GUE $\times$ GUE limit law at hard shocks in ASEP

We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ with initial data such that in the large time particle density $\rho(\cdot)$ a discontinuity (shock) at the origin is created. At the shock, the value of $\rho$ jumps from zero to one, but $\rho(-\varepsilon),1-\rho(\varepsilon) >0$ for any $\varepsilon>0$. We are interested in the rescaled position of a tagged particle which enters the shock with positive probability. We show that, inside the shock region, the particle position has the KPZ-typical $1/3$ fluctuations, a $F_{\mathrm{GUE}}\times F_{\mathrm{GUE}}$ limit law and a degenerated correlation length. Outside the shock region, the particle fluctuates as if there was no shock. Our arguments are mostly probabilistic, in particular, the mixing times of countable state space ASEPs are instrumental to study the fluctuations at shocks.