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Large fluctuations of the KPZ equation in a half-space

by Alexandre Krajenbrink, Pierre Le Doussal

Submission summary

As Contributors: Alexandre Krajenbrink
Arxiv Link: https://arxiv.org/abs/1804.08800v2
Date submitted: 2018-07-27
Submitted by: Krajenbrink, Alexandre
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Statistical and Soft Matter Physics

Abstract

We investigate the short-time regime of the KPZ equation in $1+1$ dimensions and develop a unifying method to obtain the height distribution in this regime, valid whenever an exact solution exists in the form of a Fredholm Pfaffian or determinant. These include the droplet and stationary initial conditions in full space, previously obtained by a different method. The novel results concern the droplet initial condition in a half space for several Neumann boundary conditions: hard wall, symmetric, and critical. In all cases, the height probability distribution takes the large deviation form $P(H,t) \sim \exp( - \Phi(H)/\sqrt{t})$ for small time. We obtain the rate function $\Phi(H)$ analytically for the above cases. It has a Gaussian form in the center with asymmetric tails, $|H|^{5/2}$ on the negative side, and $H^{3/2}$ on the positive side. The amplitude of the left tail for the half-space is found to be half the one of the full space. As in the full space case, we find that these left tails remain valid at all times. In addition, we present here (i) a new Fredholm Pfaffian formula for the solution of the hard wall boundary condition and (ii) two Fredholm determinant representations for the solutions of the hard wall and the symmetric boundary respectively.

Current status:
Editor-in-charge assigned


Author comments upon resubmission

Dear Editor and Referee,

We are grateful for the efforts in reviewing our manuscript. We thank the referee for his constructive comments of our paper. In the following, we believe we answer the concerns raised by the referee and list all changes we made in the resubmitted version.

Sincerely yours,

Alexandre Krajenbrink and Pierre Le Doussal

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Response to the Referee 
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The referee raised two main concerns.

1. The referee asked that we discuss more clearly the validity of the cumulant approximation. This is now elaborated in the main text Section 3.1 below Eq. 19. 

We argue that the cumulant approximation can be explained by a law of large number for the set of points generated by the Pfaffian point process in the short-time regime. The observable we are interested in (the sum of functions $\phi$ in Eq. 17) gets self-averaged as most points of the process are involved in the sum. This explains why the first cumulants dominates higher order ones. 

We would like to mention that these arguments are fully confirmed by explicit calculations which show that the cumulant expansion can be made systematic, and this, together with numerical evidence, will be presented in work in preparation, as we have now indicated
in the text - Ref  [45].

We would finally like to stress that the goal of our manuscript is to apply the cumulant approximation previously used in full-space cases of the KPZ equation to numerous half-space cases where the Pfaffian representation of the generating function is available. This is particularly helpful to unravel universal properties of the large deviations of the solution of the KPZ equation at short time. We hope this will help to get a broader picture of the behavior of the general solution for which one does not necessarily have a determinant or Pfaffian representation.

2. The other concern the referee raised is about clear pointing and cross-referencing in our manuscript. We thank the referee for this remark and added pointers to all our formulas to ensure that all derivations are made extremely clear for the reader.

List of changes

List of changes :

- We added in Section 3.1 below Eq. 19 a discussion about the validity of the cumulant approximation.  

- We added numerous cross-references and pointers all along the manuscript to ensure that all derivations are made extremely clear for the reader.

- We omitted the repetition of the expression of the new kernel for the hard-wall case by pointing in Section 5.1 to the result announced in Section 2.1 as advised by the referee.

Submission & Refereeing History

Resubmission 1804.08800v2 (27 July 2018)
Submission 1804.08800v1 (21 May 2018)

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