SciPost Submission Page
Large fluctuations of the KPZ equation in a halfspace
by Alexandre Krajenbrink, Pierre Le Doussal
This is not the current version.
Submission summary
As Contributors:  Alexandre Krajenbrink 
Arxiv Link:  https://arxiv.org/abs/1804.08800v1 (pdf) 
Date submitted:  20180521 02:00 
Submitted by:  Krajenbrink, Alexandre 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the shorttime 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 halfspace 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.
Ontology / Topics
See full Ontology or Topics database.Current status:
Submission & Refereeing History
You are currently on this page
Reports on this Submission
Anonymous Report 1 on 201875 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1804.08800v1, delivered 20180705, doi: 10.21468/SciPost.Report.529
Strengths
1. Provides new exact formula for a few halfspace cases.
2. Systematic approach for studying short time large deviation.
Weaknesses
1. Validity of the approximation is not clearly discussed.
2. There are repetitions of the same formulas without clear pointers.
3. Looks a bit like a collection of calculations.
Report
In this paper the authors study (mainly short time) large deviation properties of the KPZ equation. They use an explicit representation of a generating function of the height in the form of Fredholm determinant or Fredholm Pfaffian and applies the cumulant approximation in [33]. They show wide applicability of the method by studying a few cases of the KPZ equation in halfspace. They also provide a new Fredholm Pfaffian formula for the halfspace KPZ equation with the droplet and stationary situation.
The motivation of the study is sound. The new exact formulas for the halfspace case are new and would be useful for future studies. On the other hand, their main results about the short time large deviation is based on what they call the cumulant approximation. A problem is that it is not clear how reliable this approximation is.
They write that in [33] it was observed that such an approximation is valid for a certain cases in fullspace. There seems no guarantee that the same approximation is valid for other cases, but they do not seem to give serious discussions about the applicability of this approximation. The authors should provide clear and convincing arguments of the validity of the approximation or at least give some numerical evidence that the approximation seems to hold.
The presentation of the results are not optimal. They first present the main results in section 2. The authors should provide clearer pointers both in section 2 and in main texts. For example for the formula (5)(6), it is written that “These results are shown in Section 5”. Subsection 5.1 should be more appropriate. In addition, in subsection 5.1, the same formulas appear as (63)(64) without any notice. This should be pointed out clearly. In fact one may omit (63)(64) and refer to (5)(6). Similarly, there are some repetitions of the contents of section 2.2 and the main texts. For example (13) and (43) are the same. The connection should be clearly stated.
The paper will be reconsidered after a revision.
Requested changes
1. Give clear and convincing arguments for the validity of their approximation.
2. Give clear pointers to the repeated formulas.