Low temperature expansion for the Euclidean Φ⁴₂-measure

We study asymptotic expansions of the Euclidean Φ⁴₂-measure in the low-temperature regime. In particular, this extends the asymptotic expansions of Gaussian function space integrals developed in Schilder (1966) and Ellis and Rosen (1982) to the singular setting, where the field is no longer a function, but just a distribution. As a consequence, we deduce limit theorems, specifically the law of large numbers and the central limit theorem for the Φ⁴₂-measure in the low-temperature limit.