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Greg W. Anderson

Publications and source records attributed to Greg W. Anderson.

At least 19 recordsLinked to original sources
arXiv2014

Preservation of algebraicity in free probability

We show that any matrix-polynomial combination of free noncommutative random variables each having an algebraic law has again an algebraic law. Our result answers a question raised by a recent paper of Shlyakhtenko and Skoufranis. The result belongs to a family of results with origins outside free probability theory, including a result of Aomoto asserting algebraicity of the Green function of random walk of quite general type on a free group.

math.OA↗
arXiv2014

Tridiagonalized GUE matrices are a matrix model for labeled mobiles

It is well-known that the number of planar maps with prescribed vertex degree distribution and suitable labeling can be represented as the leading coefficient of the $\frac{1}{N}$-expansion of a joint cumulant of traces of powers of an $N$-by-$N$ GUE matrix. Here we undertake the calculation of this leading coefficient in a different way. Firstly, we tridiagonalize the GUE matrix in the manner of Trotter and Dumitriu-Edelman and then alter it by conjugation to make the subdiagonal identically equal to $1$. Secondly, we apply the cluster expansion technique (specifically, the Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical mechanics. Thirdly, by sorting through the terms of the expansion thus generated we arrive at an alternate interpretation for the leading coefficient related to factorizations of the long cycle $(12\cdots n)\in S_n$. Finally, we reconcile the group-theoretical objects emerging from our calculation with the labeled mobiles of Bouttier-Di Francesco-Guitter.

math.PR↗
arXiv2014

Asymptotically liberating sequences of random unitary matrices

A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting "asymptotic liberation" under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author and of Tulino-Caire-Shamai-Verdú.

math.PR↗
arXiv2013

A local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices

Our main result is a local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices, modeled on the local semicircle law. Our approach is to adapt some techniques from one of the recent papers of Erdös-Yau-Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica-Speicher, a self-adjointness-preserving variant of the linearization trick due to Haagerup-Schultz-Thorbjørnsen, and the Schwinger-Dyson equation. A byproduct of our work is a relatively simple deterministic version of the local semicircle law.

math.PR↗
arXiv2013

Convergence of the largest singular value of a polynomial in independent Wigner matrices

For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form "no eigenvalues outside the support of the limiting eigenvalue distribution." We build on ideas of Haagerup-Schultz-Thorbjørnsen on the one hand and Bai-Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Instead of Poincaré-type inequalities, we use a variety of matrix identities and $L^p$ estimates. The Schwinger-Dyson equation controls much of the analysis.

math.PR↗
arXiv2012

Counting colored planar maps free-probabilistically

Our main result is an explicit operator-theoretic formula for the number of colored planar maps with a fixed set of stars each of which has a fixed set of half-edges with fixed coloration. The formula bounds the number of such colored planar maps well enough to prove convergence near the origin of generating functions arising naturally in the matrix model context. Such convergence is known but the proof of convergence proceeding by way of our main result is relatively simple. Besides Voiculescu's generalization of Wigner's semicircle law, our main technical tool is an integration identity representing the joint cumulant of several functions of a Gaussian random vector. The latter identity in the case of cumulants of order 2 reduces to one well-known as a means to prove the Poincare inequality. We derive the identity by combining the heat equation with the so-called BKAR formula from constructive quantum field theory and rigorous statistical mechanics.

math.PR↗
arXiv2009

A CLT for regularized sample covariance matrices

We consider the spectral properties of a class of regularized estimators of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by banding. We prove a law of large numbers (similar to that proved in the Gaussian case by Bickel and Levina), which implies that the spectrum of a banded empirical covariance matrix is an efficient estimator. Our main result is a central limit theorem in the same regime, which to our knowledge is new, even in the Gaussian setup.

math.PR↗
arXiv2006

Interpolation of hypergeometric ratios in a global field of positive characteristic

In connection with each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author (math.NT/0407535). Most notably, all examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i.~e., shtukas not associated to any Drinfeld module.

math.NT↗
arXiv2006

Interpolation of numbers of Catalan type in a local field of positive characteristic

Let $k$ be a local field of positive characteristic. Let $L$ be a cocompact discrete subgroup of $k$. Let $U$ be an open compact subgroup of $k$. Let $\ell$, $u$ and $a$ be elements of $k$, with $a$ nonzero. We study the behavior of the product $\prod_{0\neq x\in (\ell+L)\cap a(u+U)}x$ as $a$ varies. Our main result, which blends harmonic analysis and local class field theory, provides local confirmation of a conjecture concerning global function fields recently made by the author (math.NT/0407535).

math.NT↗
arXiv2005

Digit patterns and Coleman power series

Our main result is elementary and concerns the relationship between the multiplicative groups of the coordinate and endomorphism rings of the formal additive group over a field of characteristic $p>0$. The proof involves the combinatorics of base $p$ representations of positive integers in a striking way. We apply the main result to construct a canonical quotient of the module of Coleman power series over the Iwasawa algebra when the base local field is of characteristic $p$. This gives information in a situation which apparently has never previously been investigated.

math.NT↗
arXiv2005

A two-variable refinement of the Stark conjecture in the function field case

We propose a conjecture refining the Stark conjecture St(K/k,S) (Tate's formulation) in the function field case. Of course St(K/k,S) in this case is a theorem due to Deligne and independently to Hayes. The novel feature of our conjecture is that it involves two- rather than one-variable algebraic functions over finite fields. Because the conjecture is formulated in the language and framework of Tate's thesis we have powerful standard techniques at our disposal to study it. We build a case for our conjecture by (i) proving a parallel result in the framework of adelic harmonic analysis which we dub the adelic Stirling formula, (ii) proving the conjecture in the genus zero case and (iii) explaining in detail how to deduce St(K/k,S) from our conjecture. In the genus zero case the class of two-variable algebraic functions thus arising includes all the F_q[T]-solitons previously studied and applied by the author, collaborators and others. Ultimately the inspiration for this paper comes from striking examples given some years ago by R. Coleman and D. Thakur.

math.NT↗
arXiv2002

Edited 4-Theta embeddings of Jacobians

By the Lefschetz embedding theorem, a principally polarized $g$-dimensional abelian variety is embedded into projective space by the linear system of $4^g$ half-characteristic theta functions. Suppose we {\em edit} this linear system by dropping all the theta functions vanishing at the origin to order greater than parity requires. We prove that for Jacobians the edited $4Θ$ linear system still defines an embedding into projective space. Moreover, we prove that the projective models of Jacobians arising from the elementary construction of Jacobians recently given by the author are (after passage to linear hulls) copies of the edited $4Θ$ model. Thus, for all compact Riemann surfaces, we tie together algebraic and analytic Jacobians in a new way.

math.NT↗
arXiv2002

Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring

The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrère symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed field, namely sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law due to Witt. Needless to say, we have been much influenced by the work of Tate on sum-of-residues-equals-zero and the work of Arbarello-DeConcini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

math.NT↗
arXiv2002

A note on cyclotomic Euler systems and the double complex method

In this note we give the Kolyvagin recursion in cyclotomic Euler systens a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of the chosen field satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution a la Kubert. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.

math.NT↗
arXiv2001

Abeliants and their application to an elementary construction of Jacobians

The {\em abeliant} is a polynomial rule for producing an $n$ by $n$ matrix with entries in a given ring from an $n$ by $n$ by $n+2$ array of elements of that ring. The theory of abeliants, first introduced in an earlier paper of the author, is redeveloped here in a simpler way. Then this theory is exploited to give an explicit elementary construction of the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. The standard of usefulness and aptness we strive toward is that set by Mumford's elementary construction of the Jacobian of a hyperelliptic curve. This paper has appeared as Advances in Math 172 (2002) 169-205.

math.NT↗
arXiv2001

Kronecker-Weber plus epsilon

We say that a group is {\em almost abelian} if every commutator is central and squares to the identity. Now let $G$ be the Galois group of the algebraic closure of the field $\QQ$ of rational numbers in the field of complex numbers. Let $G^{\ab+ε}$ be the quotient of $G$ universal for homomorphisms to almost abelian profinite groups and let $\QQ^{\ab+ε}/\QQ$ be the corresponding Galois extension. We prove that $\QQ^{\ab+ε}$ is generated by the roots of unity, the fourth roots of the (rational) prime numbers and the square roots of certain sine-monomials. The inspiration for the paper came from recent studies of algebraic $Γ$-monomials by P.~Das and by S.~Seo. This paper has appeared as Duke Math. J. 114 (2002) 439-475.

math.NT↗
arXiv1997

Dark Matter with Time-Dependent Mass

We propose a simple model in which the cosmological dark matter consists of particles whose mass increases with the scale factor of the universe. The particle mass is generated by the expectation value of a scalar field which does not have a stable vacuum state, but which is effectively stabilized by the rest energy of the ambient particles. As the universe expands, the density of particles decreases, leading to an increase in the vacuum expectation value of the scalar (and hence the mass of the particle). The energy density of the coupled system of variable-mass particles (``vamps'') redshifts more slowly than that of ordinary matter. Consequently, the age of the universe is larger than in conventional scenarios.

astro-ph↗