Reviewer #1: Report on "Shift in critical temperature for random spatial permutations with cycle weights" by J. Kerl The model of random spatial permutations was introduced to mimic the description of the Bose gas in terms of permutation cycles. It was conjectured that the onset of Bose-Einstein condensation coincides with the appearance of macroscopic cycles. Thus, proving BEC in the interacting gas could be done by showing that cycles of a length proportional to the number of particles appear with a probability which is nonvanishing in the thermodynamic limit. Ueltschi and coworkers used different versions of the model to confirm (or, in arXiv:0910.3558, to cast doubt on) the physicists' consensus that not only there is BEC in the interacting gas but for weak interaction the critical temperature is higher than that of the ideal Bose gas. The author proposes an extended Monte Carlo study of a variant of the model. The Hamiltonian defined on permutations is the sum of a kinetic energy term = temperature times the accumulated jump lengths and a chemical potential term = alpha times the number of cycles (the chemical potential is -alpha). In a detailed analysis it is then shown that Tc indeed increases proportionally with alpha, and the constant of proportionality is the same found in another version of the model. While the ultimate proof of BEC in the interacting Bose gas may turn out to be as difficult with permutation cycles as with other methods, the family of random spatial permutation models is interesting on its own. The present study is very careful and provides a nice example of a controlled use of the Monte Carlo method. I therefore recommend its publication in J. Stat. Phys. However, the writing can be improved. Below I list my suggestions in the order of the Sections. 1. Einstein alone can be credited for the notion of the Bose-Einstein condensation and the computation of Tc in the ideal gas. Bose suggested the statistics carrying his name for describing the gas of photons. 2.1 - The Hamiltonian should be defined more precisely right after equation (2.2): X={x\in Z^3 | x_i=0,1,.,L-1} N=L^3 ||x-y||=min {|x-y+Ln| | n\in Z^3} because of the periodic b.c. alpha\geq 0 - One should not double (or triple) the notation for expectation value and probability (cf. (2.3), (2.4) and later). Because Lambda and X are fixed, E and P suffice, as are indeed used later in the text. 2.3 - Define explicitly the winding number instead of saying that it is more subtle than it appears. Drop the pompous sentence "The distances \tilde{d}.". - There is a confusion of "left or right to Tc" in the comment on figure 3. Better to use T\geq Tc and TH(pi), there is a second, independent, choice of probability e^{-H(pi')+H(pi)} to accept pi'. Hence, the transition probability M(pi,pi')=P(pi'|pi) is the product of the two probabilities associated with the two independent choices. Note that M(pi,pi)\geq 0 automatically holds. 3.3 - Change Definition 3.11 and drop Proposition 3.12: It is confusing to call p=p(pi), introduced in Definition 3.11, the period of pi. A permutation has period p if it reappears with probability 1 (!) after every p steps. This is rightly expressed by "must occur" in the first sentence, but contradicted in the formal definition. Definition 3.11 should not be that of the period of pi, but of the aperiodicity of a chain: a chain is aperiodic if p(pi):=gcd{n: P(Pi_n=pi|Pi_0=pi)>0}=1 for every pi. Then, Proposition 3.12 is superfluous, because aperiodicity immediately follows from irreducibility. Indeed, if for an integer n_0, M^{n_0}(pi,pi')>0 for every pair pi, pi', then P(Pi_n=pi|Pi_0=pi)=M^{n}(pi,pi)>0 for all pi and all n>n_0, implying p(pi)=1. - Lemma 3.15 is equivalent to the following: pi'\in R(pi) iff pi\in R(pi'), which is a direct consequence of Definition 3.5 of R(pi). - Start Proposition 3.17 with the sentence "Let ||pi(x)-pi(y)||=1." Drop "non-trivial" from the succeeding sentence. - In the second formula of Remark 3.18 correct the misprint ell_y(pi) to ell_y(pi').