Observing symmetry-broken optimal paths of the stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media

Abstract

Consider the short-time probability distribution P(H, t) of the one-point interface height difference h(x = 0, τ = t) − h(x = 0, τ = 0) = H of the stationary interface h(x, τ ) described by the Kardar-Parisi-Zhang equation. It was previously shown that the optimal path -the most probable history of the interface h(x, τ ) which dominates the higher tail of P(H, t) -is described by any of two ramplike structures of h(x, τ ) traveling either to the left, or to the right. These two solutions emerge, at a critical value of H, via a spontaneous breaking of the mirror symmetry x ↔ −x of the optimal path, and this symmetry breaking is responsible for a second-order dynamical phase transition in the system. Here we employ a large-deviation Monte Carlo sampling algorithm in conjunction with the mapping between the KPZ interface and the directed polymer in a random potential at high temperature. This allows us to observe the optimal paths, which determine each of the two tails of P(H, t), down to probability densities as small as 10 −500 . The observed mirror-symmetry-broken traveling optimal paths for the higher tail, and mirror-symmetric paths for the lower tail, are in good quantitative agreement with analytical predictions.

Symmetries and Zero Modes in Sample Path Large Deviations

“…The KPZ equation [70], an SPDE describing nonlinear surface growth, and in particular its large deviation statistics have been the subject of various studies. Here, particularly noteworthy works are [43][44][45][46] for an investigation of a short time dynamical phase transition for the distribution of the surface height at one point in space, starting from a stationary surface. Furthermore, recently, in [15], an exact computation of the rate function for the same observable with general deterministic initial condition has been carried out; and for the flat initial condition, the exact distribution of the height at one point in space for all times has already been found in [71].…”

“…All of the works listed above deal with the KPZ equation on an unbounded spatial domain. Here, we proceed in the spirit of [43][44][45][46], but modify the setup to study continuous symmetry breaking instead of only a discrete mirror symmetry. Accordingly choosing the spatially averaged surface height as an observable necessitates considering a bounded spatial domain.…”

“…Of particular interest is the case of a dynamical phase transition, where this symmetry breaking happens spontaneously with the extremeness of the rare event under consideration as the control parameter. Relevant examples of this phenomenon in the context of sample path LDT include the one-dimensional Kardar-Parisi-Zhang (KPZ) equation [43][44][45][46] for the surface height at one point in space and with two-sided Brownian motion initial condition (leading to discrete mirror symmetry breaking), the two-dimensional [47] and three-dimensional [48] incompressible Navier-Stokes equations and a Lagrangian turbulence model [49] (all with rotational symmetry breaking). In all of these cases, due to the underlying symmetries, it turns out that it suffices to integrate a single Riccati equation, corresponding to a single reduced functional determinant evaluation, which thereby allows for a generalization of earlier results [30][31][32][33] without increasing the computational costs.…”

Short-time large deviations of the spatially averaged height of a Kardar–Parisi–Zhang interface on a ring

“…An overview of our main results for the large-deviation problem (8) on the ( H, ℓ) phase diagram. In the light blue region, which extends to all H < 0, the global minimum of the action functional ( 7) is attained at the spatially uniform solution (19). In the light orange region, the least-action solution h(x, t) is instead a spatially non-uniform path with a single maximum.…”

Non-analytic behaviour in large-deviations of the susceptible-infected-recovered model under the influence of lockdowns

“…For this purpose we have to employ methods more advanced than typical-event sampling to access states of particularly low probability in numerical simulations. Such large-deviation algorithms have been applied to a variety of models, for example to investigate various graph- [48][49][50][51], RNA- [52] and protein-properties [53][54][55], the Kardar-Parisi-Zhang equation [56][57][58] and power-grids [59,60].…”