论文标题
非线性ODE的对称性：ISING相关性的lambda扩展
Symmetries of non-linear ODEs: lambda extensions of the Ising correlations
论文作者
论文摘要
本文提供了几个插图，说明了Ising模型的两点相关函数的lambda扩展的众多特性，从而对Painlevé类型的非线性ODE进行了一些启示。我们首先表明，出于教学原因，两个点相关函数的因素也存在于两个示例，即C（0,5）和C（2,5）$ν= -k $。然后，我们以逐个典型的方式展示这些lambda扩展的一些令人困惑的属性和结构：对于无限的（代数）$λ$的（代数）值的值，这些功率序列成为代数函数，并且对于lambda的有限（lambda）的有限型型组合，它们的整体效果是零星的，它是零星的杂物（均可依赖的元素）（均已脱颖而出）（在第一和第二种k和E中，对于$λ$的通用值，这些功率系列不是d-finite，它们在代数方面是差异的。对于$λ$的无限数量的其他（理性）值，这些功率系列是全球界限的序列，因此提供了无限数量的全球界限差异代数序列的示例。最后，以两个对角线两点相关函数的产物为例，我们建议在二维ISING模型及其结构及其结构，尤其是相关的lambda扩展上，还有更多的Painlevé类型非线性ODE家族。他们可能减少复杂转化后可能减少的问题，朝卫摩的Sigma形式的ParelevéVI形式仍然是一个极其困难的挑战。
This paper provides several illustrations of the numerous remarkable properties of the lambda-extensions of the two-point correlation functions of the Ising model, sheding some light on the non-linear ODEs of the Painlevé type. We first show that this concept also exists for the factors of the two-point correlation functions focusing, for pedagogical reasons, on two examples namely C(0,5) and C(2,5) at $ν= -k$. We then display, in a learn-by-example approach, some of the puzzling properties and structures of these lambda-extensions: for an infinite set of (algebraic) values of $ λ$ these power series become algebraic functions, and for a finite set of (rational) values of lambda they become D-finite functions, more precisely polynomials (of different degrees) in the complete elliptic integrals of the first and second kind K and E. For generic values of $ λ$ these power series are not D-finite, they are differentially algebraic. For an infinite number of other (rational) values of $ λ$ these power series are globally bounded series, thus providing an example of an infinite number of globally bounded differentially algebraic series. Finally, taking the example of a product of two diagonal two-point correlation functions, we suggest that many more families of non-linear ODEs of the Painlevé type remain to be discovered on the two-dimensional Ising model, as well as their structures, and in particular their associated lambda extensions. The question of their possible reduction, after complicated transformations, to Okamoto sigma forms of Painlevé VI remains an extremely difficult challenge.
