Conformal invariance in three dimensional percolation

Abstract

The aim of the paper is to present numerical results supporting the presence of conformal invariance in three dimensional statistical mechanics models at criticality and to elucidate the geometric aspects of universality. As a case study we study three dimensional percolation at criticality in bounded domains. Both on discrete and continuous models of critical percolation, we test by numerical experiments the invariance of quantities in finite domains under conformal transformations focusing on crossing probabilities. Our results show clear evidence of the onset of conformal invariance in finite realizations especially for the continuum percolation models. Finally we propose a simple analytical function approximating the crossing probability among two spherical caps on the surface of a sphere and confront it with the numerical results.

10 pages, 7 figures, references added

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• A structural test for the conformal invariance of the critical 3d Ising model
• Observation of non-scalar and logarithmic correlations in 2D and 3D percolation
• Crossover exponents, fractal dimensions and logarithms in Landau-Potts field theories
• Geometry of bounded critical phenomena
• Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem
• Critical $1$- and $2$-point spin correlations for the $O(2)$ model in $3d$ bounded domains