Simple derivation of the (– λ H)^5/2 tail for the 1D KPZ equation

Abstract

We study the long-time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1 + 1 dimensions for the Brownian and droplet initial conditions and present a simple derivation of the tail of the large deviations of the height on the negative side λH < 0. We show that for both initial conditions, the cumulative distribution functions take a large deviations form, with a tail for −s 1 given by − log P H t <s = t 2 4 15π (−s) 5/2 . This exact expression was already observed at small time for both initial conditions suggesting that these large deviations remain valid at all times. We present two methods to derive the result (i) long time estimate using a Fredholm determinant formula and (ii) the evaluation of the cumulants of a determinantal point process where the successive cumulants appear to give the successive orders of the large deviation rate function in the larges expansion. An interpretation in terms of large deviations for trapped fermions at low temperature is also given. In addition, we perform a similar calculation for the KPZ equation in a half-space with a droplet initial condition, and show that the same tail as above arises, with the prefactor 4 15π replaced by 2 15π . Finally, the arguments can be extended to show that this tail holds for all times. This is consistent with the fact that the same tail was obtained previously in the short time limit for the full-space problem. arXiv:1802.08618v1 [cond-mat.stat-mech]

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

“…To provide the short-time probability distribution for the droplet initial condition in a half-space in presence of a hard wall, let us first define the branching height (20) and (21). The leading tails at large |H| given in the Table I are…”

“…Before we introduce the results of the simulations obtained through the importance sampling method, we will explain in some details the lattice construction of the Figure 4: (color online) The analytic expression of Φ(H) obtained in Eqs. (20) and (21) is compared with the simulation data obtained for the flat and for Brownian initial conditions. In the case of the flat initial condition, the data are rescaled according to Eq.…”

“…4 the comparison in a region around H c2 between the exact expression of the different branches of Φ(H) given in Eqs. (20) and (21) and the directed polymer simulations of length L = 128 at a time t = 1/16 for the Brownian and the flat initial conditions. Note that we rescaled the distribution for the flat initial condition according to the scaling predicted in Eq.…”

“…These results have been obtained in Refs. [17][18][19][20][21] and Refs. [23][24][25][26][27][28][29][30][31][32].…”

“…In both plots the solid lines indicate the analytical prediction displayed in Eq. (22), together with Eqs (20). and(21).…”