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

Submission summary

 As Contributors: Alexandre Krajenbrink Arxiv Link: https://arxiv.org/abs/1905.05718v4 (pdf) Date accepted: 2020-02-17 Date submitted: 2020-01-24 01:00 Submitted by: Krajenbrink, Alexandre Submitted to: SciPost Physics Academic field: Physics Specialties: Probability Statistical and Soft Matter Physics Approach: Theoretical

Abstract

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 $A=- \frac{1}{2}$.

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Published as SciPost Phys. 8, 035 (2020)

Dear Editor and Referees,

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

Sincerely yours,
Alexandre Krajenbrink and Pierre Le Doussal

List of changes

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Response to Referee 1
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The referee raised several remarks to which we answer:

(i) The dependence of the amplitude A on the parity of the rapidities $\lambda_j$ is not a typo. Each rapidity comes in pair with its opposite since we study a hard-wall geometry: the image method imposes to sum over the parity of the rapidity. Nonetheless we agree that the parentheses are confusing in Eq. (23) hence it has been modified for clarity.

(ii) We agree that the fraction requires some more space so we added parenthesis and an extra multiplication sign for clarity.

(iii) The typo has been changed, we that the referee for pointing this out.

(iv) We agree with the referee that a few words could be added to explain the continuum limit and hence we added the sentence « the momenta sums become continuous and one can use that the string momenta $m_j k_j$ correspond to free particles.» and quoted a number papers where the exact same replacement was made.

(v) We have added a classical reference from Knuth to Pfaffians which includes the Schur Pfaffian and have transformed the equation for the Schur Pfaffian from an inline equation to a numbered equation. The main remark we have concerning this Pfaffian is that for the KPZ equation in full-space, the inverse of the norm of the Bethe states are given by a Cauchy determinant whereas in the half-space it is given by a Schur Pfaffian which allows to obtain an explicit solution of the generating function of the exponential of the KPZ height in terms of a Fredholm Pfaffian.

(vi) The Mellin-Barnes summation applied in our work is inspired by the one carried out by Sasamoto and Imamura in the study of the stationary initial condition for the KPZ equation in full-space. In their case, a similar requirement on the drift of the Brownian initial condition was expressed and was shown to be lifted after the application of a Mellin-Barnes summation in conjunction with an analytic continuation of Gamma functions and the right choice of contour in the complex plane. We entirely agree with the referee that this passage deserves some more details and hence we have added a precise reference to the work of Sasamoto and Imamura to find the original application of Mellin-Barnes for this case and we have added the sentence:
« The summation over the variables $m_p$ can be done using the Mellin-Barnes summation trick similarly to Refs. [paper of Sasamoto and Imamura]. The barrier $A>(n-1)/2$ is overcome exactly as in Ref.~[paper of Sasamoto and Imamura] (see Lemma.~6 and the discussion following there) from an analytic continuation of Gamma functions in the $B_{k,m}$ factor, the introduction of a particular contour $C_0$ and a final requirement for the drift $A+1/2>0$. »