正方格点Ising模型$\tildeχ^{(5)}$ ODE的精确算法
Square lattice Ising model $\tildeχ^{(5)}$ ODE in exact arithmetic
摘要：
用精确算法得到了非齐次方程$L_{24}(Φ^{(5)}) = E^{(5)}$的24阶线性微分算子$L_{24}$和右边的$E^{(5)}$，其中$Φ^{(5)} =\tildeχ^{(5)}-\tildeχ^{(3)}/2+\tildeχ^{(1)}/120$是$n$-粒子对正方格子伊辛模型磁化率贡献的线性组合.在Bostan等人（J. Phys. A：Math. Theor. {\bf 42}，275209（2009））算子$L_{24}$（模素数）被证明可以分解成$L_{12}^{(\rm left)} \cdot L_{12}^{(\rm right)}$;在这里，我们证明了12阶算子$L_{12}^{(\rm left)}$的进一步分解是不可能的。我们使用精确的常微分方程获得的行为$\tildeχ^{(5)}$在铁磁临界点，并获得有限数量的解析延拓的$\tildeχ^{(5)}$超出其高温系列定义的主盘。与Boukraa等人（J. Phys. A：Math. Theor. {\bf 41} 455202（2008）），我们发现在无限多个分支上，$\tildeχ^{(5)}$在$w=1/2$处是奇异的。
英文摘要：
We obtain in exact arithmetic the order 24 linear differential operator $L_{24}$ and right hand side $E^{(5)}$ of the inhomogeneous equation$L_{24}(Φ^{(5)}) = E^{(5)}$, where $Φ^{(5)} =\tildeχ^{(5)}-\tildeχ^{(3)}/2+\tildeχ^{(1)}/120$ is a linear combination of $n$-particle contributions to the susceptibility of the square lattice Ising model. In Bostan, et al. (J. Phys. A: Math. Theor. {\bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(\rm left)} \cdot L_{12}^{(\rm right)}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(\rm left)}$ is possible. We use the exact ODE to obtain the behaviour of $\tildeχ^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $\tildeχ^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {\bf 41} 455202 (2008)), we find that $\tildeχ^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.
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