Surface growth is ubiquitous in many physical phenomena. Perhaps the most famous model of surface growth is the Kardar-Parisi-Zhang (KPZ) equation which describes nonequilibrium stochastic surface growth resulting from random deposition, surface relaxation and nonlinearity. The height fluctuations in this and other models can be characterized by the probability distribution P(H,t) of height in a spatial point at a finite time.

I will show how one can use the optimal fluctuation method (OFM) to evaluate the complete distribution P(H,t) at short times for different initial conditions for the KPZ equation in 1 + 1 dimensions. The central part of the distribution is Gaussian, but the tails are non-Gaussian and strongly asymmetric. We found a surprising connection between P(H,t) for the flat and stationary initial conditions. For the stationary initial condition, a singularity was recently discovered in the large deviation function of the height at a critical value of H. The singularity results from a breakdown of mirror symmetry of the optimal path of the system. We developed an effective Landau theory for this system, thereby showing that the singularity has the character of a mean-field second-order phase transition.

[1] N. R. Smith, and B. Meerson, Phys. Rev. E 97, 052110 (2018)

[2] N. R. Smith, A. Kamenev, and B. Meerson, Phys. Rev. E 97, 042130 (2018).