1999/11/04 by Peter Kleban
Computer Science · Mathematics · Physics and Astronomy · #Boundary (topology) #Boundary conformal field theory #Computer science #Conformal field theory #Conformal map #Conformal symmetry #Free boundary problem #Geometry #Homogeneous space #Mathematical analysis #Mathematics #Modular design #Modular form #Modular invariance #Percolation (cognitive psychology) #Physics #Product (mathematics) #Pure mathematics #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Theoretical physics #Topological and Geometric Data Analysis #cond-mat.stat-mech #hep-th #math-ph #math.MP #math.NT
paper · pdf · doi:10.1016/s0378-4371(00)00035-2
published as Physica A281:242-251,2000 · 12 pages,LaTeX, uses file elsart.cls from Elsevier. Submitted to Physica (proceedings of StatPhys-Taiwan 1999)
arxiv created 1999/11/04 · openalex publication_date 2000/06/01 · arxiv updated 2011/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossing probabilities transform as a vector modular form, one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form. We include consideration of the interplay between conformal and modular invariance that arises.