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 (KPZ) equation. It was previously shown that the optimal path, the most probable history of the interface h (x ,τ ) which dominates the upper 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. We simulate the interface configurations numerically by employing 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. At short times we observe mirror-symmetry-broken traveling optimal paths for the upper tail, and a single mirror-symmetric path for the lower tail, in good quantitative agreement with analytical predictions. At long times, even at moderate values of H , where the optimal fluctuation method is not supposed to apply, we still observe two well-defined dominating paths. Each of them violates the mirror symmetry x ↔−x and is a mirror image of the other.
- Publication:
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Physical Review E
- Pub Date:
- November 2021
- DOI:
- 10.1103/PhysRevE.104.054125
- arXiv:
- arXiv:2106.08705
- Bibcode:
- 2021PhRvE.104e4125H
- Keywords:
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- Condensed Matter - Statistical Mechanics;
- Mathematics - Probability
- E-Print:
- 11 pages, 9 figures