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From combinatorial maps to correlation functions in loop models

by Linnea Grans-Samuelsson, Jesper Lykke Jacobsen, Rongvoram Nivesvivat, Sylvain Ribault, Hubert Saleur

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Submission summary

Authors (as registered SciPost users): Linnea Grans-Samuelsson · Rongvoram Nivesvivat · Sylvain Ribault
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-09-06
Date submitted: 2023-08-25 11:00
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


In two-dimensional statistical physics, correlation functions of the O(N) and Potts models may be written as sums over configurations of non-intersecting loops. We define sums associated to a large class of combinatorial maps (also known as ribbon graphs). We allow disconnected maps, but not maps that include monogons. Given a map with n vertices, we obtain a function of the moduli of the corresponding punctured Riemann surface. Due to the map's combinatorial (rather than topological) nature, that function is single-valued, and we call it an n-point correlation function. We conjecture that in the critical limit, such functions form a basis of solutions of certain conformal bootstrap equations. They include all correlation functions of the O(N) and Potts models, and correlation functions that do not belong to any known model. We test the conjecture by counting solutions of crossing symmetry for four-point functions on the sphere.

Published as SciPost Phys. 15, 147 (2023)

Author comments upon resubmission

We have made clarifications on the points that were raised in Bernard Nienhuis' report.

List of changes

- On page 5, in order to clarify the origin of combinatorial maps, we have rewritten the first full paragraph after Figure (1.4). ("The main idea...")

- On page 5, we have rewritten the last paragraph before the Highlights, in order to improve the discussion of the interpretation of correlation functions in terms of local fields.

- On page 18, we have enumerated the parameters of our sum over loop configurations, in order to clarify which parameters are discrete or continuous. In particular, the angles are parameters of the loops themselves and not of the sum. The angular momentums are discrete, and the only continuous parameters of the sum are the weights of closed loops.

- On page 21, we have rewritten the last paragraph of Section 3 in order to better conclude the comparison with the Coulomb gas approach.

Submission & Refereeing History

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Resubmission 2302.08168v2 on 25 August 2023

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