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Let $(J_{ij})$ be an $n \times n$ random matrix with i.i.d Gaussian centered coefficients with $\displaystyle \mathbb{E}[J_{ij}^2] = \frac{\sigma^2}{n}$.

Let the random variable $A_n(\sigma)$ defined as the number of real solutions in $\mathbb{R}^n$ of : $$-x_i + \sum_{j=1}^n J_{ij} \phi(x_j) = 0$$ where $\phi(x) = \arctan(x)$.

The question is : what is the law of $A_n(\sigma)$ ? In particular, its expectation ?

I know how to solve "by hand" the case n=1, and n=2, but then it becomes really painful...

Any idea ?

Thank you !

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