First, I'd say that the case you describe corresponds to a wetting transition, rather than to a roughening transition: the system goes from a regime of complete wetting (the wall if fully covered by a mesoscopic layer) at small $K$ to a regime of partial wetting (the wall is covered by microscopic droplets) at large $K$.

Concerning the variant you are asking about:

• For a one-dimensional model without pinning potential, the interface will always be delocalized. It is very easy to prove, as this is simply the trajectory of a random walk on $\mathbb{Z}$ conditioned to be at $0$ at time $0$ and $N$.
• There is a roughening transition in dimension $2$ (that is, in the case of a function $h:\mathbb{Z}^2\to\mathbb{Z}$). This is a famous result first proved rigorously by Fröhlich and Spencer in this paper.
• In dimensions $d\geq 3$, there is no roughening transition, since the interface should be always localized (as can be guessed from the fact that even the rougher Gaussian free field has bounded variance when $d\geq 3$); this was proved in some cases, but I don't remember the references at the moment.

Of course, things are more interesting in the case where you have $h_i\in(-\infty,\infty)$ and a pinning potential. This case is very well understood and I can provide references if needed.

First, I'd say that the case you describe corresponds to a wetting transition, rather than to a roughening transition: the system goes from a regime of complete wetting (the wall if fully covered by a mesoscopic layer) at small (negative) $K$ to a regime of partial wetting (the wall is covered by microscopic droplets) at large (negative) $K$.

Concerning the variant you are asking about:

• For a one-dimensional model without pinning potential, the interface will always be delocalized. It is very easy to prove, as this is simply the trajectory of a random walk on $\mathbb{Z}$ conditioned to be at $0$ at time $0$ and $N$.
• There is a roughening transition in dimension $2$ (that is, in the case of a function $h:\mathbb{Z}^2\to\mathbb{Z}$). This is a famous result first proved rigorously by Fröhlich and Spencer in this paper.
• In dimensions $d\geq 3$, there is no roughening transition, since the interface should be always localized (as can be guessed from the fact that even the rougher Gaussian free field has bounded variance when $d\geq 3$); this was proved in some cases, but I don't remember the references at the moment.

Of course, things are more interesting in the case where you have $h_i\in(-\infty,\infty)$ and a pinning potential. This case is very well understood and I can provide references if needed.

First, I'd say that the case you describe corresponds to a wetting transition, rather than to a roughening transition: the system goes from a regime of complete wetting (the wall if fully covered by a mesoscopic layer) at small $K$ to a regime of partial wetting (the wall is covered by microscopic droplets) at large $K$.

• For a one-dimensional model without pinning potential, the interface will always be delocalized. It is very easy to prove, as this is simply the trajectory of a random walk on $\mathbb{Z}$ conditioned to be at $0$ at time $0$ and $N$.
• There is a roughening transition in dimension $2$ (that is, in the case of a function $h:\mathbb{Z}^2\to\mathbb{Z}$). This is a famous result first proved rigorously by Fröhlich and Spencer in this paper.
• In dimensions $d\geq 3$, there is no roughening transition, since the interface should be always localized (as can be guessed from the fact that even the rougher Gaussian free field has bounded variance when $d\geq 3$); this was proved in some cases, but I don't remember the references at the moment.

Of course, things are more interesting in the case where you have $h_i\in(-\infty,\infty)$ and a pinning potential. This case is very well understood and I can provide references if needed.

First, I'd say that the case you describe corresponds to a wetting transition, rather than to a roughening transition: the system goes from a regime of complete wetting (the wall if fully covered by a mesoscopic layer) at small $K$ to a regime of partial wetting (the wall is covered by microscopic droplets) at large $K$.

Concerning the variant you are asking about:

• For a one-dimensional model without pinning potential, the interface will always be delocalized. It is very easy to prove, as this is simply the trajectory of a random walk on $\mathbb{Z}$ conditioned to be at $0$ at time $0$ and $N$.
• There is a roughening transition in dimension $2$ (that is, in the case of a function $h:\mathbb{Z}^2\to\mathbb{Z}$). This is a famous result first proved rigorously by Fröhlich and Spencer in this paper.
• In dimensions $d\geq 3$, there is no roughening transition, since the interface should be always localized (as can be guessed from the fact that even the rougher Gaussian free field has bounded variance when $d\geq 3$); this was proved in some cases, but I don't remember the references at the moment.

Of course, things are more interesting in the case where you have $h_i\in(-\infty,\infty)$ and a pinning potential. This case is very well understood and I can provide references if needed.