The boundary disorder correlation for the Ising model on a cylinder

In response to the referee reports I have made a number of changes throughout the manuscript. I recognize that there were an unusual quantity of errors and problems of presentation, which I sincerely regret. I hope that these are corrected in the new version.

That said, one of the referees' recommendations was mainly based on an assessment of the relevance of the main result. I am afraid that in this respect I am only able to restate the arguments in the paper (that this disorder correlation plays an interesting role in the exact solution for boundary spin correlations on the cylinder), which I hope I have made clearer in the present version. I am aware that the significance of the new result is modest, and I do not wish to exaggerate it, and I do not feel competent to expand it into a full review of similar asymptotic expansions. The only consideration I would add regarding the originality of the work is that it incorporates (with some corrections) the most original part of Reference [18], an unpublished preprint which I posted in 2014 (most of the rest of that preprint involves an alternative to part of the McCoy-Wu solution which is unnecessary for the calculation of the partition function but turned out to be helpful for the study of the fermionic correlation functions in [4]).

Nonetheless, I think that the revisions provide a substantial improvement over the original version, and I hope that they meet with your approval. If not, I still wish to thank the referees for taking time and effort to review the manuscript and for helping me to improve it (whether or not it is destined for formal publication).

I have made a number of changes to the introduction to clarify the notation (including adding Figure 1 and changing the symbol for the rescaling factor to Ξ rather than ξ to make it easier to distinguish from ζ), and attempted to clarify the discussion regarding the disorder correlation and its relationship to other correlation functions (including adding references [1-2,6] and updating [5]).
- I have also added a further explanation of the notation for the theta functions after Equation (6),
- made a grammatical correction in the citation of [18] on page 3,
- and added to the text after Equation (7) to state more clearly that p,s do not depend on the boundary conditions (among the two cases under consideration).

In Section 2, I have changed the discussion of the McCoy-Wu solution to eliminate a number of details which were not relevant (as well as being poorly explained) and instead indicate directly the point of departure for the analysis. I have also added a passage around Equation (10) to explain that the critical values of z₁ and z₂ depend only on the ratio E₁/E₂ of the coupling parameters (explaining the seeming paradox that the rescaling of the aspect ratio is expressed in terms of only z₁ in Equation (14)).
- I have also eliminated some unexplained and confusing notation (M=2𝓜, N=2𝓝, and t in place of either z₁ or z₂) that I had carelessly used here and
- made a minor correction in the text between Equations (11) and (12).

I have made a number of corrections and clarifications in Section 3. In particular:
- made it clearer in the text before equation (18) which limit is being considered
- explained in more detail the origin of the coefficients expressed in (30)
- clarified that (33) is the contribution of non-small values of θ, so that only small values remain to be controlled
- added more explicit cross references and Equation (45) to explain the passage from Equation (45) to Equation (6)

In Section 4, I have changed the title as suggested by the second referee, and attempted to improve the discussion at the beginning of the section by more clearly stating the predictions of [11] which are to be compared to my results.
- I also recall the restriction 𝓜/𝓝 -> 0 (between Equations (47) and (48)), and
- correct my erroneous use of x₊, x₋ in place of z₊, z₋.