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

@article{Krajenbrink2019ReplicaBA,
  title={Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line},
  author={Alexandre Krajenbrink and Pierre Le Doussal},
  journal={arXiv: Statistical Mechanics},
  year={2019}
}
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 $A - \frac{1}{2}$ 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… Expand
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References

SHOWING 1-10 OF 108 REFERENCES
Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
We revisit the Lieb-Liniger model for $n$ bosons in one dimension with attractive delta interaction in a half-space $\mathbb{R}^+$ with diagonal boundary conditions. This model is integrable forExpand
Large fluctuations of the KPZ equation in a half-space
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 inExpand
The Lower Tail of the Half-Space KPZ Equation
We establish the first tight bounds on the lower tail probability of the half-space KPZ equation with Neumann boundary parameter $A = -1/2$ and narrow-wedge initial data at the boundary point. TheseExpand
Height Fluctuations for the Stationary KPZ Equation
We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0,X)=B(X)$\mathcal {H}(0,X)=B(X)$, for B(X) a two-sided standard Brownian motion) and show thatExpand
Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature.
TLDR
The early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry is considered and the probability distribution function Φ_{drop}(H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Expand
Simple derivation of the $(- \lambda H)^{5/2}$ tail for the 1D KPZ equation
We study the long-time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions for the Brownian and droplet initial conditions and present a simple derivation of the tail of the largeExpand
Exact lower tail large deviations of the KPZ equation
Consider the Hopf--Cole solution $ h(t,x) $ of the KPZ equation with narrow wedge initial condition. Regarding $ t\to\infty $ as a scaling parameter, we provide the first rigorous proof of the LargeExpand
Exact short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation with Brownian initial condition.
The early-time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension, starting from a Brownian initial condition with a drift w, is studied using the exact Fredholm determinantExpand
Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions.
TLDR
This work provides the first exact calculation of the height distribution at arbitrary time t of the continuum Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with flat initial conditions and obtain the generating function of the moments of the directed polymer partition sum as a Fredholm Pfaffian. Expand
When fast and slow interfaces grow together: Connection to the half-space problem of the Kardar-Parisi-Zhang class.
TLDR
The fluctuation properties at and near the boundary are described by the KPZ half-space problem developed in the theoretical literature, and the distribution at the boundary is given by the largest-eigenvalue distribution of random matrices in the Gaussian symplectic ensemble. Expand
...
1
2
3
4
5
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