# Replica approach to the KPZ equation with the half Brownian motion initial condition

@article{Imamura2011ReplicaAT, title={Replica approach to the KPZ equation with the half Brownian motion initial condition}, author={Takashi Imamura and Tomohiro Sasamoto}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2011}, volume={44} }

We consider the one-dimensional Kardar–Parisi–Zhang (KPZ) equation with the half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We employ the replica Bethe ansatz and show that the generating function of the exponential moments of the height is expressed as a Fredholm determinant. From this, the height distribution and its asymptotics are studied. Furthermore, using the replica method we also discuss the multi-point height…

## 31 Citations

Stationary Correlations for the 1D KPZ Equation

- Mathematics
- 2013

We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the…

Height Fluctuations for the Stationary KPZ Equation

- Mathematics
- 2014

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 that…

Interacting particles on the line and Dunkl intertwining operator of type A: application to the freezing regime

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- 2012

We consider a one-dimensional system of Brownian particles that repel each other through a logarithmic potential. We study two formulations for the system and the relation between them. The first,…

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- MathematicsPhysical review. E
- 2021

We present an exact solution for the height distribution of the KPZ equation at any time t in a half space with flat initial condition. This is equivalent to obtaining the free-energy distribution of…

On the equal time two-point distribution of the one-dimensional KPZ equation by replica

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- 2013

In a recent contribution, Dotsenko establishes a Fredholm determinant formula for the two-point distribution of the Kardar–Parisi–Zhang equation in the long time limit and starting from narrow wedge…

Replica analysis of the one-dimensional KPZ equation

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- 2014

In the last few years several exact solutions have been obtained for the onedimensional KPZ equation, which describes the dynamics of growing interfaces. In particular the computations based on…

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Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or…

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- Mathematics
- 2017

The coordinate Bethe Ansatz solution of the log-gamma polymer is extended to boundary conditions with one fixed end and the other attached to one half of a one-dimensional lattice. The large-time…

The KPZ equation converges to the KPZ fixed point

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- 2020

We show that under the 1:2:3 scaling, critically probing large space and time, the solution of the KPZ equation starting from a continuous function plus a finite collection of narrow wedges converges…

Directed polymer near a hard wall and KPZ equation in the half-space

- Physics
- 2012

We study the directed polymer with fixed endpoints near an absorbing wall, in the continuum and in the presence of disorder, equivalent to the KPZ equation on the half-space with droplet initial…

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