LKJ(1) is not supposed to yield uniform margins for the correlations. It is â€śflatâ€ť in the sense of placing equal prior density over every possible correlation matrix, but the space of allowable correlation matrices contains more volume where correlations are low than where correlations are high.

Despite being the identity over correlation matrices, the marginal distribution over the entries in that matrix (i.e., the correlations) is not uniform between -1 and 1. Rather, it concentrates around zero as the dimensionality increases due to the complex constraints.

I missed this statement. So, if I understand it correctly, the requirement of the positive semidefiniteness on a correlation matrix prevent it from being uniform.

Exactly. Itâ€™s also possible to gain a little bit of intuition about this by thinking about the meaning of correlations. If variates \theta_1 and \theta_2 are very strongly correlated, then if \theta_3 is strongly correlated with one it must be strongly correlated with both. So there are a bunch of disallowed matrices whenever any correlations are high. Thatâ€™s another way (beyond the relatively abstract concept of positive semidefiniteness) to understand why @spinkneyâ€™s figure that @nhuurre linked above pinches down to tiny points in the corners. Once two correlations are 1 (meaning \theta_1 \propto \theta_2 and \theta_1 \propto \theta_3), the third correlation is required to be 1.