In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.
Grassmannian flows and applications to non-commutative non-local and local integrable systems
本文给出了一类具有局部和非局部非线性的矩阵值非线性偏微分方程的线性化方法。事实上，我们一般化的线性化过程最初开发的Pöppe基于解决相应的基本线性偏微分方程，以产生一个进化的汉克尔算子的“散射数据”，然后解决一个线性Fredholm方程类似的Marchenko方程，以产生进化的解决方案的非线性偏微分系统。我们的推广涉及膨胀的散射数据的基本线性偏微分系统，以纳入相应的伴随，逆时间或逆时空数据，它也允许汉克尔算子与矩阵值内核。用这种方法，我们展示了如何线性化矩阵非线性薛定谔和修改Korteweg de Vries方程，以及这些系统的非局部逆时和/或逆时空版本。此外，我们制定了一个统一的线性化程序，将所有这些系统作为特殊情况。更进一步，我们证明了所有这样的系统是例子Fredholm格拉斯曼流。
Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation
Instituto de Ciencias Matemáticas（数学科学研究所）；Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid（马德里康普顿斯大学物理学院理论物理系）
