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?


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.


1 Answer 1


I found the calculation in Rajan, Abbott PRL 2006. They follow Sommers et al PRL 1988 and formulate the Green's fn (Stieltjes transform) as a 2d field derived from a "potential", from which one can find the eigenvalue density as a Gaussian integral. They perform the calculation for the situation in which each column has one of two possible variances.

It is even more straightforward following Ahmadian et al PRE 2015, equation 2.13.


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