Universal Distributions for Growth Processes in1+1Dimensions and Random Matrices

Abstract

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...

Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

“…This type of scaling is characteristic for one-dimensional surface-growth models of the KPZ universality class ( [8]). For our initial state the integrated current J(x, t) gives the number of particles being at sites k > x at time t. It can be shown using theorem 1.6 of [6] that…”

“…Our starting point is the rhs. of (8). The summation over (k 1 , k 2 , · · · k N ) can be done in N steps for which we choose the following sequence:…”

“…Despite their greatly reduced complexity they capture various generic nonequilibrium phenomena such as the occurrence of shocks and particle condensation [3,4,5]. A major breakthrough in the exact calculation of universal properties came with the realization that various ensembles of random matrices occur in the study of current fluctuations and related quantities in the totally asymmetric simple exclusion process (TASEP) [6,7] and also in the polynuclear growth model [8,9,10,11]. Here we use random matrix theory combined with the Bethe ansatz to study a process introduced by Priezzhev [12] which describes the stochastic fragmentation and reassembly of diffusing particle clusters in discrete time.…”

Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

“…We have recovered the dynamical exponent z = 3/2, which comes hardly as a surprise, since it is well established through theoretical arguments and Monte-Carlo simulations. Novel is the computation of scaling functions along with the insight that they depend on the geometry of the growth process [37]. If there is a non-zero curvature on the macroscopic scale, the height fluctuations are governed by the GUE TracyWidom distribution.…”

A KPZ Cocktail-Shaken, not Stirred...

“…The 2 nd KPZ Revolution, commenced in 2000 with the spectacular physical insights of Prähofer and Spohn [48] on polynuclear growth (PNG), and the creative mathematical efforts of Johansson [49] on the single-step (SS) model, both making explicit connections to the related problem of directed polymers in random media (DPRM). These works established that earlier numerically observed 1+1 KPZ height PDFs were simply zero-mean, unit-variance versions of the universal Tracy-Widom (TW) limit distributions [50], well-known from random matrix theory-RMT.…”