Mathematics Colloquia and Seminars

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The partially asymmetric exclusion process (PASEP) is an important and well-studied non-equilibrium model from statistical physics, in which particles hop left and right on a finite one-dimensional lattice with open boundaries. Typically the model has 3 hopping parameters (governing the rates at which particles enter and exit the lattice, and hop left and right). The two-species PASEP is a generalization of the PASEP in which there are two types of particles, one ``heavy'' and one ``light''. Much past work was devoted to finding combinatorial formulas for the steady state probabilities of the original PASEP. In this talk I will present our recent work on the combinatorics of the two-species PASEP. Together with Viennot, we introduced ``rhombic alternative tableaux,'' which are fillings of certain rhombic tilings, and used them to provide combinatorial formulas for the two-species PASEP. We also introduced ``k-rhombic alternative tableaux,'' with analogous combinatorial formulas for a more general k-species PASEP. Finally, in recent joint work with Corteel and Williams, we introduced ``rhombic staircase tableaux'', and provided combinatorial formulas for the more general 5-parameter two-species PASEP. This talk will be accessible to graduate students in combinatorics.