Combinatorics of Interacting Particles and Applications

我的研究项目探索位于组合学，表示论，统计物理和可积系统的交叉点的结构。总体目标是通过在它们之间插值的组合对象来探索某些格模型和正交多项式之间的联系。 本案无关该方案的核心是多种群非对称简单排斥过程（ASEP），它是一个一维精确可解的统计模型，被认为是非平衡系统的一个典型例子。ASEP在蛋白质合成中的交通流和翻译中有应用。ASEP还具有丰富的组合结构和与正交多项式（如Askey-Wilson，Macdonald和Koornwinder）的深层联系。这些显着的多项式发挥了至关重要的作用，表示论，代数组合，代数几何，并已被广泛研究，但至今仍有很多重要的悬而未决的问题。 背景麦克唐纳多项式（A型）是一个双参数族，形成了对称函数环的基础。大量的工作一直致力于了解他们的组合由于其固有的复杂性质。在过去的十年中，统计力学和麦克唐纳多项式之间的联系已经在几个方面进行了探索。特别是，它被发现的麦克唐纳多项式的专业化是成比例的多物种ASEP在一个圆上的分区函数。在最近的工作中，Corteel和威廉姆斯公式被发现，这些对象的多线队列。 最近发现，Koornwinder多项式（BC型根系的Macdonald）与开放边界的多物种ASEP的概率有很深的联系。在获得多物种情况下的概率公式方面进展甚微。在过去的工作中，我和我的合作者发现了两种情况下的公式，从而解决了十多年来一直悬而未决的问题。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。目标.主要目标包括：目标1。使用多线队列研究修改的Macdonald多项式[C2]，最近发现的准对称Macdonald多项式[C1]，以及新发现的与TAZRP的连接。目标2.研究了具有开边界的多种群ASEP和Koornwinder多项式的组合数学，目的是：（1）得到该ASEP的概率组合公式;（2）通过ASEP发现Koornwinder多项式的公式。 这些目标的成功将导致重大进展的理解显着的麦克唐纳多项式，并将进一步有重大影响的晶格模型的研究领域的可积系统。注：[C？]参考我的CCV中的出版物

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