Probing large deviations of the Kardar-Parisi-Zhang equation at short times with an importance sampling of directed polymers in random media

Probing the large deviations for the beta random walk in random medium

We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.

Kardar-Parisi-Zhang Physics in the Density Fluctuations of Localized Two-Dimensional Wave Packets

We identify the key features of Kardar-Parisi-Zhang (KPZ) universality class in the fluctuations of the wave density logarithm in a two-dimensional Anderson localized wave packet. In our numerical analysis, the fluctuations are found to exhibit an algebraic scaling with distance characterized by an exponent of 1/3, and a Tracy-Widom probability distribution of the fluctuations. Additionally, within a directed polymer picture of KPZ physics, we identify the dominant contribution of a directed path to the wave packet density and find that its transverse fluctuations are characterized by a roughness exponent 2/3. Leveraging on this connection with KPZ physics, we verify that an Anderson localized wave packet in 2D exhibits a stretched exponential correction to its well-known exponential localization.

Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions

We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The nonlinear hydrodynamic equations of the WNT are solved analytically through a connection to the Zakharov-Shabat (ZS) system using its classical integrability. This approach is based on a recently developed Fredholm determinant framework previously applied to the droplet IC. The flat IC provides the case for a nonvanishing boundary condition of the ZS system and yields a richer solitonic structure comprising the appearance of multiple branches of the Lambert function. As a byproduct, we obtain the explicit solution of the WNT for the Brownian IC, which undergoes a dynamical phase transition. We elucidate its mechanism by showing that the related spontaneous breaking of the spatial symmetry arises from the interplay between two solitons with different rapidities.

Microscopic interplay of temperature and disorder of a one-dimensional elastic interface

Elastic interfaces display scale-invariant geometrical fluctuations at sufficiently large lengthscales. Their asymptotic static roughness then follows a power-law behavior, whose associated exponent provides a robust signature of the universality class to which they belong. The associated prefactor has instead a nonuniversal amplitude fixed by the microscopic interplay between thermal fluctuations and disorder, usually hidden below experimental resolution. Here we compute numerically the roughness of a one-dimensional elastic interface subject to both thermal fluctuations and a quenched disorder with a finite correlation length. We evidence the existence of a power-law regime at short lengthscales. We determine the corresponding exponent ζ_{dis} and find compelling numerical evidence that, contrarily to available analytic predictions, one has ζ_{dis}<1. We discuss the consequences on the temperature dependence of the roughness and the connection with the asymptotic random-manifold regime at large lengthscales. We also discuss the implications of our findings for other systems such as the Kardar-Parisi-Zhang equation and the Burgers turbulence.

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

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.

Inverse Scattering of the Zakharov-Shabat System Solves the Weak Noise Theory of the Kardar-Parisi-Zhang Equation

We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic, and integrable techniques. We expand the program of the weak noise theory, which maps the large deviations onto a nonlinear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system. Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed. These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions.