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for a precise formulation of statement 2, with a proof, see page 2 of these lecture notes:
$${\rm Prob}\,\left(|\cos\theta|<\sqrt{\frac{\log d}{d}}\right)>1-\frac{1}{d},$$ where $\theta$ is the angle between two randomly chosen unit vectors in $d$ dimensions.
_{notice a typo at the top of the page in those lecture notes, corrected at the bottom, the square root should extend over the entire fraction}