Large Deviations of the $Φ^4_3$ Measure via Stochastic Quantisation

Abstract

The $Φ^4_3$ measure is one of the easiest non-trivial examples of a Euclidean quantum field theory (EQFT) whose rigorous construction in the 1970's has been one of the celebrated achievements of constructive quantum field theory. In recent years, progress in the field of singular stochastic PDEs, initiated by the theory of regularity structures, has allowed for a new construction of the $Φ^4_3$ EQFT as the invariant measure of a previously ill-posed Langevin dynamics, a strategy originally proposed by Parisi and Wu ('81) under the name stochastic quantisation. We apply the same methodology to obtain a large deviation principle (LDP) for the family of periodic $Φ^4_3$ measures at varying temperature. In addition, we show that the rate functional of the LDP and the $Φ^4_3$ action functional coincide up to a constant. We wish to highlight that while our main result had previously been obtained by Barashkov (2022), the main focus of this work is on the approach through the stochastic quantisation equation.