论文标题
与SOS模型的Gibbs测量有关的非线性二阶差异方程
A non-linear second-order difference equation related to Gibbs measures of a SOS model
论文作者
论文摘要
对于具有外场的SOS（固体固体）模型，来自Cayley树上所有整数的自旋值（梯度）吉布斯度量对应于边界定律（在Cayley树的顶点上定义的无限维矢量函数），以满足非线性功能方程。最近，发现了对方程式的一些翻译不变和高度周期性（不差）溶液。在这里，我们的目的是为SOS模型找到非高度周期性和非标准性边界定律。通过这样的解决方案，人们可以构建非概率的吉布斯度量。我们发现明确的几个不合同的边界定律。此外，我们将问题减少到解决非线性的二阶差异方程。我们对差异方程进行分析和数值分析。
For the SOS (solid-on-solid) model with an external field and with spin values from the set of all integers on a Cayley tree each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a non-linear functional equation. Recently some translation-invariant and height-periodic (non-normalisable) solutions to the equation are found. Here our aim is to find non-height-periodic and non-normalisable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalisable boundary laws. Moreover, we reduce the problem to solving of a non-linear, second-order difference equation. We give analytic and numerical analysis of the difference equation.
