  We consider a discrete bridge from $(0,0)$ to $(2N,0)$ evolving according to
the corner growth dynamics, where the jump rates are subject to an upward
asymmetry of order $N^{-\alpha}$ with $\alpha \in (0,\infty)$. We provide a
classification of the asymptotic behaviours - invariant measure, hydrodynamic
limit and fluctuations - of this model according to the value of the parameter
$\alpha$.
