BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional $$\Phi ^3_d$$ model

We consider stochastic PDEs on the d-dimensional torus with fractional Laplacian of parameter \(\rho \in (0,2]\), quadratic nonlinearity and driven by space-time white noise. These equations are known to be locally subcritical, and thus amenable to the theory of regularity structures, if and only if \(\rho > d/3\). Using a series of recent results by the second named author, A. Chandra, I. Chevyrev, M. Hairer and L. Zambotti, we obtain precise asymptotics on the renormalisation counterterms as the mollification parameter \(\varepsilon \) becomes small and \(\rho \) approaches its critical value. In particular, we show that the counterterms behave like a negative power of \(\varepsilon \) if \(\varepsilon \) is superexponentially small in \((\rho -d/3)\), and are otherwise of order \(\log (\varepsilon ^{-1})\). This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.