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Are there results/techniques pertaining to the analysis of squares of random matrices ?

More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with equal probability. Now if we want to analyze for any $u,v \in \{-1,1\}^n$, we can make a case for concentration of the value of $u^TAv$ using chernoff bound arguements. However suppose now we want to analyze the value of $u^TA^2v$. This time due to a lot of dependencies among the variables a chernoff type arguement becomes difficult or at least I cannot see it straightaway. Could someone point me to an analysis for this scenario ?

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I am a fan of these matrices as connecting physics with number theory, but honestly I saw yet no literature on their squares. Interesting! – al-Hwarizmi Jun 14 '13 at 17:48

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