A \emph{Perron number} is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any non-negative integer matrix $M$ such that some power of $M$ is strictly positive has a unique positive eigenvector whose eigenvalue is a Perron number. Doug Lind proved the converse: given a Perron number $\lambda$, there exists such a matrix, perhaps in dimension much higher than the degree of $\lambda$. Perron numbers come up frequently in many places, especially in dynamical systems.

Note that for any fixed $C > 1$ and integer $d > 0$, there are only finitely many Perron numbers $\lambda < C$ of degree $< d$, since there is obviously a bound on the discriminant of the minimal polynomial for $\lambda$, so the question is only interesting when a bound goes to infinity. In any particular field, the set of algebraic numbers that are Perron lie in a convex cone in the product of Archimedean places for the field; the central projection of any lattice to the plane $x_1 = 1$ is uniformly distributed in the intersection of the halfspace $x_1 \le C$ with this cone, so as $C \rightarrow \infty$, the distribution of Perron ratios converges to a uniform distribution in the unit disk (a contribution for each complex place) plus a uniform distribution in the interval $[-1,1]$ (for each real place), even for any fixed field, and thus for all fields of bounded degree.

But what happens when $C$ is held bounded and the degree goes to infinity? This seems related to the theory of random matrices, but I don't see any direct translation from things I've heard.

The distribution is decidedly not tending toward a uniform distribution on the disk plus a uniform distribution on the interval. Maybe the artificial bounds on the coefficients cause the high density ring.

To see if it might help separate the effect of the artificial bounds on coefficients, I tried plotting the Perron ratios restricted to $\lambda$ in subintervals. For the degree 21 sample, here is the plot of $\lambda$ by rank order:

alt text http://dl.dropbox.com/u/5390048/PerronPoints21%25281.5%252C2%2529.jpg

alt text http://dl.dropbox.com/u/5390048/PerronPoints21%25283%252C4%2529.jpg

The restriction to an interval seems to make the Perron ratios even more concentrated in dense rings. What is the explanation?

A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any non-negative integer matrix $M$ such that some power of $M$ is strictly positive has a unique positive eigenvector whose eigenvalue is a Perron number. Doug Lind proved the converse: given a Perron number $\lambda$, there exists such a matrix, perhaps in dimension much higher than the degree of $\lambda$. Perron numbers come up frequently in many places, especially in dynamical systems.

Note that for any fixed $C > 1$ and integer $d > 0$, there are only finitely many Perron numbers $\lambda < C$ of degree $< d$, since there is obviously a bound on the discriminant of the minimal polynomial for $\lambda$, so the question is only interesting when a bound goes to infinity. 

In any particular field, the set of algebraic numbers that are Perron lie in a convex cone in the product of Archimedean places of the field. For any lattice, among lattice points with $x_1 < C$ that are within this cone, the projection along lines through the origin to the plane $x_1 = 1$ tends toward the uniform distribution, so as $C \rightarrow \infty$, the distribution of Perron ratios converges to a uniform distribution in the unit disk (with a contribution for each complex place of the field) plus a uniform distribution in the interval $[-1,1]$ (with a contribution for each real place of the field).

But what happens when $C$ is held bounded and the degree goes to infinity? This question seems related to the theory of random matrices, but I don't see any direct translation from things I've heard. Choosing a random Perron number seems very different from choosing a random nonnegative integer matrix.

The distribution decidedly does not appear that it will converge toward a uniform distribution on the disk plus a uniform distribution on the interval. Maybe the artificial bounds on the coefficients cause the higher density ring.

To see if it would help separate what's happening, I tried plotting the Perron ratios restricted to $\lambda$ in subintervals. For the degree 21 sample, here is the plot of $\lambda$ by rank order:

alt text http://dl.dropbox.com/u/5390048/PerronPoints21%281.5%2C2%29.jpg

alt text http://dl.dropbox.com/u/5390048/PerronPoints21%283%2C4%29.jpg

The restriction to an interval seems to concentrate the absolute values of Perron ratios even more. The angular distribution looks like it converges to the uniform distribution on a circle plus point masses at $0$ and $\pi$.

Is there an explanation for the distribution of radii? Any guesses for what it is?