Dynamic entropic repulsion for interacting interfaces

The dynamic entropic repulsion for the Ginzburg-Landau $\nabla\phi$ interface model was discussed in [Deuschel-N. 2007] and the asymptotics of the height of the interface was identified. This paper studies a similar problem for two interfaces on the wall which are interacting with one another by the exclusion rule. Each leading order of the asymptotics of height is $\sqrt{\log t}$ as $t\to\infty$ for the system on $\integer^d,\,d\ge3$, $\log t$ for the system on $\integer^2$. The coefficient of the leading term for each interface is also identified.