The KPZ equation with flat initial condition and the directed polymer with one free end

Understand

We study the directed polymer (DP) of length $t$ in a random potential in dimension 1+1 in the continuum limit, with one end fixed and one end free.

• This maps onto the Kardar-Parisi-Zhang growth equation in time $t$, with flat initial conditions.
• We use the Bethe Ansatz solution for the replicated problem which is an attractive bosonic model.
• The problem is more difficult than the previous solution of the fixed endpoint problem as it requires regularization of the spatial integrals over the Bethe eigenfunctions.