Consider a random matrix with mean-zero, independent elements with variance defined column-wise: $M_{ij}$ with $$\mathbb{E}[M_{ij}^2]=\frac{\sigma_j^2}{N}$$
and assume that the average the variances is $\sigma^2$, say via a doubly-stochastic process with ($\mathbb{E}[\sigma_j^2]=\sigma^2$).
I know from the literature on chaotic dynamics in recurrent neural networks with asymmetric connections that the spectral radius of $M_{ij}$ is $\sigma$. (The curious can see the context below)
I would like to find a proof or a citation in the random matrix literature. Any leads?
(The same question went unanswered on this forum 7 years ago: Random matrix with non-identical variances)
Alternate Formulation: Define $\mathbf{M}=\mathbf{J}\boldsymbol{\Sigma}$, as the matrix product of a random Gaussian i.i.d. matrix, $J_{ij}~\mathcal{N}(0,\frac{\sigma^2}{N})$, times a diagonal non-negative matrix, $\boldsymbol{\Sigma}$, with mean element equal $1$. If we consider $\boldsymbol{\Sigma}$ to be deterministic with some profile, does that help?
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The situation comes up often in the non-linear dynamics of neural networks with random asymmetric connections. In particular consider the dynamics defined as
$$\dot{h_i} = -h_i + \sum_j J_{ij} \phi (h_j)$$ where J_{ij} is mean-zero Gaussian i.i.d. with variance $\frac{g^2}{N}$, and $\phi$ is a sigmoidal non-linearity, say for concreteness $\phi(h)=\tanh(h)$
Now, for $g>1$ one can show that there are non-zero fixed points, $h_i^*$. And when one comes to study the local stability of these fixed points one writes the linearized dynamics: $$\dot{\delta h}_i=-\delta h_i + \sum_j M_{ij} \delta h_j$$ where $$M_{ij} \equiv J_{ij} \phi'(h_j^*)$$
The fixed point itself $h_j^*$ is assumed Gaussian and independent of $J_{ij}$, and therefore $M_{ij}$ has the form described above with $$\mathbb{E}[M_{ij}^2] = \frac{g^2}{N}\mathbb{E}[(\phi'(h_j))^2]$$.
Such a fixed point is shown in the literature to be unstable when $g\sqrt{\mathbb{E}[(\phi'(h_j))^2]}>1$, i.e. the spectral radius of $M_{ij}$ is given by $g\sqrt{\mathbb{E}[(\phi'(h_j))^2]}$.
See Kadmon, Sompolinsky PRX 2015 among many others.
