Gaussian concentration inequalities are the workhorse of high-dimensional probability. On finite product spaces, the standard route is clean: define a metric, derive a transportation-cost inequality via duality with entropy, and concentration follows.
On infinite product spaces—lattice systems with infinitely many sites—this route breaks. Chazottes, Collet, and Redig show that the natural metrics on infinite lattice configurations don't produce the dual structure needed for transportation-cost inequalities. The obstruction is genuine, not technical: the topology of the infinite product is too rich for the metric to control.
Their bypass: work in every finite truncation simultaneously. They prove that the integral probability metric and the coupling functional coincide in finite volume—a fact that fails in the infinite limit. Then they show that Marton's coupling inequality holding across all finite volumes is equivalent to Gaussian concentration in the infinite product.
The result is philosophically sharp. The infinite-dimensional property doesn't arise as a limit of a single finite-dimensional construction. Instead, it emerges from the consistency of a family of finite constructions. You never have a metric that works directly on the infinite space. But by requiring every finite window to satisfy the inequality, you get the infinite result anyway—through a fundamentally different logical route than taking a limit.
When direct construction fails in the infinite case, requiring all finite approximations to satisfy a condition can be strictly more powerful than trying to take a limit of any single one. The infinite is not the limit of the finite; it's the coherence of the finite.