Tomas Berggren: Double integral formulas for periodic multi-level particle systems

Abstract:
Recently, important progress has been made on the asymptotic behavior of certain periodic multi-level particle systems, such as the periodic weightings of domino tilings of the Aztec diamond and periodic weightings of lozenge tilings of a hexagon. In a general setting Duits and Kuijlaars recently proved a double integral formula for the kernel of the point process in terms of matrix valued orthogonal polynomials. I will discuss a simplification in the case of certain particle systems with infinite levels. In those cases, the integrand in the double integral formula can be expressed in terms of a matrix Wiener-Hopf factorization for an associated weight function. If there is time, I will mention the solution of the factorization problem for the $ 2\times 2 $ periodic weighting of lozenge tilings in a hexagon.