We study height fluctuations in a stochastic surface which grows according to the Kardar-Parisi-Zhang (KPZ) equation. Using the optimal fluctuation method, we evaluate the distribution  of the interface height H at short times in 1 + 1 dimensions. For initial conditions drawn from the stationary distribution of the KPZ interface, a singularity was recently discovered in the large deviation function of the height at a critical value of H [1]. The singularity results from a breakdown of mirror symmetry of the optimal path of the system conditioned on H. We developed an effective Landau theory for this system [2], thereby showing that the singularity has the character of a mean-field second-order phase transition.

[1] M. Janas, A. Kamenev, and B. Meerson, Phys. Rev. E 94, 032133 (2016).
[2] N. R. Smith, A. Kamenev, and B. Meerson, Phys. Rev. E 97, 042130 (2018).