  Motivated by numerical challenges in first-order mean field games (MFGs) and
the weak noise theory for the Kardar-Parisi-Zhang equation, we consider the
problem of vanishing viscosity approximations for MFGs. We provide the first
results on the convergence rate to the vanishing viscosity limit in mean field
games, with a focus on the dimension dependence of the rate exponent. Two cases
are studied: MFGs with a local coupling and those with a nonlocal, regularizing
coupling. In the former case, we use a duality approach and our results suggest
that there may be a phase transition in the dimension dependence of vanishing
viscosity approximations in terms of the growth of the Hamiltonian and the
local coupling. In the latter case, we rely on the regularity analysis of the
solution, and derive a faster rate compared to MFGs with a local coupling. A
list of open problems are presented.
