Explicit solution of the optimal fluctuation problem for an elastic string in a random medium

Weak noise theory of the O'Connell-Yor polymer as an integrable discretization of the nonlinear Schrödinger equation

We investigate and solve the weak noise theory for the semidiscrete O'Connell-Yor directed polymer. In the large deviation regime, the most probable evolution of the partition function obeys a classical nonlinear system which is a nonstandard discretization of the nonlinear Schrödinger equation with mixed initial-final conditions. We show that this system is integrable and find its general solution through an inverse scattering method and a non-standard Fredholm determinant framework that we develop. This allows us to obtain the large deviation rate function of the free energy of the polymer model from its conserved quantities and to study its convergence to the large deviations of the Kardar-Parisi-Zhang equation. Our model also degenerates to the classical Toda chain, which further substantiates the applicability of our Fredholm framework.

Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth

We study large deviations of the one-point height H of a stochastic interface, governed by the Golubović-Bruinsma equation, ∂_{t}h=-ν∂_{x}^{4}h+(λ/2)(∂_{x}h)^{2}+sqrt[D]ξ(x,t), where h(x,t) is the interface height at point x and time t and ξ(x,t) is the Gaussian white noise. The interface is initially flat, and H is defined by the relation h(x=0,t=T)=H. We focus on the short-time limit, T≪T_{NL}, where T_{NL}=ν^{5/7}(Dλ^{2})^{-4/7} is the characteristic nonlinear time of the system. In this limit typical, small fluctuations of H are unaffected by the nonlinear term, and they are Gaussian. However, the large-deviation tails of the probability distribution P(H,T) "feel" the nonlinearity already at short times, and they are non-Gaussian and asymmetric. We evaluate these tails using the optimal fluctuation method (OFM). The lower tail scales as -lnP(H,T)∼H^{5/2}/T^{1/2}. It coincides with its analog for the Kardar-Parisi-Zhang (KPZ) equation, and we point out to the mechanism of this universality. The upper tail scales as -lnP(H,T)∼H^{11/6}/T^{5/6}, it is different from the upper tail of the KPZ equation. We also compute the large deviation function of H numerically and verify our asymptotic results for the tails.

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.

Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition

We consider the relaxation (noise-free) statistics of the one-point height H=h(x=0,t), where h(x,t) is the evolving height of a one-dimensional Kardar-Parisi-Zhang (KPZ) interface, starting from a Brownian (random) initial condition. We find that, at short times, the distribution of H takes the same scaling form -lnP(H,t)=S(H)/sqrt[t] as the distribution of H for the KPZ interface driven by noise, and we find the exact large-deviation function S(H) analytically. At a critical value H=H_{c}, the second derivative of S(H) jumps, signaling a dynamical phase transition (DPT). Furthermore, we calculate exactly the most likely history of the interface that leads to a given H, and show that the DPT is associated with spontaneous breaking of the mirror symmetry x↔-x of the interface. In turn, we find that this symmetry breaking is a consequence of the nonconvexity of a large-deviation function that is closely related to S(H), and describes a similar problem but in half space. Moreover, the critical point H_{c} is related to the inflection point of the large-deviation function of the half-space problem.

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.

Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

We determine the full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti lattice gas model at long times by uncovering and exploiting complete integrability of the underlying equations of the macroscopic fluctuation theory. These equations are closely related to the derivative nonlinear Schrödinger equation (DNLS), and we solve them by the Zakharov-Shabat inverse scattering method (ISM) adapted by D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798 (1978)JMAPAQ0022-248810.1063/1.523737 for the DNLS. We obtain explicit results for the exact large deviation function of the transferred heat for an initially localized heat pulse, where we uncover a nontrivial symmetry relation.

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.

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

The one-point distribution of the height for the continuum Kardar-Parisi-Zhang equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. The short-time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The flat and stationary initial conditions are studied in the full space, together with the droplet initial condition in the half-space.

Instanton based importance sampling for rare events in stochastic PDEs

We present a new method for sampling rare and large fluctuations in a nonequilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails.

Conformation of a semiflexible filament in a quenched random potential

Motivated by the observation of the storage of excess elastic free energy, prestress, in cross-linked semiflexible networks, we consider the problem of the conformational statistics of a single semiflexible polymer in a quenched random potential. The random potential, which represents the effect of cross-linking to other filaments, is assumed to have a finite correlation length ξ and mean strength V_{0}. We examine statistical distribution of curvature in filament with thermal persistence length ℓ_{P} and length L_{0} in the limit in which ℓ_{P}≫L_{0}. We compare our theoretical predictions to finite-element Brownian dynamics simulations. Finally, we comment on the validity of replica field techniques in addressing these questions.

Large fluctuations of a Kardar-Parisi-Zhang interface on a half line: The height statistics at a shifted point

We consider a stochastic interface h(x,t), described by the 1+1 Kardar-Parisi-Zhang (KPZ) equation on the half line x≥0 with the reflecting boundary at x=0. The interface is initially flat, h(x,t=0)=0. We focus on the short-time probability distribution P(H,L,t) of the height H of the interface at point x=L. Using the optimal fluctuation method, we determine the (Gaussian) body of the distribution and the strongly asymmetric non-Gaussian tails. We find that the slower-decaying tail scales as -sqrt[t]lnP≃|H|^{3/2}f_{-}(L/sqrt[|H|t]) and calculate the function f_{-} analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of L, L_{c}=0.60223⋯sqrt[|H|t]. The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as -sqrt[t]lnP≃|H|^{5/2}f_{+}(L/sqrt[|H|t]). We evaluate the function f_{+} using a specially developed numerical method which involves solving a nonlinear second-order elliptic equation in Lagrangian coordinates. The faster-decaying tail also involves a sharp transition which occurs at a critical value L_{c}≃2sqrt[2|H|t]/π. This transition is similar to the one recently found for the KPZ equation on a ring, and we believe that it has the same fractional order, 5/2. It is smoothed, however, by small diffusion effects.

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.

Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface

We determine the exact short-time distribution -lnP_{f}(H,t)=S_{f}(H)/sqrt[t] of the one-point height H=h(x=0,t) of an evolving 1+1 Kardar-Parisi-Zhang (KPZ) interface for flat initial condition. This is achieved by combining (i) the optimal fluctuation method, (ii) a time-reversal symmetry of the KPZ equation in 1+1 dimension, and (iii) the recently determined exact short-time height distribution -lnP_{st}(H,t)=S_{st}(H)/sqrt[t] for stationary initial condition. In studying the large-deviation function S_{st}(H) of the latter, one encounters two branches: an analytic and a nonanalytic. The analytic branch is nonphysical beyond a critical value of H where a second-order dynamical phase transition occurs. Here we show that, remarkably, it is the analytic branch of S_{st}(H) which determines the large-deviation function S_{f}(H) of the flat interface via a simple mapping S_{f}(H)=2^{-3/2}S_{st}(2H).

Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface

We study the short-time distribution P(H,L,t) of the two-point two-time height difference H=h(L,t)-h(0,0) of a stationary Kardar-Parisi-Zhang interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for L=0 at a critical value H=H_{c}. We show that |H| and L play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when L changes sign, at supercritical H. We also determine analytically P(H,L,t) in several limits away from the second-order transition. Typical fluctuations of H are Gaussian, but the distribution tails are highly asymmetric. The tails -lnP∼|H|^{3/2}/sqrt[t] and -lnP∼|H|^{5/2}/sqrt[t], previously found for L=0, are enhanced for L≠0. At very large |L| the whole height-difference distribution P(H,L,t) is time-independent and Gaussian in H, -lnP∼|H|^{2}/|L|, describing the probability of creating a ramplike height profile at t=0.

Exact short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation with Brownian initial condition

The early-time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension, starting from a Brownian initial condition with a drift w, is studied using the exact Fredholm determinant representation. For large drift we recover the exact results for the droplet initial condition, whereas a vanishingly small drift describes the stationary KPZ case, recently studied by weak noise theory (WNT). We show that for short time t, the probability distribution P(H,t) of the height H at a given point takes the large deviation form P(H,t)∼exp[-Φ(H)/sqrt[t]]. We obtain the exact expressions for the rate function Φ(H) for H<H_{c2}. Our exact expression for H_{c2} numerically coincides with the value at which WNT was found to exhibit a spontaneous reflection symmetry breaking. We propose two continuations for H>H_{c2}, which apparently correspond to the symmetric and asymmetric WNT solutions. The rate function Φ(H) is Gaussian in the center, while it has asymmetric tails, |H|^{5/2} on the negative H side and H^{3/2} on the positive H side.

Local average height distribution of fluctuating interfaces

Height fluctuations of growing surfaces can be characterized by the probability distribution of height in a spatial point at a finite time. Recently there has been spectacular progress in the studies of this quantity for the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. Here we notice that, at or above a critical dimension, the finite-time one-point height distribution is ill defined in a broad class of linear surface growth models unless the model is regularized at small scales. The regularization via a system-dependent small-scale cutoff leads to a partial loss of universality. As a possible alternative, we introduce a local average height. For the linear models, the probability density of this quantity is well defined in any dimension. The weak-noise theory for these models yields the "optimal path" of the interface conditioned on a nonequilibrium fluctuation of the local average height. As an illustration, we consider the conserved Edwards-Wilkinson (EW) equation, where, without regularization, the finite-time one-point height distribution is ill defined in all physical dimensions. We also determine the optimal path of the interface in a closely related problem of the finite-time height-difference distribution for the nonconserved EW equation in 1+1 dimension. Finally, we discuss a UV catastrophe in the finite-time one-point distribution of height in the (nonregularized) KPZ equation in 2+1 dimensions.

Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation: Starting from a parabola

We study the probability distribution P(H,t,L) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x^{2}/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory-a variant of WKB approximation-which yields the optimal history of the interface, conditioned on reaching the given height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f_{+}|H|^{5/2}/t^{1/2} and f_{-}|H|^{3/2}/t^{1/2}. The factor f_{+}(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f_{-} is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

Dynamical phase transition in large-deviation statistics of the Kardar-Parisi-Zhang equation

We study the short-time behavior of the probability distribution P(H,t) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension. The process starts from a stationary interface: h(x,t=0) is given by a realization of two-sided Brownian motion constrained by h(0,0)=0. We find a singularity of the large deviation function of H at a critical value H=H_{c}. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x↔-x of optimal paths h(x,t) predicted by the weak-noise theory of the KPZ equation. At |H|≫|H_{c}| the corresponding tail of P(H) scales as -lnP∼|H|^{3/2}/t^{1/2} and agrees, at any t>0, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of P scales as -lnP∼|H|^{5/2}/t^{1/2} and coincides with the corresponding tail for the sharp-wedge initial condition.

Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature

We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability distribution P(H,t) of the height H at a given point x takes the scaling form P(H,t)∼exp[-Φ_{drop}(H)/sqrt[t]] where the rate function Φ_{drop}(H) is computed exactly for all H. While it is Gaussian in the center, i.e., for small H, the probability distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φ_{drop}(H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.

Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation

Using the weak-noise theory, we evaluate the probability distribution P(H,t) of large deviations of height H of the evolving surface height h(x,t) in the Kardar-Parisi-Zhang equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height H at time t. We argue that the tails of P behave, at arbitrary time t>0, and in a proper moving frame, as -lnP∼|H|^{5/2} and ∼|H|^{3/2}. The 3/2 tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.