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Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

by Alexandre Krajenbrink, Pierre Le Doussal

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Submission summary

As Contributors: Alexandre Krajenbrink
Arxiv Link: (pdf)
Date submitted: 2019-10-11 02:00
Submitted by: Krajenbrink, Alexandre
Submitted to: SciPost Physics
Academic field: Physics
  • Probability
  • Statistical and Soft Matter Physics
Approach: Theoretical


We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height $h(x,t)$ on the positive half line with boundary condition $\partial_x h(x,t)|_{x=0}=A$. It is equivalent to a continuum directed polymer (DP) in a random potential in half-space with a wall at $x=0$ either repulsive $A>0$, or attractive $A<0$. We provide an exact solution, using replica Bethe ansatz methods, to two problems which were recently proved to be equivalent [Parekh, arXiv:1901.09449]: the droplet initial condition for arbitrary $A \geqslant -1/2$, and the Brownian initial condition with a drift for $A=+\infty$ (infinite hard wall). We study the height at $x=0$ and obtain (i) at all time the Laplace transform of the distribution of its exponential (ii) at infinite time, its exact probability distribution function (PDF). These are expressed in two equivalent forms, either as a Fredholm Pfaffian with a matrix valued kernel, or as a Fredholm determinant with a scalar kernel. For droplet initial conditions and $A> - \frac{1}{2}$ the large time PDF is the GSE Tracy-Widom distribution. For $A= \frac{1}{2}$, the critical point at which the DP binds to the wall, we obtain the GOE Tracy-Widom distribution. In the critical region, $A+\frac{1}{2} = \epsilon t^{-1/3} \to 0$ with fixed $\epsilon = \mathcal{O}(1)$, we obtain a transition kernel continuously depending on $\epsilon$. Our work extends the results obtained previously for $A=+\infty$, $A=0$ and $A=- \frac{1}{2}$.

Ontology / Topics

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Bethe Ansatz Kardar-Parisi-Zhang (KPZ) equation Stochastic dynamics
Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 1905.05718v4 on 24 January 2020

Reports on this Submission

Anonymous Report 3 on 2019-12-8 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1905.05718v3, delivered 2019-12-08, doi: 10.21468/SciPost.Report.1377


1. A solid progress on an important but hard problem

2. Well written


In this paper the authors study the KPZ equation on the half-line using the replica Bethe ansatz.
Compared to the KPZ equation on the whole line, studies for the model on the half-line has been restricted to a few particular special cases. In this paper the authors succeeded in generalizing
such results to the case with the Neumann boundary condition for much broader range of the
boundary parameter.

Their main results is the exact Fredholm Pfaffian expression for the Laplace transform of the height
function. With this formula, the authors discuss various aspects of the problem such as the long time limits, specializations to previously known cases and the transition regions. The paper is well written. The motivations are clearly explained, previous works are explained well, the deviations are detailed enough, and many interesting issues related to the problem are addressed.

The reviewer could recommend a publication of the article as it is (after taking into account the comments by the other referee).

  • validity: high
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2019-11-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1905.05718v3, delivered 2019-11-28, doi: 10.21468/SciPost.Report.1345


1. Leading edge results of great importance
2. Very High technical level
3. Extremely well written


The authors study the KPZ equation on the positive half-line with
a Neumann boundary condition at $x=0$, $\partial_x h(0,t) = A$,
and droplet initial condition. Inspired by a very recent theorem of Parekh, the authors solve the equivalent problem of KPZ on the half-line with Dirichlet boundary condition at $x=0$ but in which the initial condition is a Brownian motion with drift $A + 1/2$. A full solution is obtained for positive drift. The central result is an exact formula for the Laplace transform of the distribution of the exponential of the height at 0, valid at all times.This result is expressed as a (Fredholm) Pfaffian of a matrix kernel. The authors also show (in section 5) that their formula can also be written as the square-root of a (Fredholm) Determinant of a scalar kernel.
The large time asymptotics is also investigated : the generic case leads to
the GSE TW distribution; the limiting case $A + 1/2 =0$ gives the GOE.
The crossover is also studied in details, leading to a new expression
for the cross-over kernel $K^{\epsilon}$ which is conjectured to be equivalent to another kernel $K_{\rm{cross}}$ that has appeared in various recent works.

The strategy used by the authors is based on the replica Bethe Ansatz, a technique one that one the authors -- with different collaborators - successfully applied a few years ago to solve the KPZ equation (simultaneously with other groups). The main difficulty being to determine the overlap of the Bethe states with the initial condition. An exact formula is obtained in Appendix A, based on a generalization of a ``magic formula'' derived by Imamura and Sasamoto.

This paper presents leading edge results and is extremely well written. The logical line followed by the authors is very clearly stated. The authors are very explicit and very honest about their assumptions and the (present) limitations of their results. All important details are given and, with some patience, it is possible to understand and to reproduce the calculations of the manuscript. I have no specific remarks apart from:

(i) a possible typo in eq. (23): I do not understand why $A$ depends on the $\epsilon_i$'s.

(ii) It took me time to understand that in eq. (28) what is written is
a product of two fractions (and not three terms in the numerator
and three terms in the denominator). I had to go to ref [24].
May be you could add some space between the two fractions.

(iii) In the exponential in eq. (29): $ m_j \rightarrow m_p$.

(iv) May be it could be useful to add a few words to explain the
continuum limit $\sum_{k_j} \rightarrow m_j \int_{\mathbb{ R}} \frac{d k}{2 \pi}$.

(v) A crucial step is given in eqs (39) and (40). However, no reference
and no proof of the Schur Pfaffian are given. I checked
it with Mathematica for small $n_s$ but did not find a proof. It could
be nice to have some clue here.

(vi) The authors use the Mellin-Barnes summation formula before eq. (45).
It would be useful to explain where and how they overcome the barrier
$A >(n-1)/2$.

I wholeheartedly recommend this manuscript for publication.
This is an impressive work, beautifully presented.


  • validity: top
  • significance: top
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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