Gaussian comparison matrix in the moment argument

PDF p. 34: Proof of Proposition 6.3, Gaussian comparison paragraph

Assume the Axiom of Choice, as required by the adopted normal-law construction. Let $n\ge 1$ be an integer and let $G=(G_{ij}{)}_{1\le i,j\le n}$ be a real random matrix constructed as follows. The entries $G_{ij}$ with $1\le i\le j\le n$ are mutually independent and each has the standard Gaussian law $N(0,1)$, whose density is $(2\pi {)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}2}$. Set $G_{ji}=G_{ij}$ for $i<j$. Mutual independence means that, for every choice of Borel sets, the probability of the intersection of the corresponding coordinate events is the product of their probabilities.

Thus $G$ is symmetric, including its diagonal entries in the independent triangular family. The mirrored entries $G_{ij}$ and $G_{ji}$ for $i<j$ are the same variable, so the full array is not a mutually independent family when $n\ge 2$. Here $G$ denotes the comparison matrix, not the interaction graph.

The adopted standard-normal definition supplies the probability measure $N(0,1)$ under Choice (The Axiom of Choice, Standard normal and normal laws). Take one copy of this probability space for each of the finitely many pairs $i\le j$. Repeated application of the sigma-finite product-measure theorem gives a probability measure on their finite product: each factor is sigma-finite because its total mass is one, and the rectangle formula gives the product total mass one (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique). The coordinate projections are measurable by the product sigma-algebra definition (The product sigma-algebra and its finite iterates), hence are real random variables on this common probability space (Random elements and real random variables); the rectangle formula gives each projection the standard Gaussian law and gives the displayed mutual independence condition, including arbitrary subfamilies by taking the remaining Borel sets to be $\mathbb{R}$ (Independent random elements). Assigning each lower-triangular entry its reflected coordinate therefore constructs the stated real symmetric matrix. Its matrix action and the meaning of symmetry are as in Finite vertex Hilbert space in the article convention.

The Gaussian-entry moments supplier applies to precisely this triangular construction; it shows that a mirrored pair is nonconstant and cannot be independent of itself (Gaussian entry moments and triangular independence). The diagonal convention above makes explicit the literal standard-Gaussian-entry reading used here; no operator-norm estimate follows merely from this definition.

The article introduces a symmetric matrix with i.i.d. standard Gaussian entries in the proof of Proposition 6.3. The triangular interpretation above makes the independence convention compatible with symmetry.