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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 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) , n7. 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.
MSC:
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
34M55Painlevé and other special equations; classification, hierarchies
47E05Ordinary differential operators
81QxxGeneral mathematical topics and methods in quantum theory
32G34Moduli and deformations for ordinary differential equations
82-05Experimental papers (statistical mechanics)