Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Extreme Diffusion Measures Statistical Fluctuations of the Environment

We consider many-particle diffusion in one spatial dimension modeled as "random walks in a random environment." A shared short-range space-time random environment determines the jump distributions that drive the motion of the particles. We determine universal power laws for the environment's contribution to the variance of the extreme first passage time and extreme location. We show that the prefactors rely upon a single extreme diffusion coefficient that is equal to the ensemble variance of the local drift imposed on particles by the random environment. This coefficient should be contrasted with the Einstein diffusion coefficient, which determines the prefactor in the power law describing the variance of a single diffusing particle and is equal to the jump variance in the ensemble averaged random environment. Thus a measurement of the behavior of extremes in many-particle diffusion yields an otherwise difficult to measure statistical property of the fluctuations of the generally hidden environment in which that diffusion occurs. We verify our theory and the universal behavior numerically over many random walk in a random environment models and system sizes.

First-passage time for many-particle diffusion in space-time random environments

The first-passage time for a single diffusing particle has been studied extensively, but the first-passage time of a system of many diffusing particles, as is often the case in physical systems, has received little attention until recently. We consider two models for many-particle diffusion-one treats each particle as independent simple random walkers while the other treats them as coupled to a common space-time random forcing field that biases particles nearby in space and time in similar ways. The first-passage time of a single diffusing particle under both models shows the same statistics and scaling behavior. However, for many-particle diffusions, the first-passage time among all particles (the extreme first-passage time) is very different between the two models, effected in the latter case by the randomness of the common forcing field. We develop an asymptotic (in the number of particles and location where first passage is being probed) theoretical framework to separate the impact of the random environment with that of the sampling trajectories within it. We identify a power law describing the impact of the environment on the variance of the extreme first-passage time. Through numerical simulations, we verify that the predictions from this asymptotic theory hold even for systems with widely varying numbers of particles, all the way down to 100 particles. This shows that measurements of the extreme first-passage time for many-particle diffusions provide an indirect measurement of the underlying environment in which the diffusion is occurring.

Dynamics at the edge for independent diffusing particles

We study the dynamics of the outliers for a large number of independent Brownian particles in one dimension. We derive the multitime joint distribution of the position of the rightmost particle, by two different methods. We obtain the two-time joint distribution of the maximum and second maximum positions, and we study the counting statistics at the edge. Finally, we derive the multitime joint distribution of the running maximum, as well as the joint distribution of the arrival time of the first particle at several space points.

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.

Crossover from the macroscopic fluctuation theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion

We study the crossover from the macroscopic fluctuation theory (MFT), which describes one-dimensional stochastic diffusive systems at late times, to the weak noise theory (WNT), which describes the Kardar-Parisi-Zhang (KPZ) equation at early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a nonlinear system which interpolates between the derivative and the standard nonlinear Schrodinger equations in imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.

Hidden diagonal integrability of q-Hahn vertex model and Beta polymer model

We study a new integrable probabilistic system, defined in terms of a stochastic colored vertex model on a square lattice. The main distinctive feature of our model is a new family of parameters attached to diagonals rather than to rows or columns, like in other similar models. Because of these new parameters the previously known results about vertex models cannot be directly applied, but nevertheless the integrability remains, and we prove explicit integral expressions for q-deformed moments of the (colored) height functions of the model. Following known techniques our model can be interpreted as a q-discretization of the Beta polymer model from (Probab Theory Relat Fields 167(3):1057–1116 (2017). arXiv:1503.04117) with a new family of parameters, also attached to diagonals. To demonstrate how integrability with respect to the new diagonal parameters works, we extend the known results about Tracy–Widom large-scale fluctuations of the Beta polymer model.

Coulomb-Gas Electrostatics Controls Large Fluctuations of the Kardar-Parisi-Zhang Equation

We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process-an infinite particle Coulomb gas that arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straightforward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution, which demonstrate a crossover between exponential decay with exponent 3 (in the shallow left tail) to exponent 5/2 (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech. (2017) 063203JSMTC61742-546810.1088/1742-5468/aa73f8] via a WKB approximation analysis of a nonlocal, nonlinear integrodifferential equation generalizing Painlevé II which Amir et al. [Commun. Pure Appl. Math. 64, 466 (2011)CPMAMV0010-364010.1002/cpa.20347] related to the KPZ one-point distribution.