Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

摘要: Focusing on examples associated with holonomic functions, we try to bring new ideas how look at phase transitions, for which the critical manifolds are not points but curves depending a spectral variable, or even fill higher dimensional submanifolds. Lattice statistical mechanics often provides natural (holonomic) framework perform singularity analysis several complex variables that would, in most general mathematical framework, be too complex, simply could defined. In learn-by-example approach, considering Picard–Fuchs systems of two-variables ‘above’ Calabi–Yau ODEs, double hypergeometric series, show D-finite functions actually good finding properly singular manifolds. The found genus-zero curves. We then analyze algebraic varieties quite important lattice mechanics, n-fold integrals χ(n), corresponding n-particle decomposition magnetic susceptibility anisotropic square Ising model. this case, revisit set so-called Nickelian singularities turns out two-parameter family elliptic find first non-Nickelian χ(3) and χ(4), also rational underline fact these depend anisotropy model, or, equivalently, they parameter This has consequences physical nature χ(n)s appear highly composite objects. address, from birational viewpoint, emergence families curves, such problems. address question non-holonomic discussion accumulation full χ.This article is part ‘Lattice models integrability’, special issue Journal Physics A: Mathematical Theoretical honour F Y Wu's 80th birthday.