KPZ普适类的非阿贝尔Hirota-Miwa方程
Non-Abelian Hirota-Miwa Equations for the KPZ Universality Class
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中文总结 AI 辅助
本研究提出代数框架，推导菱形方程，结合Darboux变换压缩Fredholm数据，将KPZ普适类18个模型的非线性方程验证简化为少量线性条件，证明菱形方程与非阿贝尔Hirota-Miwa系统规范等价。
中文摘要 AI 辅助
本研究引入了一个代数框架，该框架为KPZ普适类可精确求解领域内跨四个标度 regime 的18个模型，生成显式的闭合矩阵微分-差分方程。通过将Fredholm行列式数据组织为有向格图上的超定线性问题，我们推导得到了一个名为菱形方程的相容性系统。从Fredholm核的平移结构中提取的基本种子数据，为该系统提供了简单解。随后，我们构造了一个Darboux变换，以将无限维Fredholm数据压缩为有限维矩阵可观测量。我们证明该 dressing 过程保持了菱形方程；因此，所得矩阵可观测量服从与初始种子数据相同的非线性结构。由此，验证任意特定模型的闭合非线性方程，简化为对其核数据检查少量线性条件。在标量约化下，该框架为Fredholm行列式生成变系数Hirota-Miwa方程，作为特例恢复了作者早期工作中的单点双线性方程。为提供必要的种子数据，带可允许传播子的乘积图构造在全离散 regime 中构建多点数据，而多项式商代数中的欧几里得除法处理顶点模型和聚合物模型。最后，我们证明菱形方程是非阿贝尔Hirota-Miwa系统的规范等价重参数化，后者是经典可积性理论中的核心系统。
英文摘要
This work introduces an algebraic framework yielding explicit, closed matrix differential-difference equations for eighteen models in the exactly solvable sector of the KPZ universality class across four scaling regimes. By organizing Fredholm determinant data into an overdetermined linear problem on a directed lattice graph, we derive a compatibility system termed the diamond equations. Elementary seed data extracted from the shift structure of the Fredholm kernel provides simple solutions to this system. We then construct a Darboux transformation to compress the infinite-dimensional Fredholm data into a finite-dimensional matrix observable. We show this dressing procedure preserves the diamond equations; consequently the resulting matrix observable obeys the same nonlinear structure as the initial seed data. Verifying a closed nonlinear equation for any specific model thus reduces to checking a handful of linear conditions on its kernel data. Under a scalar reduction, the framework produces variable-coefficient Hirota-Miwa equations for Fredholm determinants, recovering the one-point bilinear equations of the author's earlier work as specializations. To supply the necessary seed data, a product graph construction with admissible propagators builds multipoint data in the fully discrete regime, while Euclidean division in a polynomial quotient algebra handles vertex and polymer models. Finally, we demonstrate the diamond equations are a gauge-equivalent reparametrization of the non-abelian Hirota-Miwa system, a central system in classical integrability theory.
