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I think I didn't formulate it well: I don't need the eigenvectors to form a basis, and actually even don't need them to be distinct (the top one and a combination of bottom ones are used to construct a Lyapunov function for some stochastic process, so I don't need all of them). So, rather, it should be "let $v_1,\ldots,v_N$" be some corresponding eigenvectors; then they can be chosen in such a way that (2) holds.
Looks like the distributions of $Y$ and $Z$ only depend on $k$ ($=$ the number of 1-components of $v$). Do you also assume that $k$ is large? And why do you even need $n$?
Thanks a lot for your answer! I've obtained a bit simpler formula, $\mathop{\mathrm{cap}}(A_r)=\frac{2\sin(\pi\phi)}{(1+\phi)\sin\frac{2\pi\phi}{1+\phi}}$ (at least, the asymptotics is the same). See docdro.id/GFTxNxN
