Experimental mathematics on the magnetic susceptibility of the square lattice Ising model. (English) Zbl 1152.82305
Summary: We calculate very long low- and high-temperature series for the susceptibility of the square lattice Ising model as well as very long series for the five-particle contribution and six-particle contribution . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 CPU hours on computer clusters. The series for (low- and high-temperature regimes), and are now extended to 2000 terms. In addition, for , 10 000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by modulo a prime. A diff-Padé analysis of the 2000 terms series for and 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 and the (as yet unknown) ODE of are given: they are all rational numbers. We find the presence of singularities at for the linear ODE of , and for the ODE of , which are not singularities of the ‘physical’ and , that is to say the series solutions of the ODE’s which are analytic at . Furthermore, analysis of the long series for (and ) combined with the corresponding long series for the full susceptibility yields previously conjectured singularities in some , . The exponents at all these singularities are also seen to be rational numbers. We also present a mechanism of resummation of the logarithmic singularities of the leading to the known power-law critical behaviour occurring in the full , and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility .
|82B20||Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs|
|34M55||Painlevé and other special equations; classification, hierarchies|
|47E05||Ordinary differential operators|
|81Qxx||General mathematical topics and methods in quantum theory|
|32G34||Moduli and deformations for ordinary differential equations|
|82-05||Experimental papers (statistical mechanics)|