Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
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...
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| Format: | Article |
| Language: | English |
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SciPost
2020-03-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.8.3.035 |
| _version_ | 1818956197475123200 |
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| author | Alexandre Krajenbrink, Pierre Le Doussal |
| author_facet | Alexandre Krajenbrink, Pierre Le Doussal |
| author_sort | Alexandre Krajenbrink, Pierre Le Doussal |
| collection | DOAJ |
| description | 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 $A=- \frac{1}{2}$. |
| first_indexed | 2024-12-20T10:50:08Z |
| format | Article |
| id | doaj.art-959116e49c9748699152377e426c46f6 |
| institution | Directory Open Access Journal |
| issn | 2542-4653 |
| language | English |
| last_indexed | 2024-12-20T10:50:08Z |
| publishDate | 2020-03-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics |
| spelling | doaj.art-959116e49c9748699152377e426c46f62022-12-21T19:43:18ZengSciPostSciPost Physics2542-46532020-03-018303510.21468/SciPostPhys.8.3.035Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-lineAlexandre Krajenbrink, Pierre Le DoussalWe 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 $A=- \frac{1}{2}$.https://scipost.org/SciPostPhys.8.3.035 |
| spellingShingle | Alexandre Krajenbrink, Pierre Le Doussal Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line SciPost Physics |
| title | Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line |
| title_full | Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line |
| title_fullStr | Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line |
| title_full_unstemmed | Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line |
| title_short | Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line |
| title_sort | replica bethe ansatz solution to the kardar parisi zhang equation on the half line |
| url | https://scipost.org/SciPostPhys.8.3.035 |
| work_keys_str_mv | AT alexandrekrajenbrinkpierreledoussal replicabetheansatzsolutiontothekardarparisizhangequationonthehalfline |