Exact Exponents for Concentration and Isoperimetry in Product Polish Spaces

In this paper, we derive variational formulas for the asymptotic exponents of the concentration and isoperimetric functions in the product Polish probability space. These formulas are expressed in terms of relative entropies (which are from information theory) and optimal transport cost functionals (which are from optimal transport theory). Our results verify an intimate connection among information theory, optimal transport, and concentration of measure or isoperimetric inequalities. In the concentration regime, the corresponding variational formula is in fact a dimension-free bound on the exponent of the concentration function. The proofs in this paper are based on information-theoretic and optimal transport techniques. Our results generalize Alon, Boppana, and Spencer's in \cite{alon1998asymptotic}, Gozlan and L\'eonard's \cite{gozlan2007large}, and Ahlswede and Zhang's in \cite{ahlswede1999asymptotical}.