrivial function. In the cases when the flow is into the ordered phase it is going to
develop a flat region near phi=0. If the initial condition is tweaked to critical then
you flow to the fixed point. All this is well known within the standard picture.

C4) My question is what kind of initial conditions do they have in mind when they tell their
statement?

C5) What is the relevance of the initial conditions of the flow to their program of signal detection?
In Sec 5 I seem to see that they are interested in the critical situation. Why is that?

As a general statement, I beg the authors to improve the language
throughout the text. In many places the language constructions are atypical of
English language and words are wrong.

All in all I appreciate the author's effort and I agree with publishing the article provided
they take the criticisms I laid out into the account in good faith. In this case it
would certainly meet the requirements of the journal. This will also make the article
much easier to read and accessible to a wider audience.