We report on the first exact solution of the Kardar-Parisi-Zhang (KPZ) equation in one dimension, with an initial condition which physically corresponds to the motion of a macroscopically curved height profile. The solution provides a determinantal formula for the probability distribution function of the height h(x,t) for all t>0. In particular, we show that for large t, on the scale t(1/3), the statistics is given by the Tracy-Widom distribution, known already from the Gaussian unitary ensemble of random matrix theory. Our solution confirms that the KPZ equation describes the interface motion in the regime of weak driving force. Within this regime the KPZ equation details how the long time asymptotics is approached.

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 prove duality relations for two interacting particle systems: the q-deformed totally asymmetric simple exclusion process (q-TASEP) and the asymmetric simple exclusion process (ASEP). Expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents, respectively. Duality implies that they solve systems of ODEs. These systems are integrable and for particular step and half-stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems' solutions, and hence also the moments.We form Laplace transform-like generating functions of these moments and via residue calculus we compute two different types of Fredholm determinant formulas for such generating functions. For ASEP, the first type of formula is new and readily lends itself to asymptotic analysis (as necessary to reprove GUE Tracy-Widom distribution fluctuations for ASEP), while the second type of formula is recognizable as closely related to Tracy and Widom's ASEP formula Both q-TASEP and ASEP have limit transitions to the free energy of the continuum directed polymer, the logarithm of the solution of the stochastic heat equation or the Hopf-Cole solution to the

Stochastic motion of a point – known as Brownian motion – has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.

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 *