Correlation without Causation I know the famous expression "correlation does not imply causation". In a DAG, this situation might look like
$$
X \leftarrow U \rightarrow Y
$$
Here even though $X$ and $Y$ are not causally related, the presence of confounder $U$ induces a correlation between them.
I also know that two variables that are causally related can be uncorrelated, as correlation is a linear measure of association. For example, the correlation between $X$ and $Y$ with $Y = X^2$ is $0$
In the context of counterfactual formal causal reasoning, my question is: If there is no unblocked path between $X$ and $Y$, would be ever expect there to be non-zero correlation between $X$ and $Y$ in the infinite sample limit? I know that in finite samples, spurious correlations can appear simply due to chance, but asymptotically, is it possible that if there is no open causal path between two variables, that we can expect a non-zero correlation, or really, any measure of association to be positive, between them?
In short: can two d-separated variables have an expected non-zero correlation? Answers should use counterfactual causal reasoning formalisms.


In short: can two d-separated variables have an expected non-zero
correlation?

No, it is not possible.
More precisely: d-separation warrant us that, in a DAG $G$, if two variable $X$ and $Y$ are d-separated by a set of variables $Z$ it is implied that $X$ and $Y$ are independent conditional on $Z$. Note that $Z$ can be the empty set too.
Now, you speak about "correlation" and not "conditional correlation", however you speak about d-speration too. From that I suppose that the two d-separated variables that you use are so for $Z$=empty set.
Therefore, no correlation nor any kind of statistical association can appear in the population.
For example in your DAG
$$
X \leftarrow U \rightarrow Y
$$
$X$ and $Y$ are d-separated given $U$
Moreover you write

For example, the correlation between $X$ and $Y$ with $Y = X^2$ is $0$

I guess the idea in your mind but this statement is not true in general. Indeed if $X$ have distribution $U[0,1]$ this correlation is $>0$.
