We develop a scaling theory for KPZ growth in one dimension by a detailed study of the polynuclear growth (PNG) model. In particular, we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape. These distribution functions are computed using the partition function of Gaussian random matrices in a cosine potential.PACS numbers: 64.60. Ht, 68.35.Ct, 81.10.Aj Growth processes lead to a rich variety of macroscopic patterns and shapes [1]. As has been recognized for some time, growth may also give rise to intriguing statistical fluctuations comparable to thermal fluctuations at a critical point. One of the most prominent examples is the Kardar-Parisi-Zhang (KPZ) universality class [2]. In essence one models a stable phase which grows into an unstable phase through aggregation, as for example in Eden type models where perimeter sites of a given cluster are filled up randomly. In real materials, mere aggregation is often too simplistic an assumption and one would have to take other dynamical modes, such as surface diffusion, at the stable/unstable interface into account [3]. In our letter we remain within the KPZ class.From the beginning there has been evidence that in one spatial dimension KPZ growth processes are linked to exactly soluble models of two-dimensional statistical mechanics. Kardar [4] mapped growth to the directed polymer problem. The replica trick then yields the Bose gas with attractive δ-interaction which in one dimension can be solved through the Bethe ansatz [5]. In [6], considerably generalized in [7], for a particular discrete growth model the statistical weights for the local slopes were mapped onto the six vertex model. To solve the six vertex model one diagonalizes the transfer matrix, again, through the Bethe ansatz, which also allows for a study of finite size scaling [8]. Unfortunately none of these methods go beyond what corresponds to the free energy in the six vertex model and the associated dynamical scaling exponent β = 1/3.In this letter we point out that within the KPZ universality class the polynuclear growth (PNG) model plays a distinguished role: it maps onto random permutations, the height being the length of the longest increasing subsequence of such a permutation, and thereby onto Gaussian random matrices [9,10]. We use these mappings to obtain an analytic expression for certain scaling distributions, which then leads to an understanding of how the self-similar height fluctuations depend on the initial conditions and to a more refined scaling theory for KPZ growth.PNG is a simplified model for layer by layer growth [1]. One starts with a perfectly flat crystal in contact with its super-saturated vapor. Once in a while a supercritical nucleus is formed, which then spreads laterally by further attachment of particles at its perimeter sites. Such islands coalesce if they are in the same layer and further islands may be nucleated upon already existing ones. The PNG model ignores the lateral lattice structure and assumes that the...

We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us [34] and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.

We determine the stationary two-point correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann-Hilbert problem related to the Painlevé II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode coupling approximation.

Abstract:The time-integrated current of the TASEP has non-Gaussian fluctuations of order t 1/3 . The recently discovered connection to random matrices and the Painlevé II RiemannHilbert problem provides a technique through which we obtain the probability distribution of the current fluctuations, in particular their dependence on initial conditions, and the stationary two-point function. Some open problems are explained.

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