TITLE: Perron number distribution
QUESTION [65 upvotes]: 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.
My question:

What is the limiting distribution of Galois conjugates of Perron numbers $\lambda$ in
  some bounded interval, as the degree goes to infinity?

I'm particularly interested in looking at the limit as the length of the interval goes to
0.  One way to normalize this is to look at the ratio $\lambda^g/\lambda$, as $\lambda^g$
ranges over the Galois conjugates. Let's call these numbers \emph{Perron ratios}.
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.
I tried some crude experiments, by looking at randomly-chosen polynomials of a fixed degree whose coefficients are integers in some fixed range except for the coefficient of $x^d$
which is $1$, selecting from those the irreducible polynomials whose largest real root is Perron.  This is not the same as selecting a random Perron number of the given degree
in an interval.  I don't know any reasonable way to do the latter except for small enough $d$ and $C$ that one could presumably find them by exhaustive search.
Anyway, here are some samples from what I actually tried.
First, from among the 16,807 fifth degree polynomials with coefficients in the range -3 to 3, there are $3,361$ that define a Perron number.  Here is the plot of the Perron ratios:
alt text http://dl.dropbox.com/u/5390048/PerronPoints5%2C3.jpg
Here are the results of a sample of 20,000 degree 21 polynomials with coefficients
between -5 and 5.  Of this sample, 5,932 defined Perron numbers:
alt text http://dl.dropbox.com/u/5390048/PerronPoints21.jpg
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.

Are there good, natural distributions for selecting random integer polynomials? Is there a
  way to do it without unduly prejudicing the distribution of roots?

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/CDF21.jpg
(If you rescale the $x$ axis to range from $0$ to $1$ and interchange $x$ and $y$ axes,
this becomes the plot of the sample cumulative distribution function of $\lambda$.)
Here are the plots of the Perron ratios restricted to the intervals $1.5 < \lambda  < 2$
and $3 < \lambda < 4$:
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?

REPLY [11 votes]: There are many contexts in which one can establish that a family of polynomials, or even a single polynomial, has roots which are distributed according to some measure which approximates the uniform measure on the unit circle.
Let $f(x) = a_N x^N + \ldots + a_1 x + a_0$ be a polynomial with real coefficients, and assume that $a_N \ne 0$. Consider the quantity:
$$L_N(f) = \log \left( \sum |a_k| \right) - \frac{1}{2} \log |a_N| - \frac{1}{2} \log |a_0|.$$
A theorem of Hughes and Nikeghbali (The zeros of random polynomials cluster uniformly near the unit circle, Compositio 144 (1998)) says (informally) that if $L_N(f)$ is small compared to $N$, then the roots are distributed uniformly along the unit circle. An ingredient of their result is a theorem of Erdos-Turan (On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950)) which (under the same hypothesis) shows that the arguments of the roots are distributed uniformly. Using this, we can answer:

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

If one takes the coefficients to lie in some bounded range, say $[-5,5]$, then $L_N(f)$ has order $O(\log(N))$, which is certainly $o(N)$. Hence the random polynomials generated in this way have roots which are distributing in the unit circle. On the other hand, the polynomials being chosen are those which have a unique largest root of size at most five and usually around $5$, and hence the renormalized roots will appear to cluster around the circle of radius $1/5$. The clustering becomes more pronounced when one restricts to polynomials where the largest root lies in $[5,5-\lambda]$ for smaller $\lambda$ (Hence Thurston's graph with more restricted intervals are more "ring-like." (There is no point mass along the real axis by the theorem of Erdos-Turan.)

Are there good, natural distributions for selecting random integer polynomials? Is there a way to do it without unduly prejudicing the distribution of roots?

Given a large dimensional probability space, a natural way to generate random elements according to the distribution is to use a random walk Metropolis-Hastings algorithm. Concretely, one can start with a random Perron polynomial with largest root $< 5$, and then perturb the coefficients according to some distribution (say a normal distribution with small variance). If the new polynomial is also Perron with largest root $< 5$, choose this polynomial, or else keep the same polynomial. This Markov process should --- under suitable conditions --- generate random polynomials (in the aggregate) according to the required distribution. For example, let's run this algorithm and compare it to the roots of random monic polynomials with coefficients in $[-5,5]$ which have a unique largest root. The roots of polynomials with coefficients in $[-5,5]$ are illustrated in the first graph, and the result of the Metropolis-Hastings algorithm for all Perron polynomials with largest root less than $5$ is given in the second graph:

Notice that the roots of the first graph cluster around the unit circle, as explained above. However, the roots of the second graph are clustering around the boundary. Indeed, this turns out to reflect reality (see the theorem below).

What is the limiting distribution of Galois conjugates of Perron numbers $\lambda$ in some bounded interval, as the degree goes to infinity?

Let's first consider the easier problem of describing "real Perron polynomials" --- that is, monic polynomials with real coefficients which have a unique largest root $\le r$ (necessarily real). The actual Perron numbers form integral lattice points in this space (although not all lattice points, just the ones corresponding to irreducible polynomials; however, the reducible lattice points form a thin subset). If one fixes the degree $N$ and increases the radius $r$, then (because the regions involved are suitable nice) the lattice points are distributed more or less uniformly inside the space (Davenport's Lemma). However, if one fixes $r$ and lets $N \rightarrow \infty$, it is no longer so clear whether the distribution of lattice points can be approximated by the corresponding real region. Studying lattice points in non-convex regions (or even convex ones) often leads to pretty thorny number  theoretic issues, but one can at least hope (and compare with experiment) that the real geometry gives some indication of the truth of the matter. To this end, one has the following:
Theorem: Fix a real positive integer $r > 0$. Let $\Omega_N$ denote the space of monic polynomials with real coefficients which have a real root $\alpha$ such that $r > \alpha > |\sigma \alpha|$ for all conjugates $\sigma \alpha \ne \alpha$. Then, as $N \rightarrow \infty$, the roots of a random polynomial in $\Omega_N$ are distributed uniformly along the circle $|z| = r$.
The proof of this theorem uses the theorem of Hughes and Nikeghbali mentioned above. The point is that one has to obtain estimates of how the quantities $|a_i|$ vary over the space $\Omega_N$, which reduces, in the end, to evaluating and/or estimating various integrals over $\Omega_N$. For example, consider the example above of degree $21$ polynomials with a largest root $< 5$. The model given by the OP choose $a_{21} \in [-5,5]$. It turns out, however, that the expected value of the constant term $a_{21}$ over $\Omega_{21}$ (with $r = 5$) is
$$ \frac{3^2 \cdot 5^{21} \cdot 7 \cdot 13 \cdot 17 \cdot 19}{2^{17} \cdot 11} \sim 8.748 \times 10^{13}.$$
This is pretty big compared to $[-5,5]$! 
Other statistics
There are a few other interesting probabilities one can compute using integrals over spaces like $\Omega_N$ and related spaces. For example, one can consider all monic polynomials of degree $2N$ with the property that their roots all have absolute value at most one. This defines a compact region of (the coefficient space) $\mathbf{R}^N$, and so it has a natural uniform measure. Then the probability that a random such polynomial has no real roots in the interval $[-1,1]$ (equivalently, is positive on $[-1,1]$, equivalently, has no roots on all of $\mathbf{R}$) is equal to
$$\sim  \frac{2C}{\sqrt{2\pi} \cdot (2N)^{3/8}},$$
where the constant $C$ is equal to
$$C = 2^{-1/24} e^{- 3/2 \cdot \zeta'(-1)} =  1.24514 \ldots.$$
Curiously enough, there is a theorem of Dembo, Poonen, Shao, and Zeitouni that a random polynomial (in the more usual sense) whose coefficients are chosen
with (say) identical normal distributions with zero mean is positive in $[-\infty,\infty]$
 with probability $N^{-b + o(1)}$ and positive in $[-1,1]$ with probability $N^{-b/2 + o(1)}$
 for some universal constant~$b/2$, which 
 they estimate be $0.38 \pm  0.015$. On the other hand, 
the exponent occurring above is $3/8 = 0.375$. Is there any direct relationship between
these theorems? For example, does this suggest that $b = 3/4$? (The result of DPSZ actually holds for a very wide range of distributions with the same undetermined $b$, but it does not apply to our very rigid context.)
Further Remarks
There's some further analysis one can make of the space $\Omega_{N}$, or more generally the space of all monic polynomials with real coefficients whose roots have absolute value at most $r$. For example, one can show that a "random" polynomial of degree $N$, subject to the constraint that all its roots have absolute value at most $r$, will be a Perron polynomial (i.e. have a unique largest root) with probability exactly $1/N$ if $N$ is odd and $1/(N+1)$ if $N$ is even. Some of this analysis will appear in my thesis (I will add a link here when it is written!). As my advisor said, "Any question that Thurston asks is probably worth thinking about."