We consider the solution of the stochastic heat equationwhose logarithm, with appropriate normalization, is the free energy of the continuum directed polymer, or the Hopf-Cole solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions, the crossover distributions, which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time).The proof is via a rigorous steepest-descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion process with antishock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes.

We introduce a new disorder regime for directed polymers in dimension 1 + 1 that sits between the weak and strong disorder regimes. We call it the intermediate disorder regime. It is accessed by scaling the inverse temperature parameter β to zero as the polymer length n tends to infinity. The natural choice of scaling is βn := βn −1/4 . We show that the polymer measure under this scaling has previously unseen behavior. While the fluctuation exponents of the polymer endpoint and the log partition function are identical to those for simple random walk (ζ = 1/2, χ = 0), the fluctuations themselves are different. These fluctuations are still influenced by the random environment, and there is no self-averaging of the polymer measure. In particular, the random distribution of the polymer endpoint converges in law (under a diffusive scaling of space) to a random absolutely continuous measure on the real line. The randomness of the measure is inherited from a stationary process A β that has the recently discovered crossover distributions as its one-point marginals, which for large β become the GUE Tracy-Widom distribution. We also prove existence of a limiting law for the four-parameter field of polymer transition probabilities that can be described by the stochastic heat equation.In particular, in this weak noise limit, we obtain the convergence of the point-to-point free energy fluctuations to the GUE Tracy-Widom distribution. We emphasize that the scaling behaviour obtained is universal and does not depend on the law of the disorder.

Our understanding of the one-dimensional KPZ equation, alias noisy Burgers equation, has advanced substantially over the past five years. We provide a nontechnical review, where we limit ourselves to the stochastic PDE and lattice type models approximating it.Recently there has been spectacular progress on exact solutions for strongly interacting stochastic particle systems in one spatial dimension. Very roughly, these are twodimensional field theories with some similarity to two-dimensional systems of equilibrium statistical mechanics and 1 + 1 dimensional quantum field theories, including quantum spin chains. In these stochastic models the static correlations decay exponentially, but, because of conservation laws, the dynamic behavior shows non-trivial scaling properties. The physical interest in such models results from being representatives of huge universality classes. This is in direct analogy to the two-dimensional Ising model. Its exact solution is restricted to nearest neighbor couplings and zero external magnetic field. But the thereby predicted critical behavior is representative for a much larger class of models, including real two-dimensional ferromagnets to which, at first sight, the Ising model would be a very crude approximation.Our review attempts to answer "What is currently known about the one-dimensional Kadar-Parisi-Zhang equation?". This will cover mathematically rigorous results, but also important conjectures and intriguing replica computations. In addition, we discuss discrete models which in a particular scaling limit converge to the KPZ equation. They constitute the universality class of the KPZ equation alluded to in the title. Such models have been, and still are, an important tool to arrive at information on the KPZ equation itself. Our program has obvious disadvantages. It mostly ignores in which temporal order discoveries were made. It also misses largely the amazing progress on other models in the KPZ universality class. Fortunately there are excellent reviews available [1-8], which cover further aspects of the story.1.1 Interface motion. In our article we focus on the KPZ universality class for growing fronts and interfaces in two-dimensional bulk systems. With the hindsight of theoretical activities over many years, let us first recall the physical set-up under consideration. One considers a thin film, for which two bulk phases can be realized such that they are in contact along an interfacial curve. One of the phases is stable, while the other one is metastable with a very long life time. The dynamics of the bulk phases has no conservation laws and is rapidly space-time mixing. Therefore, while fluctuating, the bulk phases are not changing in time and the only observable motion is at the interface, where the transition from metastable to stable is fast, while the reverse transition is very much suppressed. Thereby the stable phase is expanding into the metastable one, accompanied by characteristic interfacial fluctuations. If both phases would be stable, then there is no net mo...

We consider a large class of 1 + 1-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf-Cole solutions to the KPZ equation.

Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter beta, and for a given beta and realization of the noise the path evolves in a Markovian way. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all beta > 0 and for almost all realizations of the white noise the path measure has the same Holder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).Comment: 21 page