The limiting distributions of observables in the KPZ universality class exhibit tail exponents of 3/2 and 3. In this talk we will review recent work studying the upper tail exponent 3/2 in the moderate deviations regime of several KPZ models at finite size, including the stochastic six vertex model, the ASEP and a class of non-integrable interacting diffusions.

Joint work with C. Noack and P. Sosoe.