六顶点模型、圈模型与铺砌模型：可积性与组合学

可积性与组合学

Integrability and combinatorics

本文探讨了将可积系统方法应用于组合学问题，并以交错符号矩阵枚举和对称多项式理论为例进行了阐述。

随机张量模型：组合学、几何学、量子引力和可积性

Random Tensor models: Combinatorics, Geometry, Quantum Gravity and Integrability

在这篇论文中，我们探讨了随机张量模型的不同方面。这些模型已经被引入来模仿随机矩阵模型在物理学、数学和组合学中令人难以置信的成功。在对随机矩阵模型的几个方面进行了简短的介绍并回顾了称为群场论的物理动机之后，我们开始探索随机张量模型的世界及其与几何，量子引力和组合学的关系。我们首先以自然的方式定义这些模型，并讨论它们的几何和组合。在这些初步探索之后，我们开始将随机矩阵方法推广到随机张量，以便至少在某些特定情况下描述随机张量模型的数学和物理性质。

图组合学通过随机游动和二聚体堆的可积性

Integrability of graph combinatorics via random walks and heaps of dimers

研究了内顶点为偶数价的可平面图的测地线距离依赖性的离散非线性方程的可积性。这个方程是由图和开花树之间的双射得出的，并且用随机游动的生成函数来表示。我们明确地构造了这个方程的守恒量的无限集，也涉及到随机游动生成函数的适当组合。它们守恒的证明，即它们最终与测地距离无关，依赖于随机游动和二聚体堆之间的联系。对于具有固定数目外腿的图，用生成函数确定了守恒量的值。还讨论了守恒量集的替代等价选择，并给出了一些应用。

Twenty Vertex model and domino tilings of the Aztec triangle

We show that the number of configurations of the 20 Vertex model on certain domains with domain wall type boundary conditions is equal to the number of domino tilings of Aztec-like triangles, proving a conjecture from [P. Di Francesco and E. Guitter, Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings, Elec. Jour. of Combinatorics 27 (2020), no. 2, P2.13]. The result is based on the integrability of the 20 Vertex model and uses a connection to the U-turn boundary 6 Vertex model to re-express the number of 20 Vertex configurations as a simple determinant, which is then related to…

Twenty-Vertex model with domain wall boundaries and domino tilings

We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phas…

Integrability from supersymmetric duality: a short review

Integrable models of statistical mechanics play a prominent role in modern mathematical physics, especially in conformal field theory, knot theory, combinatorics, topology, etc. In this brief review, we discuss a program of constructing integrable lattice spin models with the nearest neighbor interaction using methods inspired by the supersymmetric gauge theory computations, called gauge/YBE correspondence. After a brief introduction to the topic, we review some recent examples of this correspondence and the role of special functions and symmetries. Finally, we discuss future directions of res…