TY - JOUR

T1 - Conformal covariance of connection probabilities and fields in 2D critical percolation

AU - Camia, Federico

N1 - Publisher Copyright:
© 2023 Wiley Periodicals LLC.

PY - 2024/3

Y1 - 2024/3

N2 - Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well-defined scaling limit for every (Formula presented.). Moreover, the limiting functions (Formula presented.) transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and (Formula presented.) for any (Formula presented.). This implies that (Formula presented.) and (Formula presented.), for some constants C2 and C3. We also define a site-diluted spin model whose n-point correlation functions Cn can be expressed in terms of percolation connection probabilities and, as a consequence, have a well-defined scaling limit with the same properties as the functions (Formula presented.). In particular, (Formula presented.). We prove that the magnetization field associated with this spin model has a well-defined scaling limit in an appropriate space of distributions. The limiting field transforms covariantly under Möbius transformations with exponent (scaling dimension) 5/48. A heuristic analysis of the four-point function of the magnetization field suggests the presence of an additional conformal field of scaling dimension 5/4, which counts the number of percolation four-arm events and can be identified with the so-called “four-leg operator” of conformal field theory.

AB - Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well-defined scaling limit for every (Formula presented.). Moreover, the limiting functions (Formula presented.) transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and (Formula presented.) for any (Formula presented.). This implies that (Formula presented.) and (Formula presented.), for some constants C2 and C3. We also define a site-diluted spin model whose n-point correlation functions Cn can be expressed in terms of percolation connection probabilities and, as a consequence, have a well-defined scaling limit with the same properties as the functions (Formula presented.). In particular, (Formula presented.). We prove that the magnetization field associated with this spin model has a well-defined scaling limit in an appropriate space of distributions. The limiting field transforms covariantly under Möbius transformations with exponent (scaling dimension) 5/48. A heuristic analysis of the four-point function of the magnetization field suggests the presence of an additional conformal field of scaling dimension 5/4, which counts the number of percolation four-arm events and can be identified with the so-called “four-leg operator” of conformal field theory.

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U2 - 10.1002/cpa.22171

DO - 10.1002/cpa.22171

M3 - Article

AN - SCOPUS:85173524426

SN - 0010-3640

VL - 77

SP - 2138

EP - 2176

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 3

ER -