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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 : $$-x_i + \sum_{j=1}^n J_{ij} \phi(x_j) = 0\mbox{ for all }1\leq i \leq n$$ 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?

  • 1
    Have you really solved it for $n=1$ ? What is your result ?2011-07-08
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    In your equation, must this hold for all $i$ independently, or are implicitly summing over all $i$?2011-11-18
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    @AWalker: I assumed that it must hold for each $i$. The equation could be written in vectorial form ${\bf x} = {\bf J} atan({\bf x})$ where $atan(\cdot)$ is applyied to each element. Seems difficult.2011-11-18

1 Answers 1