Spatial correlations of the 1D KPZ surface on a flat substrate

Abstract

We study the spatial correlations of the one-dimensional KPZ surface for the flat initial condition. It is shown that the multi-point joint distribution for the height is given by a Fredholm determinant, with its kernel in the scaling limit explicitly obtained. This may also describe the dynamics of the largest eigenvalue in the GOE Dyson's Brownian motion model. Our analysis is based on a reformulation of the determinantal Green's function for the totally ASEP in terms of a vicious walk problem.Surface growth has been an important subject of physics both from practical and fundamental aspects. While a good control of it is crucial in recent atom-scale technology, a rich variety of interesting surface pattern has attracted much attention of theoretical studies [1,2]. It is also important from the point of view of noneqilibrium statistical mechanics.It is in general difficult to obtain detailed information about the properties of surface by analytical methods. However in one spatial dimension some surface growth models are known to be exactly solvable. They are very special in many respects, but give us a lot of insight to understand the properties of surfaces in nature. The Kardar-Parisi-Zhang (KPZ) equation, introduced in [3], is one of the minimal models in the theory of surface growth which have both nonlinear and noise effects. A lot of models were shown to belong to the same universality class as the KPZ equation, i.e., the KPZ universality class [4]. But the analysis had been mainly restricted to the exponents for some time.A next breakthrough comes from an observation that some surface growth models, in particular the polynuclear growth (PNG) model, are related to combinatorial problem of Young tableaux [5]. In [6,7], the height fluctuation of the surface in the KPZ universality *

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