Perturbation theory for the ${\mathrm{\Phi }}_3^4$ measure, revisited with Hopf algebras

We give a relatively short, almost self-contained proof of the fact that the partition function of the suitably renormalised ${\mathrm{\Phi }}_3^4$ measure admits an asymptotic expansion, the coefficients of which converge as the ultraviolet cut-off is removed. We also examine the question of Borel summability of the asymptotic series. The proofs are based on Wiener chaos expansions, Hopf-algebraic methods, and bounds on the value of Feynman diagrams obtained through BPHZ renormalisation.