Advances in Mathematics ( IF 1.7 ) Pub Date : 2021-09-08 , DOI: 10.1016/j.aim.2021.107952 Helen Jenne 1
We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to a problem in Donaldson-Thomas and Pandharipande-Thomas theory. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.
中文翻译:
双二聚体模型的组合学
我们证明平面二部图的三部双二聚体配置的配分函数满足与线性代数的 Desnanot-Jacobi 恒等式相关的递归。近 20 年前 Kuo 建立了二聚体分配函数的类似身份,并应用于随机平铺理论和簇代数理论。这项工作的部分动机是这些领域的应用潜力。此外,我们讨论了 Donaldson-Thomas 和 Pandharipande-Thomas 理论中问题的应用。证明我们的重复性需要对 Kenyon 和 Wilson 的推广工作;具体来说,解除他们的假设,即图的节点是黑色和奇数或白色和偶数。