This paper shows that a standard large‑deviation upper bound that holds on compact sets can fail on some closed sets in entropic optimal transport. The authors build an explicit example with a continuous cost and non‑atomic marginals where the entropic minimizers converge to an optimal transport plan whose support is not compact. In that example the known compact‑set upper bound still holds, but the corresponding closed‑set upper bound breaks down. They also prove there is no full large‑deviation principle at the natural speed 1/ε with any lower semicontinuous rate function for their model.

Entropic optimal transport (also called the static Schrödinger problem) is a version of optimal transport that adds an entropy penalty. For two fixed probability marginals µ and ν and a cost function C(x,y), one minimizes the expected cost plus ε times the relative entropy (a measure of distance between probability measures) to a reference product measure. For each positive ε this problem has a unique minimizer Pε. The paper studies what happens to Pε when the regularization parameter ε goes to zero, a regime often described informally as “zero temperature.” Earlier work (by Bernton, Ghosal and Nutz) showed that an upper bound for the probabilities of compact sets holds in this limit. The question addressed here is whether that bound can be extended from compact sets to all closed sets.

To answer that question the authors construct a counterexample. They first make a discrete block model in which each block contains many off‑diagonal states at the same cost. Because there are many indistinguishable states in a block, combinatorial multiplicity lowers the effective exponential penalty for those states. This produces a limiting optimal plan with noncompact support and a sequence of entropic minimizers that concentrate mass on a closed set in a way that violates the closed‑set upper bound: there is a closed set F for which the candidate rate says the mass should be exponentially small, yet the actual limsup of ε log Pε(F) is larger than that prediction. The discrete construction is then “lifted” to a non‑atomic setting by replacing each discrete label with a small compact interval and keeping the cost constant on label rectangles; entropy identities show the continuous problem inherits the discrete behaviour.