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Denote $\mathcal I_m=\{-m,-m+1,\dots,0,\dots,m-1,m\}$ where $m\geq0$ is an integer.

Pick a uniformly random vectors $$\hat a=(a_n,a_{n-1},\dots,a_1)\in(\mathcal I_{m_1}+\sqrt{-1}\cdot\mathcal I_{m_2})^n$$ $$\hat b=(b_n,b_{n-1},\dots,b_1)\in(\mathcal I_{m_1'}+\sqrt{-1}\cdot\mathcal I_{m_2'})^n$$ ($a_i,b_j$ are uniformly distributed (not gaussian) in a region complex plane) where integers $m_1,m_2,m_1',m_2'$ satisfy $0\leq m_1'\leq m_1$ and $0\leq m_2'\leq m_2$.

What is the probability that $\langle\hat a,\hat b\rangle=0$ holds?

I am not looking for exact answer. I am only looking for sharp asymptotics and approximations.

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  • $\begingroup$ For $m_1, m_2$ fixed, $n \rightarrow \infty$ it should be amenable to a local limit theorem. If interested y0u might browse Petrov, 'sums of i.I.d random variables' $\endgroup$
    – user83457
    Commented Feb 7, 2017 at 15:56

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