Definition of average $\langle \langle \cdot \rangle \rangle$

Question

I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in this paper.

In the paper, $J_{ij}$ are i.i.d random variables, for $1 \le i < j \le N$. This means that their distributions $\mu_{ij} = P\circ J_{ij}^{-1}$ are all the same; maybe I can write $\mu_{ij} = \mu$. Then, the authors define $\langle \langle \cdot \rangle \rangle$ as "the average over the $J_{ij}$'s". What does this mean, exactly? As far as I understand, this should be: $$\langle \langle f\rangle \rangle = \frac{1}{N(N-1)}\sum_{1 \le i < j \le N}\int_{\mathbb{R}}f(x)d\mu_{ij}(x)$$ but since these are identically distributed, this actually becomes: $$\langle \langle f\rangle \rangle = \int_{\mathbb{R}}f(x)d\mu(x).$$ Is this the right definition? Moreover, the paper says that examples of distributions for these $J_{ij}$ random variables are: $$\mu_{ij} = P(dJ_{ij}) = \frac{1}{\sqrt{2\pi}J}e^{-\frac{J_{ij}^{2}}{2J^{2}}}dJ_{ij} \quad \mbox{and} \quad P(dJ_{ij}) = \frac{1}{2}(\delta(J_{ij}-J)+\delta(J_{ij}+J))dJ_{ij}$$ So, silly question: why they explicitly make use of the indices $i,j$ in those distributions if these are identically distributed? It gives the impression that each distribution is different from one another, in my opinion.

Answer 1

There are two classes of random variables, a set of integers $\sigma_i\in\{+1,-1\}$, $i=1,2,\ldots N$, and an $N\times N$ matrix with i.i.d. elements $J_{ij}$, $i,j,=1,2,\ldots N $. The notation $\langle\cdots\rangle$ denotes an averages of the $\sigma_i$ variables at fixed $J_{ij}$ variables, while $\langle\langle\cdots\rangle\rangle$ denotes an average over the $J_{ij}$ variables at fixed $\sigma_i$ variables.

Explicitly, one has the definition

$$\langle\langle f(\{J_{ij}\})\rangle\rangle=\left(\prod_{i,j=1}^N \int P(J_{ij})dJ_{ij}\right) f(\{J_{ij}\}),$$ where $\{J_{ij}\}$ denotes the set of matrix elements and $P(J_{ij})$ is their probability distribution function (the same for each element).

_{Even more explicitly, for $N=2$, $$\langle\langle f(J_{11},J_{12},J_{21},J_{22})\rangle\rangle=\int P(J_{11})dJ_{11} \int P(J_{12})dJ_{12}\int P(J_{21})dJ_{21}\int P(J_{22})dJ_{22} \;f(J_{11},J_{12},J_{21},J_{22}).$$}

Q: Why do the authors explicitly make use of the indices $i,j$ to define the distribution of $J_{ij}$ if these are identically distributed?

Well, they use the symbol $J$ with indices to denote the standard deviation of $J_{ij}$, so they prefer to keep the indices when they define the distribution of a matrix element.