Combinatorics of Interacting Particles and Applications

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The proposed project explores structures that lie at the intersection of combinatorics, representation theory, statistical physics, and integrable systems. The proposed research also has many applications: the key object of this work, the asymmetric simple exclusion process (ASEP), has been studied as a model for traffic flow, as well as in biological processes such as translation in protein synthesis and kinetic biopolymerization. In the longer term, the proposed project has the potential to have an important impact on these applications. Some of the topics of the proposed project are accessible to younger researchers, with several projects intended for work with students. Workshop organization, outreach, and activities aimed at improving climate in STEM are also planned. This project is jointly funded by the Combinatorics program and the Established Program to Stimulate Competitive Research (EPSCoR).The principal goal of this project is to study the remarkable connection between particle models such as ASEP and orthogonal polynomials. The first part of this work concerns Macdonald polynomials of type A as an application of the combinatorics of the ASEP on a circle, through recently discovered formulas that use multiline queues to connect probabilities of the ASEP to symmetric and nonsymmetric Macdonald polynomials. It is proposed to use those new formulas to explore Schur positivity of modified Macdonald polynomials: one line of attack is to study the quasisymmetric Macdonald polynomial. The second part of this project concerns the extension of the results obtained for Macdonald polynomials of type A to the type BC setting to discover formulas both for Koornwinder polynomials and probabilities of the multispecies ASEP with open boundaries. Thus far such formulas exist only for the two-species ASEP case corresponding to a special case of Koornwinder polynomials, studied in earlier works. It is proposed to merge those results with the multiline queue approach to find a new object that incorporates boundary conditions and admits any number of species.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.