Erratum: Height distribution tails in the Kardar–Parisi–Zhang equation with Brownian initial conditions (2017 J. Stat. Mech. 103207)

. For stationary interface growth, governed by the Kardar – Parisi – Zhang ( KPZ ) equation in 1 + 1 dimensions, typical ﬂuctuations of the interface height at long times are described by the Baik – Rains distribution. Recently Chhita et al ( 2016 arXiv:1611.06690 ) used the totally asymmetric simple exclusion process ( TASEP ) to study the height ﬂuctuations in systems of the KPZ universality class for Brownian interfaces with arbitrary diusion constant. They showed that there is a one-parameter family of long-time distributions, parameterized by the diusion constant of the initial random height proﬁle. They also computed these distributions numerically by using Monte Carlo ( MC ) simulations. Here we address this problem analytically and focus on the distribution tails at short times. We determine the ( stretched exponential ) tails of the height distribution by applying the optimal ﬂuctuation method ( OFM ) to the KPZ equation. We argue that, by analogy with other initial conditions, the ‘ slow ’ tail holds at arbitrary times and therefore provides a proper asymptotic to the family of long-time distributions studied in Chhita et al ( 2016 arXiv:1611.06690 ) . We verify this hypothesis by performing large-scale MC simulations of a TASEP with a parallel-update rule. The ‘ fast ’ tail, predicted by the OFM, is also expected to hold at arbitrary times, at suciently large heights.

The unexpectedly high sensitivity of the long-time behavior of P t (H) to initial conditions prompted interest in the domains of attraction of the three basic (GUE TW, GOE TW and BR) long-time distributions, and in the possibility of additional asymptotic distributions [1,23,24]. In particular, Chhita, Ferrari and Spohn [1] studied the case when the interface at t = 0 is described by a Brownian motion with arbitrary diusion constant. For one particular value of this diusion constant these initial data correspond to the stationary case, whereas a vanishing diusion constant corresponds to flat interface [1]. The authors [1] showed, via analytical arguments and MC simulations of the totally asymmetric exclusion process (TASEP), that the typical height fluctuations at long times (correspondingly, fluctuations of integrated current in the TASEP) are described by a new one-parameter family of distributions, parameterized by the diusion constant of the Brownian interface at t = 0. Except for the two special values of the diusion constant, mentioned above, this family of distributions is known only numerically [1].
The last two years have witnessed a growing interest in large deviations of the KPZ interface height, described by distant tails of P t (H) [23,[25][26][27][28][29][30]. Even when exact representations for P t (H) are available, extracting the tails from them is technically challenging [27][28][29][30]. A viable alternative is provided by the optimal fluctuation method (OFM). Before the advent of exact representations, Kolokolov and Korshunov [31][32][33] used the OFM to evaluate the tails of P t (H) for the flat initial condition. The OFM is a variant of WKB approximation (after Wentzel, Kramers and Brillouin) that goes back to references [34][35][36][37] in condensed matter physics. It is also known as the instanton method in turbulence [38][39][40], the macroscopic fluctuation theory in lattice gases [41], and WKB methods in reaction-diusion systems [42,43]. In the context of the KPZ and Burgers equations the OFM is also known as the weak noise theory [44][45][46].
The crux of the OFM is the 'optimal fluctuation': the most likely history of the interface height h(x, t) and the most likely realization of the noise ξ(x, t) which dominate the contribution to P t (H). The OFM involves a saddle-point evaluation of the path integral for the KPZ equation (1), constrained by the specified large deviation and the initial and boundary conditions. A necessary (but not always sucient) condition for the OFM validity demands that the 'classical action', resulting from the saddle-point calculation, be suciently large. By now all three basic initial conditions (and the parabolic initial condition which interpolates between the flat and the droplet cases) have been studied with the OFM [23,25,26,31,33]. One general outcome of these studies (see also [27,28,30] where exact representations were used instead) concerns the slow tail (λH > 0) of P t (H). This tail behaves as a stretched exponential,

J. Stat. Mech. (2017) 103207
where f = f 0 = 8 √ 2/3 for the droplet and flat initial conditions (and, more generally, for a broad class of deterministic initial conditions [23]), and f = f 1 = 4 √ 2/3 for stationary initial condition. Importantly, these asymptotics correctly describe the corresponding tails of the typical fluctuations at long times, governed by the GUE and GOE TW distributions, and by the BR distribution, respectively.
In this work we employ the OFM to determine analytically the λH > 0 tails of the family of height distributions P t (H, σ) observed in [1] for general Brownian initial interfaces. Here the parameter σ is the square root of the ratio of the diusion constant of the Brownian interface and the unique diusion constant that describes stationary initial condition. As we show, the λH > 0 tails have the following form: The function f (σ) is given by equation (51) below, and its graph is shown in figure 3. As to be expected, f (σ) satisfies the relations f (0) = f 0 and f (1) = f 1 , where f 0 and f 1 were introduced immediately after equation (3). By analogy with all previously studied initial conditions, we argue that equations (4) and (51) hold at all times t > 0. Therefore, they should correctly describe the λH > 0 tails of the family of distributions studied in [1]. To test this prediction we performed a large scale MC simulations of the TASEP. In order to resolve the slow distribution tail for dierent σ, we modified the simulation algorithm of [1] by using a parallel, rather than a random sequential, update rule. Our simulation results are in very good agreement with equations (4) and (51), as can be seen in figure 7 below. We also briefly address the λH < 0 large-deviation tail of the height distributions P t (H, σ). In the leading order, this tail is independent of ν and σ and given by equation (54) below. As time increases, this H 5/2 /t 1/2 tail is 'pushed', by an expanding H 3 /t tail, toward progressively larger heights.
Here is how we structured the rest of the paper. In section 2 we outline the OFM formalism for the KPZ equation. In section 3 we briefly describe the short-time behavior of typical height fluctuations, where the KPZ nonlinearity is small. Section 4.1 addresses the λH > 0 tail of P t (H, σ) and presents a derivation of equations (4) and (51)-the main analytic result of this work. In section 4.2 we obtain the large-deviation λH < 0 tail of P t (H, σ). In section 4.3 we discuss the behavior of the whole distribution P t (H, σ) at long times. In section 5 we present the results of our TASEP simulations and compare them with equations (4) and (51). Section 6 presents a brief summary of this work. A more detailed description of our MC simulations is presented in two appendices.

Optimal fluctuation method
Let T be the observation time of a specified interface height h(x = 0, t = T ) = H 4 . Rescaling time, t/T → t, the coordinate, x/ √ νT → x, and the interface height, |λ|h/ν → h, we can rewrite equation (1) as

J. Stat. Mech. (2017) 103207
where = Dλ 2 T 1/2 /ν 5/2 is the rescaled noise magnitude [25], and we have assumed without loss of generality that λ < 0 5 . From now on, we will use the rescaled variables and the λ < 0 convention unless stated otherwise. We want to evaluate the probability distribution P T (H, σ) of observing h(0, 1) = H , under the condition that the initial interface h(x, 0) is a random realization of two-sided Brownian motion with an arbitrary rescaled diusion constant 0 < σ 2 < ∞: where B(x) is the two-sided Wiener process with diusion constant 1, pinned at x = 0. The particular case σ = 1 corresponds to the stationary interface of equation (5). For 1, that is at short times, the whole distribution P T (H, σ) can be determined via a saddle-point evaluation of a path integral for equation (5). This procedure brings a minimization problem for the action functional [23,25,26,31,[44][45][46]. It is crucial that, suciently far in the tails of P T (H, σ), ε does not have to be small, and the saddlepoint approximation holds at long times. This is the essence of the OFM for the KPZ equation. Specifics of the random (Brownian) initial conditions is in that one has to minimize the joint action functional s = s dyn + s in , where is the dynamical contribution to the action, and is the 'fluctuation cost' of initial height profiles [26,47]. The variational procedure closely follows that of [26], where the particular case of σ = 1 was considered. The ensuing Euler-Lagrange equation can be recast into Hamilton's equations for two canonically conjugated fields: the optimal interface profile h(x, t) and the optimal realization of the noise ξ(x, t), which we call p(x, t): where is the Hamiltonian. The condition leads to a non-trivial boundary condition for p(x, t = 1): [25,31] 5 Flipping the sign of λ is equivalent to flipping the sign of h.
where the temporary parameter Λ should be ultimately expressed via H and σ.
The boundary condition at t = 0 follows from the variation of the joint action functional s over h(x, t = 0), and takes the form where the first term on the lhs comes from the variation of s dyn (via integration by parts in time of the term ∂ t h), whereas the second term comes from the variation of s in . The boundary condition (14) generalizes the boundary condition for σ = 1 obtained in [26]. A nonzero probability demands that p(x, t) and ∂ x h(x, 0) vanish suciently rapidly at |x| → ∞. Finally, the initial interface is pinned at the origin: Once the optimal path is determined, we can evaluate s = s dyn + s in , where s dyn can be rewritten as This determines P T (H, σ) up to a pre-exponential factor As we already mentioned, at T → 0 the action s(H, σ) yields the complete large deviation function of P T (H, σ). But even at large T, suciently far tails of the distribution are still described correctly.

Edwards-Wilkinson regime
At small H the optimal path can be determined via a regular perturbation expansion in the powers of H, or Λ [25,48]. The perturbation series is h( This brings an iterative set of coupled linear partial dierential equations for h i and p i that can be solved order by order [25]. The leading order of this perturbation expansion describes the Edwards-Wilkinson region of the P t (H, σ), where the KPZ nonlinearity can be neglected: with the boundary conditions After a standard algebra one can find p 1 and h 1 in terms of Λ and σ and compute the joint action s. Using the condition h 1 (0, 1) = H, one finally obtains Combined with equation (17), this result shows that the body of the short-time height distribution is a Gaussian with a σ-dependent variance which describes the Edwards-Wilkinson scaling H 2 1/2 ∼ t 1/4 . For σ → 0 and σ = 1 we obtain s = π/2 H 2 and s = √ π H 2 /2, respectively. These are well-known results for the Edwards-Wilkinson regime of flat and stationary interface, respectively [51]. At σ 1 the cost of creating a non-flat interface is very high, so the optimal initial interface is almost flat. Here the joint action s is dominated by the dynamic contrib ution s dyn . At σ → ∞ the action s goes to zero, and the variance of H grows indefinitely. This is to be expected, as for large σ the fluctuation cost s in of creating any interface at t = 0 becomes low. As a result, at σ → ∞, the fluctuation cost s in dominates the joint action s. Fluctuations are only needed in this case to create the optimal initial profile. The following evolution of h(x, t) proceeds almost deterministically. As we will see, qualitatively similar features, at σ 1 and σ 1, are also observed in the λH > 0 distribution tail.
In the Edwards-Wilkinson regime the optimal paths h(x, t) and p(x, t) (not shown here) are, at all times, symmetric functions of x. Higher orders of the perturbation expansion give higher cumulants of the distribution, as was shown in [25] for the flat initial condition. Although the KPZ nonlinearity manifests itself already in the second order of the perturbation expansion, the reflection symmetry x ↔ −x of the optimal paths is protected in all orders of the perturbation theory. Therefore, within its convergence radius, the perturbation series for s(H, σ) comes from a unique solution for the optimal path which obeys the reflection symmetry.

λH > 0 tail
For stationary growth, σ = 1, the OFM solution ceases to be unique at a finite H < 0 (to remind the reader, we set λ < 0 in this paper) [26]. At a critical value H = H c −3.7 two additional solutions with a broken reflection symmetry appear. Each of them is a mirror reflection of the other, and they have a lesser action than the (still existing) symmetric solution. As a result, the short-time large deviation function of the hight s(H, 1) exhibits a singularity at H = H c which has the character of a (mean-field) second-order phase transition. Very recently, the second-order transition at H = H c , at short times, has been reproduced from an exact representation of the height statistics [30]. were found analytically [26]. A crucial building block of these asymptotic solutions is a traveling pair of a localized pulse of p (which we call soliton) and a fluctuation-driven (that is, p-driven) shock of the interface slope At σ = 1 one should expect a critical line H = H c (σ) of the phase transition to appear. We relegate the determination of the critical line for future work. Here we focus on the −H 1 tail of the height distribution P T (H, σ) at arbitrary σ. For suciently large negative Λ, numerical solutions 6 show that traveling soliton-shock solutions continue to play a crucial role here, although the complete solution is more complicated than in the case of σ = 1. To present the traveling soliton-shock solutions, let us first dierentiate equation (9) with respect to x and rewrite the OF equations as coupled continuity equations for u and p: One of the two traveling soliton-shock solutions is of the C is an arbitrary constant, ∆u ≡ u + − u − , and u − and u + are the asymptotic values of u(ξ) at −∞ and +∞, respectively. This soliton-shock pair travels to the right with velocity c. Importantly, the solution (26) and (27) belongs to the invariant manifold of equations (9) and (10).
The mirror-symmetric soliton-shock solution travels to the left; it can be obtained by replacing u − by −u − and u + by −u + . As particular solutions of equations (24) and (25), equations (26)- (28) have been known for some time [44,45]. Here we will expose their role in the problem of height statistics specified by the boundary conditions in space and time. We will also show how these solutions should be amended by additional fluctuationless solutions, and how the constants u − , u + and C are determined by Λ and σ, and ultimately by H and σ. For concreteness, we will do it for the solution traveling to the right. https://doi.org/10.1088/1742-5468/aa8c12 As in the case of σ = 1 [26], at very large negative H, the optimal initial condition involves a large-amplitude narrow negative pulse of p, located at x = −L, where a negative plateau of h becomes a 'ramp', see figure 1. Correspondingly, the initial interface slope u u 0 = const for −L < x < 0 and zero elsewhere, as indicated by the dashed lines in figures 2(a) and (b). The parameters L and u 0 , and the plateau value H 0 = −Lu 0 are to be determined, alongside with other attributes of the solution. At large |H| the characteristic widths of the p-pulse and of the u-jumps, which all scale as 1/∆u, are small, see equations (26 and (27). (As we will ultimately see, these widths scale with H as 1/ |H| 1.) In contrast, the 'hydrodynamic' length scale of the problem, which scales as L ∼ |H| 1, is large. Therefore, at the hydrodynamic scale, the p-pulse and the u-jumps can be treated as point-like objects, in a close analogy to fluid mechanics of compressible flow where, for example, shocks can be treated as discontinuities [52,53]. In particular, p(x, t = 0) is negligible in the vicinity of x = 0, and the boundary condition (14) yields Here is how the optimal fluctuation develops in time. The u-jump, which is initially at x = 0, travels to the right, with velocity u 0 /2, as a shock of the Burgers equation At the hydrodynamic length scale the shock can be described by the inviscid Burgers equation, also called the Hopf equation [53]: The u-jump, which is initially at x = −L, in general splits into several 'elementary excitations'. As we will see shortly, for σ < 1 these elementary excitations are three distinct u-shocks. They are indicated by the numbers 1, 2 and 3 in figure 2(a), whereas the shock initially at x = 0 is indicated by number 4. The fluctuation-driven u-shock (27), accompanied by the p-soliton (26), is the shock number 2.
The u-shock 1 has u = 0 on its left and u = u − on its right. The fluctuation-driven u-shock 2 has u = u − on its left and u = u + on its right. The u-shock 3 has u = u + on its left and u = u 0 on its right. Like the u-shock 4, the u-shocks 1 and 3 are ordinary shocks of the Hopf equation (32). The p-soliton (26), which accompanies the fluctuation-driven u-shock 2, rapidly emerges from the initial p-pulse at x = −L.
For σ > 1 the u-jump, initially at x = −L, splits into a single u-shock and two socalled rarefaction waves [52,53], each of them obeying the equation u(x, t) = (x + L)/t, see figure 2(b). The u-shock is fluctuation-driven; it is accompanied by the p-soliton as described by equations (26) and (27).
The reasons for this flow structure become clear once the constants u − , u + and L are expressed via Λ and σ. For that we need three relations. One of them is the conservation law ∞ −∞ p(x, t) dx = Λ. When applied to equation (26), it yields An additional conservation law is The constant on the right hand side can be evaluated at t = 0 by using equation (14) and taking into account the fact that p(x, 0) is strongly localized around x = −L. As a result, the constant is equal to −u 2 0 /σ 2 . Now, evaluating the left hand side of equation (34) on the soliton-shock solution (26) and (27), we obtain Finally, at t = 1 the p-soliton, which travels with the velocity c from equation (28), must arrive at x = 0 7 . This yields a kinematic condition, Equations (33), (35) and (36) yield Notice that Λ < 0 in the negative tail of H that we are studying. The signs of, and the relation between, the quantities u − , u + and u + determine the flow structure presented in figures 2(a) and (b) [53].
Using equations (30) and (37), we can evaluate the action s in terms of Λ and σ. Indeed, equations (8) and (16) yield respectively. Here s dyn comes from the p-soliton (26). As a result, To express Λ in terms of H and σ we use the condition (12) together with equation (9) at x = 0, where the terms ∂ 2 x h and p can be neglected. This yields which should be equal to H. The calculations of u(x = 0, t) are dierent for σ < 1 and σ > 1. For σ < 1 we have u − < 0, therefore the u-shock 1 (see figure 2) travels to the left and is inconsequential for our calculation. As u + > u 0 , the excess u + − u 0 travels to the right as the shock 3. As a result, where τ = 2σ 2 (3σ 2 + 1) −1 < 1 is the time it takes the shock 3 and the p-soliton to travel from x = −L to the origin. Then equations (39) and (40) yield and equation (38) becomes For σ > 1 we have u + < u 0 . As a result, two rarefaction waves (obeying the same relation u(x, t) = (x + L)/t) develop on the left and on the right of the fluctuationdriven shock. There are also two weak discontinuities, see figure 2(b). The spatial arrangement of the fluctuation-driven shock and the rarefaction waves do not change until t = 1. Therefore, the left rarefaction wave is inconsequential for the purposes of evaluating h(0, 1). In the relevant regions of space we have As a result, Somewhat surprisingly, this formula coincides with equation (41), obtained for σ < 1. As a result, Λ versus H is still described by equation (42), and the action (38) is described by equation (43) for all σ. Equation (43) brings us to the main analytic result of this work, announced in equation (4), where A graph of the function f (σ) is shown in figure 3. For σ = 1 we obtain f = (4/3) √ 2, reproducing the BR distribution tail which is observed, for the stationary initial condition, at all times [17,18,26,30].
For σ = 0 we obtain f = (8/3) √ 2. This describes the λH 1 tail of the GOE Tracy-Widom distribution which is observed, at all times, for flat initial condition [16,25,31]. Indeed, at σ → 0, the fluctuational cost (8) of a non-flat interface at t = 0 becomes prohibitively high. As a result, the optimal initial interface is almost flat, and the leading-order contribution to the action mostly comes from the dynamics.
The results of this subsection hold if the characteristic soliton/shock width is much smaller that the hydrodynamic length scale L, as we assumed. For σ < 1 this condition demands |H| 1, whereas for σ 1 it is more stringent: |H| σ 2 . Although the function f (σ) does not have any singularity at σ = 1, the stationary case σ = 1 is special in terms of the optimal path. Here the optimal initial u-jump does not split into multiple shocks, neither does it produce rarefaction waves. Instead, the p-pulse and the u-jump directly, and rapidly evolve into a traveling soliton-shock pair (26) and (27) with u + = u 0 and u − = 0, as was observed in [26]. This simple behavior appears because, at σ = 1, the initial condition (14) obeys equation (29) everywhere except at the origin, where the delta-function appears. The ordinary u-shock at x = 0 'takes care' of the delta, and the ordinary shock and the fluctuation-driven shock travel with the same velocity, creating the 'shock-antishock' pair [44,45], see also [49,50].
In the dimensional units of equation (1) the λH > 0 tail is as announced in equation (4).

λH < 0 large-deviation tail
The λH < 0 large-deviation tail has quite a dierent nature. Here the optimal path respects the reflection symmetry x ↔ −x at any H. As in [23,25,26,33], the leadingorder approximation to the OFM solution can be obtained in the framework of 'inviscid hydrodynamics', that is by neglecting the diusion terms in equations (9), (10) and (14). In this approximation the 'fluctuation cost' of the initial profile, s in from equation (8), is negligible. As a result, the initial condition (14) becomes simply p(x, t = 0) = Λδ(x), and the parameter σ drops out of the problem. This case was solved in [23,26]. The resulting λH < 0 large deviation tail coincides, in the leading order, with the corresponding tail for the (deterministic) droplet initial condition: s s dyn = 4 √ 2H 5/2 /(15π) [23,27,29,30]. In dimensional units − ln P t (H, σ) 4 2|λ| 15πD Therefore, at fixed σ > 0, and for suciently large H, the λH < 0 large-deviation tail is, in the leading order, independent of ν and σ. The optimal path of the interface is dierent from that for the droplet: it includes a fluctuationless u-shock with a nontrivial dynamics [26].

P t (H, σ) at short and long times
As we have seen, at short times the (Gaussian) body of the distributions P t (H, σ) is described by equation (22), while the λH > 0 and λH < 0 tails are described by equations (4) and (54), respectively. At long times the Gaussian distributions give way to the non-Gaussian distributions observed in [1]. As the parameter ∼ √ t is now large, the OFM does not apply to typical fluctuations. However, it continues to apply suciently far in the tails. It has been shown that the H 3/2 /t 1/2 tails of the GUE and GOE TW distribution (for the droplet and flat interface), and the similar H 3/2 /t 1/2 tail of the BR distribution, persist at all times [23, 25-28, 30, 31] and are correctly described by the OFM. Therefore, it is natural to assume that the same is true for the more general random initial conditions considered in [1] and in the present work. That is, one should expect that the leadingorder asymptotic (4) and (51) provides a correct long-time description of the λH > 0 tail of P t (H, σ). In the next section we put this assumption into test by performing large-scale MC simulations of the TASEP.
The other distribution tail, λH < 0, has a more complicated structure at long times. Here a dierent tail appears at moderately large heights and 'pushes' the large-deviation tail (54) toward progressively larger heights as time increases. For the three basic initial conditions this emerging non-OFM tail behaves as H 3 /t: it is the tail of the corre sponding TW/BR distribution. The 'inviscid' large-deviation tail (54), however, is still present at |H| t 1. The ensuing dynamical two-tail structure was conjectured for a whole class of initial conditions in [23,25,26]. Recently the two-tail structure has been obtained analytically, and explicitly, for the droplet case [29] by using an exact representation for P t (H) derived in [15]. The crossover between the inner H 3 /T tail and outer H 5/2 /T 1/2 tail occurs at H ∼ t. It is natural to assume that, at long times, the λH < 0 tail of P t (H, σ) has a similar dynamical two-tail structure for the general Brownian initial conditions. We also note that the large-deviation tail (54) may be specific for the KPZ equation.

Monte-Carlo simulations of the TASEP with Brownian initial conditions
We tested our analytical predictions for the λH > 0 tail in MC simulations. In appendix A we employ nonlinear fluctuating hydrodynamics (NLFH) to establish a duality between the TASEP and the KPZ equation. In appendix B we present analytical expressions, for the parallel-update TASEP, for the quantities that appear in the NLFH description. Using the TASEP-KPZ duality and the analytical expressions, we accessed the height distributions P t (H, σ) of the KPZ interface by simulating a TASEP with particle density ρ = 1/2 and parallel update rule [54]. In a parallel-update TASEP all particles attempt to hop to the left at the same time with probability q ∈ (0, 1), see figure A1 in appendix A. The move is disallowed if the target site is occupied.
Evolving the TASEP configuration in time is a major bottleneck in terms of the computer memory and speed. Pre-caching techniques allow to overcome this bottleneck for a sequential memory readout. Therefore, using a parallel (sequential read) instead of a random sequential (random read) update is crucial for reducing the computation time. Another problem is caused by the fact that the initial configurations of the TASEP become more and more clustered with an increase of σ. For large σ the clusters become a major obstacle to reaching the long-time asymptotic statistics, so they need to be dissolved. For a random sequential update the cluster dissolution rate is constant, while for a parallel update it can be increased by increasing the hopping probability q. Note that the random sequential update is recovered in the limit of q → 0 (when time J. Stat. Mech. (2017) 103207 is properly rescaled), while in the limit of q → 1 the dynamics become deterministic. In our simulations we used the hopping probability q = 1/2 for 0 σ 1, and q = 15/16 for σ 1.
Our first objective was to access the late-time height (correspondingly, the time-integrated current) distribution which is expected to have the following scaling behavior: To suppress finite-size corrections of order L −1 to the surface growth speed (correspondingly, to the time-integrated current), the height/current statistics were recorded in periodic systems of size L = 10 9 . The statistics were extracted at every lattice site  using the translational invariance of the system. For each σ, the distributions were obtained from at least 500 independent MC samples when starting from an initial state drawn from the σ-distribution described by equations (B.1) and (B.8). To verify that the asymptotic regime was reached, we tested the expected scaling behavior (55) at dierent times. As shown in figure 5(a) very good time collapse is observed within statistical errors. The rescaled curves match, without any adjustable parameters, the BR and the GOE TW distributions for the stationary and flat initial conditions, respectively.
Having established the validity of equation (55), we calculated F (σ) for dierent σ and observation times t. As can be seen from figure 6, the measured MC data suces to resolve the asymptote ln F (σ) (x) f (σ)x 3/2 . Finally, we fitted the slopes f (σ) in the regime 8 − ln F (σ) (x) 15 and compared them to the analytical prediction, equation (51). As demonstrated in figure 7, the simulations and theory are in a very good agreement.

Summary
For a whole class of initial conditions, the λH > 0 tail of the one-point height distribution of the KPZ equation in 1 + 1 dimensions persists at all times t > 0. Here we exploited this persistence by using the optimal fluctuation method, which is asymptotically exact in the limit of t → 0. This enabled us to determine the optimal path of the conditioned process and calculate the λH > 0 tail analytically for a family of Brownian initial interfaces parameterized by their diusion constants. The λH > 0 tail is given by equations (4) and (51). Our extensive MC simulations with a parallel-update TASEP strongly support these results.
We also determined the λH < 0 tail of the height distribution, equation (54). This H 5/2 /t 1/2 tail is independent of the diusion constant of the initial interface. By analogy with other initial conditions, we expect that at long times this tail is 'pushed' towards progressively larger heights.
In conclusion, our results show yet again that the OFM can be remarkably successful in the description of large fluctuations in macroscopic systems far from equilibrium. Note that the substitution ∂ x h(x, t) = u(x, t) is motivated by the exact mapping of the TASEP to a surface growth process [3-5, 59, 60] that is known as the single-step model, illustrated in figure A1.
To capture nonlinear interactions (λ = 0) in the time-integrated current one has to account for the drift term by integrating along the directed path from (x, 0) to (x + j (ρ)t, t) [61,62]: (A.6) At long times the current distributions, when starting from an initial state with a given σ, are expected to show the scaling behavior By virtue of our analytic results, the positive tail of F (σ) has the following form: with f (σ) given by equation (51). Expressing the interface height via the time-  . Mapping the TASEP dynamics to a surface growth. Shown is a TASEP configuration evolving in time. The color-coded particles will hop at the next time step. Mapping a particle to an up-slope (• → ) and a hole to a downslope (• → ) one obtains, for each particle/hole configuration, a height profile. If a particle hops to the left a diamond is added to the surface between the initial and final position of the particle.

Appendix B. Current fluctuations in TASEP with parallel update rule
Here we present some analytical relations for the parallel-update TASEP, which allow to establish the current distributions without any adjustable parameters. In the parallel-update TASEP all particles attempt to hop to the left at the same time with probability q ∈ (0, 1), see figure A1. The move is disallowed if the target site is occupied. The exact stationary probability distribution P ( n) of a configuration n is given by the two-cluster approximation [63] P ( n) = Consequently, to derive the stationary distribution one has to solve the master equation for P (1, 0) which reduces to a quadratic equation and yields [64,65] P st (1, 0) = 1 2q With the stationary distribution at hand, one can calculate the system's current as a function of the density, its second derivative and the compressibility as We set the density to ρ = 1/2, so that the drift term vanishes, j (1/2) = 0, whereas the current-distribution scaling factor (A.10) is at its maximum, leading to the fastest convergence to the asymptotic regime equation (A.7). Translating the initial configurations, drawn from the distribution (B.1), into height profiles (see figure A1), we can express the Brownian interface diusion constant σ 2 as As σ is defined as the square root of the ratio of σ 2 and the diusion constant describing the stationary state, the initial σ-states are drawn from the distribution (B.1) using where n t k is the occupation number of site k at time t.