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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 $\chi$ of the square lattice Ising model as well as very long series for the five-particle contribution ${\chi }^{\left(5\right)}$ and six-particle contribution ${\chi }^{\left(6\right)}$. 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), ${\chi }^{\left(5\right)}$ and ${\chi }^{\left(6\right)}$ are now extended to 2000 terms. In addition, for ${\chi }^{\left(5\right)}$, 10 000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by ${\chi }^{\left(5\right)}$ modulo a prime. A diff-Padé analysis of the 2000 terms series for ${\chi }^{\left(5\right)}$ and ${\chi }^{\left(6\right)}$ 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 ${\chi }^{\left(5\right)}$ and the (as yet unknown) ODE of ${\chi }^{\left(6\right)}$ are given: they are all rational numbers. We find the presence of singularities at $w=1/2$ for the linear ODE of ${\chi }^{\left(5\right)}$, and ${w}^{2}=1/8$ for the ODE of ${\chi }^{\left(6\right)}$, which are not singularities of the ‘physical’ ${\chi }^{\left(5\right)}$ and ${\chi }^{\left(6\right)}$, that is to say the series solutions of the ODE’s which are analytic at $w=0$. Furthermore, analysis of the long series for ${\chi }^{\left(5\right)}$ (and ${\chi }^{\left(6\right)}$) combined with the corresponding long series for the full susceptibility $\chi$ yields previously conjectured singularities in some ${\chi }^{\left(n\right)}$, $n\ge 7$. 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 ${\chi }^{\left(n\right)}$ leading to the known power-law critical behaviour occurring in the full $\chi$, and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility $\chi$.
##### MSC:
 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)