In this paper we consider two models in the Kardar-Parisi-Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class of initial data (which we call generalized step Bernoulli initial data) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic line, across which the fluctuation exponent changes from 1/2 to 1/3. On the characteristic line, the current fluctuations converge to the general (rank k) Baik-Ben-Arous-Péché distribution for the law of the largest eigenvalue of a critically spiked covariance matrix. For k = 1, this was established for the ASEP by Tracy and Widom; for k > 1 (and also k = 1, for the stochastic six-vertex model), the appearance of these distributions in both models is new.

In this paper we consider N × N real generalized Wigner matrices whose entries are only assumed to have finite (2 + ε)-th moment for some fixed, but arbitrarily small, ε > 0. We show that the Stieltjes transforms m N (z) of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when η = ℑ(z) is almost of order N −1 . As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian Orthogonal Ensemble (GOE) in the large N limit.

Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time T , and show that they are of order T 1/3 along a characteristic line. Upon scaling by T 1/3 , we establish that these fluctuations converge to the long-time height fluctuations of the stationary KPZ equation, that is, to the Baik-Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn in 2005 (who established the same result for the TASEP), and the work of Balázs and Seppäläinen in 2010 (who established the T 1/3 scaling for the general ASEP).Second, we introduce a class of translation-invariant Gibbs measures that characterizes a oneparameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point) in the free energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size T and show that they too are of order T 1/3 along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995 stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity.Upon scaling the height fluctuations by T 1/3 , we again recover the Baik-Rains distribution in the large T limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992. 11 3. Observables for Models With Double-Sided