Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Abstract

Abstract. We calculate very long low- and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ (5) and six-particle contribution χ (6). These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. The series for χ (low- and high-temperature regime), χ (5) and χ (6) are now extended to 2000 terms. In addition, for χ (5) , 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime. A diff-Padé analysis of the 2000 terms series for χ (5) and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of “additional ” singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at