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 : $$-x_i + \sum_{j=1}^n J_{ij} \phi(x_j) = 0\mbox{ for all }1\leq i \leq n$$ 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!
Edit: I have a conjecture but I do not know if it is true:
$$\lim_{n \to \infty} \frac{1}{n}\log \mathbb{E}[A_n(\sigma)] = C(\sigma)$$ with $C(\sigma)=0$ for $\sigma <1$ and $C(\sigma)=O((\sigma-1)^2)$ for $\sigma \to 1^+$.
What do you think of this?