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We show that the empirical process associated with a system of weaklyinteracting diffusion processes exhibits a form of noise-induced metastability.The result is based on an analysis of the associated McKean--Vlasov freeenergy, which, for suitable attractive interaction potentials, has at least twodistinct global minimisers at the critical parameter value $\beta=\beta_c$. Onthe torus, one of these states is the spatially homogeneous constant state, andthe other is a clustered state. We show that a third critical point exists atthis value. As a result, we obtain that the probability of transition of theempirical process from the constant state scales like $\exp(-N \Delta)$, with$\Delta$ the energy gap at $\beta=\beta_c$. The proof is based on a version ofthe mountain pass theorem for lower semicontinuous and $\lambda$-geodesicallyconvex functionals on the space of probability measures $\mathcal{P}_2(M)$equipped with the $2$-Wasserstein metric, where $M$ is a complete, connected,and smooth Riemannian manifold.

Barriers of the McKean--Vlasov energy via a mountain pass theorem in the space of probability measures