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Finite-size corrections for universal boundary entropy in bond percolation

by Jan de Gier, Jesper Lykke Jacobsen, Anita Ponsaing

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Submission summary

Authors (as registered SciPost users): Jesper Lykke Jacobsen · Anita Ponsaing · Jan de Gier
Submission information
Preprint Link:  (pdf)
Date accepted: 2016-12-06
Date submitted: 2016-11-24 01:00
Submitted by: Ponsaing, Anita
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Combinatorics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We compute the boundary entropy for bond percolation on the square lattice in the presence of a boundary loop weight, and prove explicit and exact expressions on a strip and on a cylinder of size $L$. For the cylinder we provide a rigorous asymptotic analysis which allows for the computation of finite-size corrections to arbitrary order. For the strip we provide exact expressions that have been verified using high-precision numerical analysis. Our rigorous and exact results corroborate an argument based on conformal field theory, in particular concerning universal logarithmic corrections for the case of the strip due to the presence of corners in the geometry. We furthermore observe a crossover at a special value of the boundary loop weight.

Author comments upon resubmission

Dear Editor,

We enclose our revised manuscript and give detailed responses to the referee reports below.

Reply to referee 1:

Thank you for your kind comments. We have included the proposed reference and fixed up the typos.

Reply to referee 2:

Thank you for your detailed and valuable report. As several results are scattered in existing literature, we made a conscious choice to try and unify these by explicitly indicating links to other papers and explain differences in notation.

1 To avoid a precise explanation which would interrupt the flow we now omit the bra-ket notation in (10) and simply define the norm as the sum over components.

2 Fixed

3 Added, thanks for this suggestion

4 We have included a sign factor $\varepsilon_{n,x}$ in (20) and (24)

5 Fixed

6 Fixed in both places

7 Fixed -- come to think of it $a_1$ should always equal 1, so $j$ is in ${2,...,n}$.

8 Fixed

9 Thank you for the simpler expression, which has been included... however we believe that our original expression was correct and equal to this one.

10 Interesting question which had crossed our minds as well. We suspect that there should be a differential equation for the reflecting case, but preliminary attempts to find one indicate that it will not be as simple as the hypergeometric equation for the periodic case. We leave this for future study.

11 Yes. Thanks for catching this. A sentence has been added to Proposition 1.

12 Fixed

13 Fixed. Thanks for catching this!

14 This is an astute observation and a good question to which we currently don't have a clear answer.

15 We have added a whole new section (4.3) discussing the CFT argument for the even periodic case.

16 The case of distinct connectivities is indeed very interesting and we believe an extension to this case is possible. Like the odd periodic case we refer this to future study.

17 Yes, and yes. Fixed with an extra sentence below what is now equation (118).

List of changes

- Sec 1, para 5: Added sentence about the entanglement entropy of the stochastic raise-and-peel model
- Eq (10): Removed bra-ket notation (Ref 2 point 1)
- Eq (10): Capitalised $\Psi$, cf Ref 1 point 1
- Eq (12): Changed, cf Ref 2 points 2&3
- Sec 1.3, para after eq (12): Added "precisely"
- Eq (20): Added sign factor, cf Ref 2 point 4
- Prop 1: Added caveat, cf Ref 2 point 11
- Sec 2.2.1, final para: Added reference, cf Ref 1 requested change
- Eq (24): Added sign factor, cf Ref 2 point 4
- Sec 2.2.2, final para: Added `be', cf Ref 1 point 2
- Proof of Prop 3, para 1: changed $x^{n-1}$ to $x^n$, cf Ref 2 point 5
- Proof of Prop 3, para 1: added $\tau$, cf Ref 2 point 6
- Proof of Prop 3, para 2: changed specification of $a_j$, cf Ref 2 point 7
- Proof of Prop 3, final para: added $x^n$, cf Ref 2 point 8
- Proof of Prop 3, final para: added $\tau$, cf Ref 2 point 6
- Eq (45): changed prefactor, cf Ref 2 point 9
- Sec 4.1, before eq (46): Added reference
- Eq (48): changed $\mathcal{O}(n^{-3})$ to $\mathcal{O}(n^{-2})$, cf Ref 2 point 12
- Eq (52): changed $\log(x)$ to $\log|x|$, cf Ref 2 point 13
- Section 4.3 has been added, explaining conformal reasoning behind the results for the even periodic case. Cf Ref 2 point 15
- Sec 5, para 2: Added some words about precision of numerical data
- App D: Incorporated information for $x\leq-1$, cf Ref 2 point 17

Published as SciPost Phys. 1, 012 (2016)

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