very mild effect here. Basically, it just shifts the position of the critical point and induces a first-order phase transition for very strong attractions.

  ------------- ------------ ------------- --------------------- --------------- ------------- --------------
                                                                                               
                 $\beta_n$    $\nu_\bot$    $\beta_n/\nu_\perp$        $z$        $\theta_n$       $\eta$
  Model 1        $0.32(2)$     $0.97(5)$         $0.34(2)$          $1.55(5)$     $0.215(15)$   not measured
  Model 2        $0.325(5)$   $\approx 1$        $0.33(2)$        $\approx 1.5$   $0.215(5)$    $\eta = 0.8$
  ------------- ------------ ------------- --------------------- --------------- ------------- --------------

  : Critical exponents for the MN2 class in $d=1$.

Model 2
-------

In order to verify the robustness and eventual universality of the previous results, we have performed a second study of a different model. It is a restricted solid-on-solid (RSOS) model, a variant of the single-step model introduced in [@krugs], in the presence of a wall. A similar model has been recently studied in the context of synchronization transitions [@Ahlers], to study MN1 type of transitions. Initially the wall is located at $h_w =0$ and a grooved interface is placed beneath it, [*i.e.*]{} the interface has negative height at all the positions.

The dynamics proceeds as follows: At each time step, a site is randomly picked from a one-dimensional lattice of length $L$ and its height decreased two units, $h(i) \to h(i)-2$, provided that 