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

We calculate very long low- and high-temperature series for the susceptibility chi of the square lattice Ising model as well as very long series for the five-particle contribution chi((5)) and six-particle contribution chi((6)). 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 chi (low- and high-temperature regimes), chi((5)) and chi((6)) are now extended to 2000 terms. In addition, for chi((5)), 10 000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by chi((5)) modulo a prime. A diff-Pade analysis of the 2000 terms series for chi((5)) and chi((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 chi((5)) and the (as yet unknown) ODE of chi((6)) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of chi((5)), and w(2) = 1/8 for the ODE of chi((6)), which are not singularities of the 'physical' chi((5)) and chi((6)), that is to say the series solutions of the ODE's which are analytic at w = 0. Furthermore, analysis of the long series for chi((5)) (and chi((6))) combined with the corresponding long series for the full susceptibility. yields previously conjectured singularities in some chi((n)), n >= 7. 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 chi((n)) 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 chi.