a simple answer follows if $A$ is a unitary $n\times n$ matrix; then
$$P(u^H Av)=P(U_{11})\propto(1-|U_{11}|^2)^{n-2}\rightarrow e^{-n|U_{11}|^2}\;{\rm for}\;\; n\gg 1$$
Carlo Beenakker
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