Combinatorics of Interacting Particles and Applications

相互作用粒子的组合学及其应用

• 批准号: RGPIN-2021-02568
• 负责人: Mandelshtam, Olya
• 金额: $ 1.31万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=749570
• 关键词: Combinatorics; Interacting; Particles; Applications

My research program explores structures that lie at the intersection of combinatorics, representation theory, statistical physics, and integrable systems. The overarching goal is to explore the connections between certain lattice models and orthogonal polynomials through the combinatorial objects that interpolate between them. Relevance. At the center of this proposal is the multi-species asymmetric simple exclusion process (ASEP), which is a 1D exactly solvable stat mech model that is considered a paradigmatic example of non equilibrium systems. The ASEP has applications to traffic flow and translation in protein synthesis. The ASEP also has rich combinatorial structure and deep connections to orthogonal polynomials such as Askey-Wilson, Macdonald, and Koornwinder. These remarkable polynomials play a crucial role in representation theory, algebraic combinatoric, and algebraic geometry, and have been widely studied, though as yet a great many important open questions remain. Background. Macdonald polynomials (of type A) are a two-parameter family that forms a basis for the ring of symmetric functions. A vast body of work has been devoted to understanding their combinatorics due to their inherently complicated nature. The link between statistical mechanics and Macdonald polynomials has been explored in several contexts over the past decade. In particular, it was found that a specialization of the Macdonald polynomial is proportional to the partition function of the multi-species ASEP on a circle. In recent work, with Corteel and Williams formulas were found for these objects in terms of multiline queues. It was found recently that Koornwinder polynomials (Macdonald of type BC root system) are deeply connected to the probabilities of the multi-species ASEP with open boundaries. Extremely little progress has been made on obtaining formulas for probabilities in the multi-species case. In past work, I and my collaborators discovered formulas in the two-species case, thus solving questions that had been open for over ten years. The connections of Macdonald and Koornwinder polynomials to the ASEP and other lattice models has motivated the objectives of this research program. Objectives. The main goals include: Objective 1. Studying the modified Macdonald polynomials using multiline queues [C2], the recently discovered quasisymmetric Macdonald polynomials [C1], and a newly discovered connection to the TAZRP. Objective 2. Studying the combinatorics of the multi-species ASEP with open boundaries and Koornwinder polynomials in order to (1) obtain combinatorial formulas for probabilities of this ASEP, and (2) discover formulas for Koornwinder polynomials via the ASEP.  The success of these objectives will result in major advancement in the understanding of the remarkable Macdonald polynomials and will furthermore have significant impact on the study of lattice models in the field of integrable systems. Note: [C?] refers to publications in my CCV

我的研究项目探索组合学、表示论、统计物理学和可积系统的交叉点，总体目标是通过插值的组合对象来探索某些晶格模型和正交多项式之间的联系。该提案的中心是多物种不对称简单排除过程（ASEP），它是一个一维精确可解的统计机械模型，被认为是ASEP 还具有丰富的组合结构以及与 Askey-Wilson、Macdonald 和 Koornwinder 等正交多项式的深层联系。麦克唐纳在表示论、代数组合和代数几何中的作用已被广泛研究，但仍有许多重要的悬而未决的问题。多项式（A 型）是一个双参数族，由于其固有的复杂性，大量工作致力于理解它们的组合数学统计力学和麦克唐纳多项式之间的联系。在过去的十年中，人们已经在多个背景下进行了探索，特别是，在最近的工作中发现麦克唐纳多项式的特化与多物种 ASEP 的配分函数成正比。 Corteel 和 Williams 公式是针对这些对象在多行队列方面找到的，最近发现 Koornwinder 多项式（BC 型根系统的 Macdonald）与具有开放边界的多物种 ASEP 的概率密切相关，但进展甚微。在过去的工作中，我和我发现了两种物种情况下的概率公式，从而解决了合作者十多年来所面临的问题。 Macdonald 和 Koornwinder 多项式与 ASEP 和其他晶格模型的联系激发了本研究计划的目标 主要目标包括： 目标 1. 使用多行队列研究修改的 Macdonald 多项式 [C2]，即最近发现的准对称 Macdonald。多项式 [C1]，以及新发现的与 TAZRP 的联系 目标 2. 研究多物种 ASEP 的组合学。开放边界和 Koornwinder 多项式，以便 (1) 获得该 ASEP 概率的组合公式，以及 (2) 通过 ASEP 发现 Koornwinder 多项式的公式。这些目标的成功将导致对卓越的 Macdonald 的理解取得重大进展。多项式，并将进一步对可积系统领域的格模型研究产生重大影响。 注：[C?] 指的是我在 CCV 中的出版物。