Equilibrium Mean-Field-Like Statistical Models with KPZ Scaling

We have considered three different “one-body” statistical systems involving Brownian excursions, which possess for fluctuations Kardar–Parisi–Zhang scaling with the critical exponent ν = 13 . In all models imposed external constraints push the underlying stochastic process to a large deviation regime. Specifically, we have considered fluctuations for: (i) Brownian excursions on non-uniform finite trees with linearly growing branching originating from the mean-field approximation of the Dumitriu–Edelman representation of matrix models, (ii) (1+1)D “magnetic” Dyck paths within the strip of finite width, (iii) inflated ideal polymer ring with fixed gyration radius. In the latter problem cutting off the long-ranged spatial fluctuations and leaving only the “typical” modes for stretched paths, we ensure the KPZ-like scaling for bond fluctuations. To the contrary, summing up all normal modes, we get the Gaussian behavior. In all considered models, KPZ fluctuations emerge in presence of two complementary conditions: (i) the trajectories are pushed to a large deviation region of a phase space, and (ii) the trajectories are leaning on an impenetrable boundary.