  We study transition paths in energy landscapes over multi-categorical Potts
configurations using the mean-field approach introduced by Mauri et al., {\em
Phys Rev Lett 130, 158402 (2023)}. Paths interpolate between two fixed
configurations or are anchored at one extremity only. We characterize the
properties of `good' transition paths realizing a trade-off between exploring
low-energy regions in the landscape and being not too long, such as their
entropy or the probability of escape from a region of the landscape. We unveil
the existence of a phase transition separating a regime in which paths are
stretched in between their anchors, from another regime, where paths can
explore the energy landscape more globally to minimize the energy. This phase
transition is first illustrated and studied in detail on a mathematically
tractable Hopfield-Potts toy model, then studied in energy landscapes inferred
from protein-sequence data.
