2026/07/18 by Maja Gwozdz
#math.OC #math.PR
We construct a compact metric space with an atomless probability measure and a bounded Lipschitz cost for which the entropic optimal-transport minimisers have no zero-temperature weak limit. More precisely, Pε does not converge as ε\downarrow0. In the example, every unregularised minimiser is singular with respect to μ⊗μ, so that the entropy on the optimal face is identically +∞. We describe the cluster set by Clust(Pε)=\Pw:w∈\mathcal W\, where Pw is the mixture of the two zero-cost graph couplings with weight w, and where \mathcal W⊂[0,1] is a non-degenerate compact interval. We then compute two explicit points w-<w+ in this interval. This shows that compactness, atomlessness, and Lipschitz regularity of the cost do not imply zero-temperature convergence. We also present a compactness theorem for the general problem. If C∈ L1(μ⊗ν) is continuous and bounded from below on Polish spaces, then the zero-temperature cluster set is a nonempty weakly compact connected subset of the optimal face. In the proof, we apply the cluster-point theorem of Bernton, Ghosal, and Nutz and the continuity of ε↦πε. Finally, we give local and exterior first-order criteria for full convergence and cluster membership. We show that nonconvergence is possible, but only through a connected continuum of optimal plans.