Summary: We study Swendsen-Wang dynamics for the critical \(q\)-state Potts model on the square lattice. For \(q=2,3,4\), where the phase transition is continuous, the mixing time \(t_{\text{mix}}\) is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large \(q\), where the phase transition is discontinuous, the authors recently showed that \(t_{\text{mix}}\) is highly sensitive to boundary conditions: \(t_{\text{mix}} \geq \exp (cn)\) on an \(n\times n\) box with periodic boundary, yet under free or monochromatic boundary conditions, \(t_{\text{mix}} \leq \exp (n^{o(1)})\).
In this work we classify this effect under boundary conditions that interpolate between these two (torus vs. free/monochromatic). Specifically, if one of the \(q\) colors is red, mixed boundary conditions such as red-free-red-free on the 4 sides of the box induce \(t_{\text{mix}} \geq \exp (cn)\), yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.