# Why the renormalization constant of the regularized $2D$ noise diverges as a logarithm

Motivation and overview

I'm trying to understand the theory of regularity structures and in particular, following this paper I'm looking to the $$\Phi^4_d$$ model on the $$d$$-dimensional torus $$T^d$$ i.e. the SPDE $$\partial_t \varphi(z)=\Delta \varphi(z) + \varphi^3(z) +\xi(z)$$ where $$z=(t,x_1,\cdots, x_d)$$ and $$\xi(z)$$ is the space time white noise.

If we call $$K(t,x)$$ the heat kernel on the torus i.e. $$K(t,x):= \frac{1}{(4\pi t)^{\frac{d}{2}}}\sum_{k \in 2\pi \mathbb{Z}^d}e^{-\frac{(x-k)^2}{t}}$$ we can define $$Z(t,x):=\int_{} K(t-s,x-y) \xi(dz)$$

We define $$\rho_\delta$$ the smooting kernel on scale $$\delta$$ i.e. a mollifiers such that $$|K * \rho_\delta| \lesssim \frac{1}{(\|z\|_s+\delta)^d}$$ where $$\|z\|_s:=\sqrt{t}+\sum_{j=1}^d x_i$$ is the scaled norm and $$*$$ is the space-time convolution.

We can (at least formally) define $$Z_\delta= Z * \rho_\delta$$ At this point it is possible to calculate $$Z_\delta^3$$ as an iterated Ito integral and we have $$Z^3_\delta(y)= \int \prod_{j=1}^3 K* \rho_\delta(y-z_j) \xi(dz_1) \xi(dz_2) \xi(dz_3) + 3 \int \left(\int (K* \rho_\delta(y-z) )^2 dz\right)K *\rho_\delta(y-z_1) \xi(dz_1)$$ My doubt is about the second integral.

My problem

The second integral can be rewritten as $$3 C_\delta Z_\delta$$ where $$C_\delta= \int_{(0,s) \times T^d} (K* \rho_\delta)^2 dz$$ In the linked paper it is stated that this constant, when the space dimension is $$2$$ and $$\delta \to 0$$ diverges logarithmically in delta. I don't understand how it is possible given the bound $$|K * \rho_\delta| \lesssim \frac{1}{(\|z\|_s+\delta)^2}$$ If instead $$d=3$$ the same integral should diverge as $$\frac{1}{\delta}$$.

TL DR

why is $$\lim_{\delta \to 0} \int_{(0,s) \times T^2} (K* \rho_\delta)^2 dz \approx \lim_{\delta \to 0} \log(\delta^{-1})$$