A conformal bootstrap approach to critical percolation in two dimensions

We are grateful to the first referee for his technical remarks, and to the second referee for his insightful suggestions that led us to perform significant clarifications.

Replies to the first referee report:

1. Our way of inserting tables and figures in the text seems better to us. We will not change it unless given a good reason to do so.

3. 4. and 5. We have added in the appendix outputs of the numerical conformal boostrap obtained by varying the number of states and the number of drawing of the sample points. We added a sentence in the last paragraph of page 5 giving more details of how we set precision of the conformal boostrap.

6. and 7. We considered 47 values of l's while taking 4 corrections b_1..b_4
and c_1..c_4. We also considered smaller lattice size with similar
fits and checked that finite size corrections are negligible for L=8192. For each fit, the quality is very high (reduced chi-square much lower than one).
The fit is done with a least-squares Levenberg-Marquardt algorithm. All this is specified in the new version.

8. We had written 2 X 10^4 randomly generated graphs which is meaning
2 X 10^4 independent configurations. At the time of writing the
letter, we had if fact 10^5 configurations for most values of Q
except for some values for which there was only 2 X 10^4
configurations. We have now 10^5 configurations for all values of Q
so 2 X 10^4 is replaced by 10^5 in the text. We also replaced
"randomly generated graphs" by "independent configurations"

Concerning the 10^13, we wrote explicitly N L^2 \simeq 10^13 and it
was clearly stated that L^2 is the number of measurements for each
graph (ie configuration). We do not see how this can mislead a reader
?

9. and 11. We include figures with the difference of the two types of data (Monte Carlo and Bootstrap) with error bars. The fit is obtained by minimizing the differences.

10. We checked in details only for the case q=1 (ie percolation) in the whole complex plane since this is the fastest case. Doing the same for large values of q would have taken a lot of computational resources. We do not believe that we would have gained much more information.

Replies to the second referee report:

1. & 6. The regular plurals 'spectrums' and 'ansatzes' can be found in dictionaries and encyclopedias, alongside the more common but irregular forms 'spectra' and 'ansätze'. Life would be easier, especially for non-Western scientists, if Latin and German plurals were eliminated whenever possible. Following the referee's advice, we however reverted to 'ansätze'.

2. We have replaced the notion of having a gap with the distinction between continuous and discrete spectrums.

3. We replaced the expression 'ground state' with 'leading state' (of a given spectrum), and defined it when it first occurs.

4. We have added a comment on modular invariance at the end of Section 1.

5. We have added a comment on unitarity after (2.4). We see no other reason why conformal dimensions might be real. But the theories we are considering must be non-unitary for generic values of c.

7. We corrected the definition of diagonal and even spin spectrums, by writing them as conditions on primary states.

Other changes:

1. We deleted the assumption |\beta| \leq 1 from (1.1), because we are not sure that it is the right assumption when q is not real. Instead, we added the assumption \frac12 \leq \beta^2 \leq 1 after (1.5).

2. We deleted the distinction between the cases |\beta|^2 \leq \frac12 and |\beta|^2 \geq \frac12 in (2.5), as it was the result of a mistake.

3. We added references [16]-[19] at the end of Section 2.1.

4. We added a sentence at the end of Section 3.3 to point out analogies of our interpretation with previous works .