### 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 150 000 CPU hours on computer clusters. The series for χ (low- and high-temperature regimes), χ^{(5)} and χ^{(6)} are now extended to 2000 terms. In addition, for χ^{(5)}, 10 000 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 w = 1/2 for the linear ODE of χ^{(5)}, and w^{2} = 1/8 for the ODE of χ^{(6)}, which are not singularities of the 'physical' χ^{(5)} and χ^{(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 χ^{(5)} (and χ^{(6)}) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ^{(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 χ^{(n)} leading to the known power-law critical behaviour occurring in the full χ, and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility χ.

Original language | English |
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Article number | 455202 |

Number of pages | 51 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 41 |

Issue number | 45 |

DOIs | |

Publication status | Published - 14 Nov 2008 |

Externally published | Yes |

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## Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*41*(45), [455202]. https://doi.org/10.1088/1751-8113/41/45/455202