Abstract
AbstractWe study the multitime distribution in a discrete polynuclear growth model or, equivalently, in directed last‐passage percolation with geometric weights. A formula for the joint multitime distribution function is derived in the discrete setting. It takes the form of a multiple contour integral of a block Fredholm determinant. The asymptotic multitime distribution is then computed by taking the appropriate KPZ‐scaling limit of this formula. This distribution is expected to be universal for models in the Kardar‐Parisi‐Zhang universality class. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
Highlights
The function grows out from the corner of the first quadrant along up-right directions, so it is a model of local random growth
A multi-time distribution function is derived in [1] for the related continuous time TASEP in a periodic setting, and the asymptotic limit is computed in the relaxation time-scale, when the TASEP is affected by the finite geometry
It should be possible to study the limit of Poissonian last-passage percolation (Poissonized Plancherel) (q → 0) from our formula in Theorem 2, but this would entail taking a limit to an infinite Fredholm determinant before the large time asymptotics are computed
Summary
Decorate points of Z2 with independent and identically distributed random weights ω(m, n) that are non-negative. A multi-time distribution function is derived in [1] for the related continuous time TASEP in a periodic setting, and the asymptotic limit is computed in the relaxation time-scale, when the TASEP is affected by the finite geometry. Correlation function of the two-time distribution has been studied in [2, 17] The distribution of this growth model under a different asymptotic scaling, related to the slow decorrelation phenomenon, has been explored in [4, 8, 16, 20]. We expect the limiting multi-time formula in Theorem 1 to be universal within a large class of models. It should be possible to study the limit of Poissonian last-passage percolation (Poissonized Plancherel) (q → 0) from our formula in Theorem 2, but this would entail taking a limit to an infinite Fredholm determinant before the large time asymptotics are computed
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