Mathematics Colloquia and Seminars

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The asymmetric simple exclusion process (ASEP) and the Heisenberg-Ising model are two well-studied models in interacting particle systems and one-dimensional statistical mechanics, respectively. The ASEP is a continuous-time Markov process on $\Z$ described by the hopping rate $0 < p < 1$ and $p \neq 1/2$. Particles jump to the right with this probability and to the left at rate $q=1-p$. In the $XYZ$ spin chain, Heisenberg hypothesized the $XYZ$ Hamiltonian to govern the dynamics of spin interactions. Bethe proposed an ansatz to find eigenvectors and eigenvalues of the isotropic case, now known as the Bethe Ansatz. Tracy and Widom used ideas of the Bethe Ansatz to show integrability of the ASEP by discovering explicit formulas for the transition probability of the ASEP for the step initial condition. I will discuss the ASEP in detail, and if time permits, its relationship to the $XYZ$ model.