# Renormalization group map on hierarchical models

I have already addressed this problem on my previous question but I still have trouble understanding Brydges' RG maps on his lecture notes, so I'll try to elaborate my question a little better.

Let $$\Lambda \subset \mathbb{Z}^{d}$$ be finite. We have a positive-definite $$|\Lambda|\times |\Lambda|$$ matrix $$C$$ which can be decomposed as a sum $$C=C_{1}+\cdots+C_{N}$$ where, for each $$j=1,...,N$$, $$C_{j}$$ is again positive-definite. Let $$\varphi:\Omega \to \mathbb{R}^{|\Lambda|}$$ be a random vector $$\varphi = (\varphi(x))_{x\in \Lambda}$$ with distribution $$\mu_{C}$$ being a Gaussian measure on $$\mathbb{R}^{|\Lambda|}$$ associated to $$C$$. Moreover, for each $$j=1,...,N$$, $$\xi_{j}:\Omega \to \mathbb{R}^{|\Lambda|}$$ is a random vector $$\xi = (\xi(x))_{x\in \Lambda}$$ whose distribution $$\mu_{j}$$ is a Gaussian measure associated to $$C_{j}$$. In addition, to each $$j=0,1,...,N$$ we define new random vectors as partial sums $$\varphi_{j} :=\sum_{k>j}^{N}\xi_{k}$$, where $$\varphi_{N}:=0$$ and $$\varphi_{0}:=\varphi$$.

Given $$X\subset \Lambda$$, let $$\mathcal{N}_{j}(X)$$ be the algebra of functions measurable with respect to the $$\sigma$$-algebra generated by $$\{\varphi_{j}(x), x\in X\}$$ and $$\tilde{\mathcal{N}}_{j}(X)$$ is the algebra of functions measurable with respect to the $$\sigma$$-algebra generated by $$\{\xi_{j+1}(x),\varphi_{j+1}(x), x\in X\}$$. If $$X=\Lambda$$, we write $$\mathcal{N}_{j}(\Lambda) \equiv \mathcal{N}_{j}$$ and similarly for $$\tilde{\mathcal{N}}_{j}$$.

Now, take $$F^{\Lambda}$$ to be a function which is $$\tilde{\mathcal{N}}_{j}$$-measurable (equation (2.20) in Brydges notes). Then, Brydges evaluates the expectation: $$\begin{eqnarray} \mathbb{E}_{j+1}F^{\Lambda} \tag{1}\label{1} \end{eqnarray}$$ which should be just the expectation with respect to the measure $$\mu_{j+1}$$. This is where I got in trouble: the measure $$\mu_{j+1}$$ is on $$\mathbb{R}^{|\Lambda|}$$, but as far as I'm concerned, $$F^{\Lambda}$$ is a random variable defined on some underlying probability space, whose $$\sigma$$-algebra is generated by $$\{\xi_{j+1}(x),\varphi_{j+1}(x),x\in\Lambda\}$$. So I don't understand the meaning of (\ref{1}).

My guess is that Brydges is using some sort f isomorphic version of the actual $$F^{\Lambda}$$. In fact, because $$F^{\Lambda}$$ is $$\tilde{\mathcal{N}}_{j}$$-measurable, there exists a function $$f: \mathbb{R}^{2|\Lambda|}\to \mathbb{R}$$ such that $$F^{\Lambda}(\cdot) = f(\xi_{j+1},\varphi_{j+1})(\cdot)$$ and this $$f$$ can be integrated with respect to $$\mu_{j+1}$$ as in (\ref{1}). But I don't know if my guess is correct and, if it is, I don't know how to justify why can $$F^{\Lambda}$$ be "replaced" by $$f$$ in (\ref{1}).

I think Brydges is (tacitly) assuming that $$\Omega=\mathbb{R}^{|\Lambda|}$$. This, of course, is bound to create confusion between elements of $$\mathbb{R}^{|\Lambda|}$$ and random elements of $$\mathbb{R}^{|\Lambda|}$$.