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The distribution of the bulk of the spectrum is an example of the circular law. For the model you selected (where each entry is uniformly chosen at random from an interval), the law was first proven by Bai (at least in the case where the entries are normalised to have mean zero), building upon previous work of Girko; the non-central case (non-zero mean) was recently established by Chafai.

The non-central case is a rank one perturbation of the central case (by the matrix all of whose entries are all equal to the mean) and should therefore cause one exceptional eigenvalue.
(In the central case it can be shown that the spectral radius is close to the radius of the disk, so there are basically no exceptional eigenvalues.) If the mean of each entry is $\mu$, then the rank one perturbation has an eigenvalue at $\mu n$, so one expects the exceptional eigenvalue to linger near this number. In your case, $\mu = 0.05$ and $n = 301$, so the exceptional eigenvalue should linger near $15.05$.

This has however not yet been proven; Chafai's result only shows that the number of exceptional eigenvalues far from the disk is almost surely o(n) for an n x n matrix. Actually this

There is not a bad open problem to paper of Silverstein which makes the above heuristics precise; see also the earlier work on (showing that there is exactly one exceptional eigenvalue, and pinning down its distribution), as it is almost within reach of existing technologyAndrew.

There is quite a bit of recent literature on the circular law, see for instance this survey by Van Vu and myself, or my lecture notes on this topic.

This

The distribution of the bulk of the spectrum is an example of the circular law. For the model you selected (where each entry is uniformly chosen at random from an interval), the law was first proven by Bai (at least in the case where the entries are normalised to have mean zero), building upon previous work of Girko; the non-central case (non-zero mean) was recently established by Chafai.

The non-central case is a rank one perturbation of the central case (by the matrix all of whose entries are equal to the mean) and should therefore cause one exceptional eigenvalue.
(In the central case it can be shown that the spectral radius is close to the radius of the disk, so there are basically no exceptional eigenvalues.) If the mean of each entry is $\mu$, then the rank one perturbation has an eigenvalue at $\mu n$, so one expects the exceptional eigenvalue to linger near this number. In your case, $\mu = 0.05$ and $n = 301$, so the exceptional eigenvalue should linger near $15.05$.

This has however not yet been proven; Chafai's result only shows that the number of exceptional eigenvalues far from the disk is almost surely o(n) for an n x n matrix. Actually this is not a bad open problem to work on (showing that there is exactly one exceptional eigenvalue, and pinning down its distribution), as it is almost within reach of existing technology.

There is quite a bit of recent literature on the circular law, see for instance this survey by Van Vu and myself, or my lecture notes on this topic.

1

This is an example of the circular law. For the model you selected (where each entry is uniformly chosen at random from an interval), the law was first proven by Bai (at least in the case where the entries are normalised to have mean zero), building upon previous work of Girko; the non-central case (non-zero mean) was recently established by Chafai.

The non-central case is a rank one perturbation of the central case and should therefore cause one exceptional eigenvalue. (In the central case it can be shown that the spectral radius is close to the radius of the disk, so there are basically no exceptional eigenvalues.) This has however not yet been proven; Chafai's result only shows that the number of exceptional eigenvalues far from the disk is almost surely o(n) for an n x n matrix. Actually this is not a bad open problem to work on (showing that there is exactly one exceptional eigenvalue, and pinning down its distribution), as it is almost within reach of existing technology.

There is quite a bit of recent literature on the circular law, see for instance this survey by Van Vu and myself, or my lecture notes on this topic.