Annotated Bibliography
The story begins with
The range of multiplicative functions on $\mathbb C[x],
\mathbb R[x]$ and $\mathbb Z[x]$. Proc. London
Math. Soc. Vol. 96(3): 697-737, 2008 · doi:
10.1112/plms/pdm037 · arXiv:
math.NT/0509591.
While this wasn't my first publication, it is the beginning of the
journey which got me to where I am today. This paper is a somewhat
abbreviated version of my PhD thesis. My advisor (Jeff Vaaler) and
one of his previous students (Shey-Jay Chern) had computed the volume
of a certain set of coefficient vectors of polynomials to produce an
asymptotic for the number of integer polynomials with fixed degree and
bounded, but large height (a height is simply a measure of complexity
of polynomials or other arithmetic objects; in this case the pertinent
height is a quantity called Mahler measure). Their computation
revealed the surprising fact that the volumes of interest were
rational numbers which could be expressed as products of simpler
rational numbers. The simplicity of this result seemed orthogonal to
its lengthy and highly technical proof. For my thesis, I was tasked
to find a similar formula for the related set of reciprocal
polynomials (these are polynomials whose coefficient vectors read the
same forwards as backwards, and they play a special role in the study
of Mahler measure). I ultimately did this, but my own personal goal
was to explain the simple product formulation arrived at previously by
my advisor.
The final explanation was that Chern and Vaaler's volume, as
well as my reciprocal analog, could both be expressed as the Pfaffian
of related antisymmetric matrices. In this paper, I generalize this
product formulation to volumes of polynomials formed from other
heights besides Mahler measure and its reciprocal cousin, and explain
the general Pfaffian mechanism responsible for it.
Stepping backwards in time a bit, we come to my first published
paper
The distribution of Mahler's measures of
reciprocal polynomials. Int. J. Math. Math.
Sci Vol. 52: 2773-2786, 2004 ·
doi:10.1155/S0161171204312469 · arXiv:
math.NT/0311255.
This paper is similar to the previous, except that the central result
here is volume calculations for coefficient vectors of complex
polynomials with bounded height, whereas the previous paper dealt
primarily with real polynomials. In this paper, I describe the
relevant volume of polynomials in terms of the determinant of a matrix
of inner products associated to the relevant height. The central
technical achievement here was the rediscovery of an 1883 formula of
Andreief (which according to deBruijn was 'almost certainly' known by
Cauchy) on integrals whose integrands are determinants. This
determinantal formula predated the Pfaffian formulation in
the The range of multiplicative functions on $\mathbb C[x],
\mathbb R[x]$ and $\mathbb Z[x]$ (where the
inner product is replaced by a certain
skew-symmetric inner product).
The next paper in this line of research is
Averages over Ginibre's ensemble of random real
matrices; Int. Math. Res. Not.
Vol. 2007: 1-15, 2007 · Article ID:rnm015,
doi:10.1093/imrn/rnm015 · arXiv: math-ph/0605006.
This paper was my first foray into random matrix theory, and
(arguably) contains the central observation which lead to the
solvability of Ginibre's ensemble of real random matrices. Ginibre's
ensemble consists of $N \times N$ matrices with iid standard normal
entries. Presented here is the calculation of the partition function
of this ensemble in terms of the Pfaffian of a square matrix formed
from a skew-symmetric inner product. In fact, this calculation is
essentially identical to the volume calculation presented in my thesis
(and at the time I wrote this paper, I saw it as a cheap way to get
another publication on my CV). The main idea at for both the volume
calculation for polynomials of bounded height, and the determination
of the partition function for Ginibre's real ensemble is that, after a
change of variables, both can be realized as a sum of integrals whose
integrands contain factors of the absolute value of a Vandermonde
determinant. The sum is over all possible numbers of real
roots/eigenvalues. This decomposition into subdomains was the central
stumbling block to the solvability of Ginibre's real ensemble. And,
were it not for Chern and Vaaler's original observation of the product
formulation for their volume calculation, Ginibre's ensemble might
still be unsolved.
I mailed the Averages paper to several people whom I thought might be
interested. One of these people was Brian Rider, which eventually got
me a (second) postdoc at Colorado (my first postdoc was in Vancouver
with Peter Borwein and David Boyd). I also mailed the paper to Percy
Deift who happened to be at Caltech for the semester. Percy and
Alexei Borodin responded by invited me to visit them to explain my
work. This led to the following publication (and it's precursor
manuscript)
(with Alexei Borodin). The Ginibre ensemble of
real random matrices and its scaling limits;
Comm. Math. Phys.
Vol. 291: 177-224, 2009 · doi:
10.1007/s00220-009-0874-5 ·
arXiv: math-ph/arxiv:0805.2986;
(with Alexei Borodin). Correlation functions of
asymmetric real matrices; July 2007 ·
arXiv: math-ph/arxiv:0706.2670.
I view this work as something of a second PhD thesis;
this one in random matrix theory. Alexei and Percy convinced me that
the the Pfaffian formulation for the partition function must lead to a
Pfaffian point process on the eigenvalues.
As before, the complexity introduced by the variable number of real
eigenvalues presented the central algebraic complication.
Historically, people had looked primarily at the correlations
restricted to one of the subdomains formed by specifying the number of
real eigenvalues (and then integrating out some number of real and
some number of complex conjugate pairs of eigenvalues), thinking that
this must simplify the problem. However, the situation is actually
simpler when one sums these 'partial correlation functions' over all
possible subdomains. This was the central algebraic observation
presented in the above unpublished manuscript, and it was done by
applying Eric Rains' Pfaffian version of the $\det(I + AB) = \det(I +
BA)$ trick (introduced in this context by Craig Tracy and Harold
Widom) to the the Pfaffian formula for the partition function for
Ginibre's real ensemble.
I am convinced this is the 'right' way to derive the correlation
functions for $\beta=1$ ensembles.
Besides the derivation of the Pfaffian point process, the paper with
Alexei also centered on the scaling limits of the matrix kernel(s).
This was a very busy time, since there were a number of competing
research groups vying for some level of credit for solving Ginibre's
real ensemble. In fact, once I produced the Pfaffian version for the
partition function in the Averages paper, Peter Forrester and
Taro Nagao quickly published the skew-orthogonal polynomials necessary
to simplify the matrix kernel for the ensemble. This was useful,
since it allowed Alexei and I to quickly write the entries in the
matrix kernel(s) in terms of the partial sums of the exponential
function. (It is worth reflecting on this fact; skew-orthogonal
polynomials do not in general, satisfy a Christoffel-Darboux type
relation which would allow the kernel asymptotics to easily be derived from
the asymptotics of the polynomials. Somewhat miraculously, however,
the entries of the matrix kernel for Ginibre's real ensemble were
explicitly summable in terms of partial sums of the exponential and
hyperbolic trigonometric functions, all of whose asymptotics had been
well-studied). It should be remarked that there are actually several
matrix kernels for this ensemble encoding information about the
real/real, real/complex and complex/complex interactions between
eigenvalues, and each of these has a different scaling limit in the
bulk and at the edge. So, while the analysis was simplified by
relying on previously known asymptotics, there was still a lot of work
to do.
Alexei and I initially did the algebraic steps necessary to analyze
Ginibre's real ensemble for even square matrices. The odd case,
and indeed the odd case for all $\beta=1$ ensembles, is more complicated
due to the fundamental fact that Pfaffians are only defined for even
square (antisymmetric) matrices. The following paper contains a
detailed derivation of the correlations for the odd case using Rains'
Pfaffian identity and Tracy and Widom's basic outline.
Correlation functions for $\beta$=1
ensembles of matrices of odd
size; J. Stat. Phys. Vol. 136: 17-33, 2009
· doi: 10.1007/s10955-009-9771-8
· arXiv:math-ph/arxiv:0811.1276.
The next paper in this line of reasoning came about during
discussions with Brian Rider while I was a postdoc at Colorado.
(with Brian Rider and Yuan Xu). A solvable mixed
charge ensemble on the line: Global results;
Accepted for publication in Probab. Theor. Relat.
Fields · doi:
10.1007/s00440-011-0394-z ·
arXiv:math-ph/arxiv:1007.2246.
What if, instead of looking at Ginibre's real ensemble, where the
eigenvalues are either real or complex conjugate pairs, we envisioned
a situation where we forced the complex conjugate pairs together onto
the real line? This would create a situation where the complex
conjugate pairs are replaced with double eigenvalues on the line.
This can be better visualized as a system of charged particles, some
with charge 1 and some with charge 2, in the presence of the harmonic
oscillator potential. The same trick which kept track of the variable
number of real eigenvalues before, keeps track of the variable number
of charge 1 particles, and the partition function can be written as
the Pfaffian of an antisymmetric matrix formed from the sum of two
skew-symmetric inner products: the inner product for the Gaussian
Orthogonal Ensemble and the inner product for the Gaussian Symplectic
Ensemble. The algebraic identities in place, we were soon stymied in
trying to derive the skew-orthogonal polynomials that would
(hopefully) allow for an in-depth analysis of the statistics of the
particles. Sometime later I joined the faculty of the University of Oregon,
and had the opportunity to talk with Yuan Xu about various problems I
was interested in. I showed him the skew-symmetric inner product
Brian and I had derived, and within a few days Yuan had derived the
skew-orthogonal polynomials! Yuan also made the observation that the
skew-orthogonal polynomials were in fact just one in a one-parameter
family, and asked what I thought of this. After a few days of
thinking, I realized that this parameter was a natural quantity, which
really should have appeared at the start of the project; this
quantity, the fugacity, determines in some sense how easy it is
to exchange a particle of charge two for two particles of charge 1.
Moreover, the partition function as a function of the fugacity, is
also the probability generating function for the number of charge 1
particles.
The skew-orthogonal polynomials in hand, we were able to find a closed
form for the density of the number of real particles, and a version of
the Central Limit Theorem for this random variable in the limit as the
total charge goes to infinity. We were also able to derive the global
density of each kind of particle in this limit. We were unable to
derive a closed form for the scaling limit of the kernel(s). However,
my PhD student Chris Shum, has since derived the scaling limit for the
kernel(s) in the analogous circular ensemble. By universality, the
bulk kernel(s) should be the same in both cases.
The primary algebraic tool necessary to derive the Pfaffian
formulation for the partition function of the two-component ensemble
was the confluent Vandermonde determinant. The confluent
Vandermonde determinant is applicable in a wider array of ensembles
than just that one, and the next two papers are the beginning of a
line of research that I am still engaged in.
The partition function of multicomponent
log-gases; J. Phys. A: Math. Theor.
Vol. 45: 165002, 2012. ·
doi: 10.1088/1751-8113/45/16/165002
· arXiv:math-ph/arXiv:1201.0223 ;
Ensemble averages when $\beta$ is a square
integer; Monatsh. Math. Vol. 166,
No. 1: 121-144, 2012 · doi:
10.1007/s00605-011-0371-8 ·
arXiv:math-ph/arxiv:1008.4362.
The first of these two papers gives a hyperpfaffian formula for the
partition function of ensembles when $\beta$ is a square integer. This
very naturally generalizes the well-known $\beta=1$ and $\beta=4$
situations. From another viewpoint, this ensemble can also be thought
of as an interacting particle system on the line (or circle) where the
charged particles all have the same integer charge, at inverse
temperature $\beta=1$. Slightly more explicitly, if the magnitude of
the charge is $L$, and the particles are placed in a potential
corresponding to some finite measure, then the partition function is
given as the hyperpfaffian of an $L$-form whose coefficients are the
integrals (with respect to the potential measure) of the various $L
\times L$ Wronskians of a fixed family of polynomials. The
hope, which heretofore has not been realized, is that a hyperpfaffian
formula for the partition function will lead to a hyperpfaffian point
process on the eigenvalues/particles. Some of the formulas in this
work are rediscoveries of similar formulas (derived via a different
method in a related but different context) by Jean-Gabriel Luque and
Jean-Yves Thibon.
The second of these papers was an attempt to generalize the
hyperpfaffian formula for the partition function for fixed integer
charges to a formula for the partition function for systems composed
of particles with different (positive integer) charge magnitudes, where the
sum total of all charges in the system is a fixed integer, and the
inverse temperature is $\beta=1$. This is a zero-current version of
the grand canonical partition function. Here we allow fugacity
variables which determine the expected ratios of the numbers of the
various particles. The appropriate generalization of the
hyperpfaffian in this situation is the Berezin integral, and the main
result says that the partition function is the Berezin integral (with
respect to the volume form) of the exponential of a sum of forms (in
the sense of the Grassmann algebra) comprised of integrals of
Wronskians of a family of polynomials (these are exactly the same
forms which appear in the square $\beta$ paper).
In Ensemble averages when $\beta$ is a square integer I noticed
that a formula in which the Vandermonde determinant can be expressed
as a Pfaffian, allows for (among other things) a Pfaffian point
process for $\beta=2$ ensembles. I mentioned this formula during a
talk at MSRI during the 2010 special program on random matrices.
Peter Forrester approached me shortly thereafter and indicated that
this formula might be useful to answer a question he had considered
previously, but had been unable to answer. This led to the paper
(with Peter Forrester). A generalized plasma and
interpolation between classical random matrix
ensembles; J. Stat. Phys. Vol. 143,
No. 2: 326-345, 2011. · doi:
10.1007/s10955-011-0173-3 ·
arXiv:math-ph/arxiv:1012.0597.
The model system here is one with two species of particles restricted
to the circle whose interaction energy is not proportional to the
product of the charges. This provides a certain
type of interpolation between $\beta=2$ and $\beta=4$ circular
ensembles. The joint density of particles here is also of interest as
a trial wave function in the anomalous quantum Hall effect. The
Pfaffian point process for $\beta=2$ ensembles discovered in my
previous paper combined with the well-known Pfaffian point process for
$\beta=4$ ensembles allowed this generalized plasma model to be
solved.
A final, and recent, paper in this line of research is the
following written with a current postdoc, Max Yattselev, at the
University of Oregon.
(with Maxim Yattselev). Universality for
ensembles of matrices with potential theoretic
weights on domains with smooth boundary;
J. Approx. Theor. Vol. 164: 682-708,
2012. · doi: 10.1016/j.jat.2012.02.001 ·
arXiv:math/arxiv:1108.3052
This paper is actually a return to some of the questions I originally
asked in my thesis. The model considered here is an ensemble of
charged particles (corresponding to a &beta=2$ ensemble) restricted to
the plane, and in the presence of an oppositely charged conducting
region whose charge density is given by the equilibrium measure (in
the sense of potential theory in the complex plane). Another model
with the same statistics is the roots of a complex polynomial chosen
at random from the unit ball of a height which corresponds to the
exponentiated equilibrium measure of the conducting region as
identified with a compact set in the complex plane. The main result
in this paper is that, if the boundary of this conducting region is
sufficiently smooth, then the limiting statistics of the particles (or
roots) is essentially independent of the actual shape of the region.
More specifically, we prove that if the domain is sufficiently smooth,
the kernel in the scaling limit is (essentially) independent of the
shape of the domain. The universal kernel is explicitly given and
shown to be a generalization of the universal reproducing kernel for
Bergman orthogonal polynomials derived by Doron Lubinsky. Max and I
are currently investigating similar type universality results for
domains with non-smooth boundaries (specifically, at corners and
cusps).
My other primary research direction diverged relatively early from the
above sequence of publications. In 2004, I attended a workshop on the
intersection of random matrix theory and number theory at the Newton
Institute in Cambridge. During lunch one day, I sat next David
Farmer, who asked me about my thesis problem. I told him that I was
trying to compute the volume of a certain set of coefficient vectors
of polynomials. He immediately asked me if I thought I could compute
the volume of the set of (coefficient vectors of) polynomials of
degree $N$ with all roots on the unit circle. I immediately responded
that I thought it shouldn't be too difficult, and that I would look
into it. The problem was a bit more difficult than I originally
anticipated, and I roped a fellow graduate student, Kathleen Petersen,
into the project. In
(with Kathleen L. Petersen). Conjugate
reciprocal polynomials with all roots on the unit
circle; Canad. J. Math. Vol. 60(5):
1149-1167, 2008 · doi: 10.4153/CJM-2008-050-8 ·
arXiv: math.NT/0511397,
we compute the volume of this set, as well as show that the group of
isometries is dihedral and that the set is homeomorphic to a ball, and
geometrically has the structure of a colored simplex, where the
colorings are given by cyclically ordered partitions of the integer
$N$, which represent the various possible multiplicities for the
roots. By the time we were ready to submit, we both landed
postdoctoral positions in Canada (albeit at different institutions) so
we thought the Canadian journal was an appropriate venue.
A year or so after its publication, I returned to thinking about the
set of polynomials of fixed degree and all roots on the unit circle,
with the hope of producing easy-to-check necessary conditions and
sufficient conditions for polynomials to have all roots on the unit
circle. During this time I was invited for a couple weeks to a
workshop on heights at the Schrodinger Institute in Vienna, where I
ended up sharing an office with my thesis advisor. I told
him what I was working on, and we quickly determined some very easy to
check conditions, which in spite of their almost trivial proofs,
appeared to be new. The next week I attended a conference on
number theory and polynomials at the Heilbronn Institute in Bristol,
and I presented our results there. Nobody there had seen these
results before (the audience included many specialists on polynomials,
including Andre Schinzel) so we thought the results should be
published, and the conference proceedings seemed like a good choice. This
resulted in
(with Jeffrey D. Vaaler). Self-inversive
polynomials with all zeros on the unit circle.
London Mathematical
Society Lecture Note Series: Number Theory and
Polynomials Vol. 352: 312-321 ·
Editors: James McKee and Chris Smyth · 2008
(refereed).
On several occasions, I have been contacted by engineers and
scientists in other domains regarding this paper. Unfortunately,
our conditions for a polynomial to have all roots on the unit circle,
while easy to check, works only for a small proportion of polynomials
which actually have all their roots on the unit circle (with the
proportion going to 0 as the degree increases). Nonetheless, it is
nice to know that a result, even one with an easy proof, may be useful
in a wider setting!
Not long afterward this paper was submitted, I was invited to a
conference in Banff on the intersection of number theory and topology.
My research isn't squarely in either of these areas, but as David Boyd
was organizing the conference and he was one of my postdoctoral
advisors, I got an invitation. Moreover, Kate Petersen was going to
be there, and I thought it might be a good opportunity to continue
talking about various topics we had previously been looking into. The
closest thing to topology I had to talk about was the geometry of the
set of degree $N$ polynomials with all roots on the unit circle.
After my talk, Ted Chinburg asked a question about how the roots of
polynomials of low height, chosen from this set of polynomials,
behaved. In fact, I don't remember the exact question Ted asked, but
Kate and I started brainstorming about the question, which led us,
eventually, to the following related question: Are algebraic numbers
on the unit circle in an imaginary quadratic extension of $\mathbb Q$
equidistributed with respect to the (Weil) height? The answer turns
out to be yes.
(with Kathleen Petersen). Equidistribution of
algebraic numbers of norm one in quadratic number
fields; Int. J. Number Theory Vol. 7,
Issue 7: 1841-1861, 2011 · doi:
10.1142/S1793042111004666 ·
arXiv:math.NT/arxiv:1004.0986
In writing up these results, we realized the correct equidistribution
question in this context is whether algebraic numbers $\gamma$ of norm 1 in a
quadratic extension are equidistributed when ordered with respect to
the minimal norm of an algebraic integer $\alpha$ such that $\gamma =
\alpha^2/N(\alpha)$. This ordering is the same as that given by the
Weil height, and allowed us to answer the analogous question for real
quadratic extensions of $\mathbb{Q}$.
We are currently working on extending this sort of equidistribution
result to arbitrary number fields.
Finally, we get to a paper that looks something like an outlier, in
that it doesn't fit squarely into either of the research stories I've
told. (In fact, it is somewhat related to both stories, but the
connection isn't worth expounding upon here). The paper
(with Kevin Hare and David McKinnon). Patterns
and periodicity in a family of
resultants; J. Théor. Nombres Bordeaux
Vol. 21: 215-234, 2009 · doi: 10.5802/jtnb.667
·download from archive.numdam.org,
began in 2004 when I was still a grad student at Texas. I had to that
point solved part of my thesis problem, but had yet to stumble across
the Pfaffian formula that would eventually throw me in the direction
of random matrix theory. When my first job search (with the intention
of graduating in 2004) was unsuccessful, I decided that, in order to
keep from having to quit mathematics in shame, that I would prove
something big. I set my sights on Lehmer's conjecture, which can be
very loosely expressed as: there are no 'almost' cyclotomic
polynomials ('cyclotomicness' being determined by Mahler measure).
Currently, there is a lower bound for the Mahler measure of
non-cyclotomic integer polynomials due to Edward Dobrowolski, which
basically says that a polynomial which is very nearly, but not,
cyclotomic, must have very large degree. The central lemma necessary
for this lower bound is as follows: Suppose $f(x)$ is a monic
irreducible non-cyclotomic degree $N$ integer polynomial and $f_m(x)$
is the monic integer polynomial whose roots are the $m$th power of
those of $f(x)$. Dobrowolski's lemma says that if $p$ is a prime, then
$p^N$ divides the resultant of $f$ and $f_p$ (and moreover, this
resultant is non-zero). One path to the proof of Lehmer's conjecture
would be to show that the resultants of $f$ and $f_n$ are always (or
at least often) very large. In fact, all of the resultants of $f_n$
and $f_m$ for $m,n$ up to some bound, figure into Dobrowolski's lower
bound, and except for the resultants of $f$ and $f_p$, only the
trivial lower bound of $\left|\mathrm{Res}(f_m, f_n)\right| \geq 1$ is
used. Thus, I set out to show that some of the $\mathrm{Res}(f_m,
f_n)$ were very large.
The central result of this paper is that, if $p$ is a prime, and $m <
n$, then $p^{N(m-1)}$ divides the resultant of $f_{p^m}$ and
$f_{p^n}$. Unfortunately, the prime powers are too rarefied to improve
Dobrowolski's lower bound.
In order to get a feel for the size of $\mathrm{Res}(f_m, f_n)$, I looked at
$p$-adic valuations for (that is the exponent of the largest power of
$p$ dividing) these resultants for various primes $p$ and various integer
polynomials. In all cases I looked at, this quantity appeared to be
periodic in both $m$ and $n$. When visiting Kevin Hare at the
University of Waterloo, I showed him some examples of this phenomenon,
and we, together with David McKinnon, established this periodicity.
Suffice it to say that none of this led to a proof of Lehmer's
conjecture, though I occasionally return to this problem to see if any
new progress can be made. Nonetheless, not getting a job that year
ended up being a positive development, in the sense that I redoubled
my efforts, and in the end derived the Pfaffian formula that began me
down the research path that I am on.