Transition surface densities as a function of the resonant index k for the different set-ups plotted in Figs. 2, 3, and the two grids in Fig. 4. We estimate the transition surface densities on the planet-mass m_p independent regime for the first setup and for the three-planet case. For the varying h₀ data, we model the h₀ dependence as and record the value for h₀ = 0.033. The black points correspond to the analytical estimate Eq. (21) for equal-mass planets, m_p = m₁ = m₂ = 10⁻⁵ M_⊙ and h₀ = 0.033.