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Large fluctuations of the KPZ equation in a half-space

by Alexandre Krajenbrink, Pierre Le Doussal

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Submission summary

As Contributors: Alexandre Krajenbrink
Arxiv Link: (pdf)
Date submitted: 2018-05-21 02:00
Submitted by: Krajenbrink, Alexandre
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


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.

Ontology / Topics

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Fredholm Pfaffians Fredholm determinants Hard wall boundary conditions Kardar-Parisi-Zhang (KPZ) equation Neumann boundary conditions
Current status:
Has been resubmitted

Submission & Refereeing History

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Submission 1804.08800v1 on 21 May 2018

Reports on this Submission

Anonymous Report 1 on 2018-7-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1804.08800v1, delivered 2018-07-05, doi: 10.21468/SciPost.Report.529


1. Provides new exact formula for a few half-space cases.

2. Systematic approach for studying short time large deviation.


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.


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 half-space. They also provide a new Fredholm Pfaffian formula for the half-space KPZ equation with the droplet and stationary situation.

The motivation of the study is sound. The new exact formulas for the half-space 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 full-space. 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.

  • validity: good
  • significance: good
  • originality: high
  • clarity: good
  • formatting: reasonable
  • grammar: excellent

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