asymptotics is slower. Upon increasing the attractiveness of the wall the transients become longer. The estimate for the critical point is the same one as before (this is due to the fact that the free interface is not affected by variations of the attractiveness parameter). Again, our best estimates for the critical exponents are $\beta_n/\nu_\bot =0.32$ and $\theta_n=0.215$ (see Figure (3)). For $b=-0.4$ we still observe a second-order phase transition with a crossover to the mentioned exponents. For $b=-0.5$ transition becomes first-order but it still occurs at $a=a_c$. Within the active phase all the sites are closed to the wall and the order parameter is $1$, but it suddenly changes to $0$ upon decreasing $a$ and hysteresis is observed for slightly subcritical values $a$ (the interface is pinned for up to long times). Therefore, a tricritical point must exist between $0.4$ and $0.5$. We have identified it at $b=-0.42(1)$. The tricritical behavior has not been analyzed.

Let us stress that, contrarily to what happens for the MN1 class, where the presence of an attractive wall induces a new and rich phenomenology (including a broad region of phase coexistence and directed-percolation type of transitions [@Review]), the addition of “attractiveness” has a 