Theory of Kinetically-Constrained-Models Dynamics

This is an interesting paper about the spin dynamics of Fredricksen-Andersen (FA) models on Bethe lattices. The persistence function is shown to obey an equation consistent with mode-coupling theory (MCT) and the implications of this equation are studied, including connection to existing results from bootstrap percolation. The results are verified by numerical simulations.

Some aspects of the presentation could be clearer.

Given the strengths noted above, these results deserve publication in scipost. However, the authors should consider the following points.

- Some of the previous works for FA models on Bethe lattices (eg [27-32]) should be cited in the introduction (instead of later in Sec 2). Their relation to the current work should be explained in more detail, to give context.

- Some of the results of this work are quoted in Ref[34] which is now published as Phys Rev E 110, 044312 (2024). In that work they are attributed to arxiv:2212.05132, which seems to be a precursor of this work, albeit with a different title. I would suggest to include a comment in the introduction to explain the relationship between Ref[34] and this work.

- In the first paragraph of section 2.1, the authors use several terminologies for the state of spin i. Either negative/positive (presumably corresponding to $s_i=\mp1$) or up/down or excited/unexcited. It would be useful to introduce a single terminology from the start, and stick to it.

- I am not sure if the" zero-switch"/"one-switch" sets defined above eq(6) are new in this work, or taken from the literature. A comment would be helpful.

- Section 2.6 draws heavily on Ref 34, which is itself quite long and technical. The explanations that appear between equations (21) and (22) of this work are interesting but not easy to follow. For example:

. I am not sure what means an "equilibrium state" in this context. The authors may want to provide a precise definition, or perhaps just a description in words. For a fixed instance of the Bethe lattice then it seems to me that there are natural "states" such that : each state has a set of frozen spins, and all other spins s_i are negative with probability p and positive with probability 1-p. (These would correspond to communication classes within the (reducible) Markov chain that describes the system). However, I am not sure if the authors refer to these states, or some other ones.

. Please also define the $m_i$. In the example "states" above then all the soft spins would have $m_i = 1-2p$, and the spin correlations also factorise. It seems that the authors have found a more complicated situation than this, this deserves a (short) physical explanation so that the paper makes sense without requiring the reader to read all of Ref[34] in detail.

- It would be useful in the conclusion to briefly return to the question raised in the introduction about the connection of static and dynamic quantities. Do the results of this work offer any new perspectives on this question? For example, my understanding of Sec 2.6, is that this model has a close connection between statics and dynamics, but on the other hand the spin-glass susceptibility is regular at dynamical arrest. Could this scenario be generic in some larger class of models, and can be attributed to some underlying physical/mathematical structure?

- In appendices I would suggest to write out in full the "MT" which I assume stands for "main text". Also below eq (C.1) I would suggest to include a pointer to the definition of $F$ in (A.1), so that each appendix can be read independently.

Ask for minor revision

1. Studies the dynamics of a class of kinetically constrained models (f-spin facilitated Fredrikson-Andersen models, part of a family of KCMs which are very relevant for understanding glasses).
2. By placing the models in a z-tree (rather than on a lattice with loops) the problem becomes effectively mean field, and therefore amenable to a technique like mode-coupling theory (and specifically schematic MCT in this case).
3. The paper obtains the MCT exponents exactly, thus completing the MCT study of such Bethe-lattice KCMs.
4. The main result is that of eq.18/19, with table 1 giving the exponent as a function of f and z.

The paper delivers what it purports to do, so in that sense I cannot see any weaknesses. It is written in a way that will be accessible to experts in this field (especially those like me that have been around it since the time MCT was more fashionable).

This paper studies the dynamics of a class of kinetically constrained models (f-spin facilitated Fredrikson-Andersen models). These are part of a family of KCMs which are very relevant for understanding glasses (cf. East model or constrained lattice gases). In an f-FA model a spin can flip if f of its neighbours are in a specific state (say up), this kinetic constraint leading to slow relaxation when that state is scarce (e.g. at low T when there are few of the energetically expensive ups). KCMs in low dimensional square lattices, and with low f (like the FA or East models) behave in a very non mean-field way. Interestingly, when place in mean-field like geometries, such as a Cayley tree, they can display mean-field like dynamics, in particular for larger f (when the KCM is more constrained) when the KCM has an ergodicity breaking singularity as some finite average density of facilitating sites (i.e. finite temperatures).

Early work had shown schematic MCT like behaviour of KCMs in trees, but this programme of work was not completed. This is what this paper mostly does. Specifically, by considering the persistence function (which is an alternative to the autocorrelation function to quantify relaxation, and often simpler to study) the paper finds the MCT exponents of f-FA in z-trees exactly, thus completing the MCT study of such Bethe-lattice KCMs. The main result is that of eq.18/19, with table 1 giving the exponent as a function of f and z. There are more results, like doing the same in the case of random pinning (which is known in other mean-field systems to lead to a higher order MCT singularity), and they also argue on convincing grounds that their results for the persistence also capture the behavior of the autocorrelator in this kind of geometries.

The paper is easy to follow for those familiar with MCT methods. The main text deals with the essential aspects of the approach that leads to the central results, with other technical details relegated to the appendices. Overall this is a solid piece of work that fills gaps in the mean-field study of KCMs. I recommend publication as is.

Publish (meets expectations and criteria for this Journal)