I study the distribution of algebraic numbers, mathematical statistical physics and roots/eigenvalues of random polynomials/matrices.

### Projects in progress

1The distribution of values of the non-archimedean absolute Vandermonde determinant and the non-archimedean Selberg integral (with Jeff Vaaler). The Mellin transform of the distribution function of the non-archimedean absolute Vandermonde (on the ring of integers of a local field) is related to a non-archimedean analog of the Selberg/Mehta integral. A recursion for this integral allows us to find an analytic continuation to a rational function on a cylindrical Riemann surface. Information about the poles of this rational function allow us to draw conclusions about the range of values of the non-archimedean absolute Vandermonde.

2Non-archimedean electrostatics. The study of charged particles in a non-archimedean local field whose interaction energy is proportional to the log of the distance between particles, at fixed coldness $\beta$. The microcanonical, canonical and grand canonical ensembles are considered, and the partition function is related to the non-archimedean Selberg integral considered in 1. Probabilities of cylinder sets are explicitly computable in both the canonical and grand canonical ensembles.

3Adèlic electrostatics and global zeta functions (with Joe Webster). The non-archimedean Selberg integral/canonical partition function are examples of Igusa zeta functions, and as such local Euler factors in a global zeta function. This global zeta function (the exact definition of which is yet to be determined) is also the partition function for a canonical electrostatic ensemble defined on the adèles of a number field. The archimedean local factors relate to the ordinary Selberg integral, the Mehta integral, and the partition function for the complex asymmetric $\beta$ ensemble. The dream would be a functional equation for the global zeta function via Fourier analysis on the idèles, though any analytic continuation would tell us something about the distribution of energies in the adèlic ensemble.

4Pair correlation in circular ensembles when $\beta$ is an even square integer (with Nate Wells and Elisha Hulbert). This can be expressed in terms of a form in a grading of an exterior algebra, the coefficients of which are products of Vandermonde determinants in integers. Hopefully an understanding of the asymptotics of these coefficients will lead to scaling limits for the pair correlation function for an infinite family of coldnesses via hyperpfaffian/Berezin integral techniques. This would partially generalize the Pfaffian point process arising in COE and CSE. There is a lot of work to do, but there is hope.

5Martingales in the Weil height Banach space (with Nathan Hunter). Allcock and Vaaler produce a Banach space in which $\overline{\mathbb Q}^{\times}/\mathrm{Tor}$ embeds densely in a co-dimension 1 subspace, the (Banach space) norm of which extends the logarithmic Weil height. Field extensions of the maximal abelian extension of $\mathbb Q$ correspond to $\sigma$-algebras, and towers of fields to filtrations. Elements in the Banach space (including those from $\overline{\mathbb Q}^{\times}/\mathrm{Tor}$) represent random variables, and the set up is ready for someone to come along and use martingale techniques—including the optional stopping time theorem—to tell us something about algebraic numbers.

## Publications

• Christopher D. Sinclair, Non-archimedean Electrostatics, , Vol., No.:, 2020.  link, DOI, arXiv
• Christopher D. Sinclair and Maxim L. Yattselev, The reciprocal Mahler ensembles of random polynomials, Random Matrices: Theory and Applications, Vol.8, No.4:1950012:1-38, 2019.  link, DOI, arXiv
• Christopher Sinclair and Maxim Yattselev, Root statistics of random polynomials with bounded Mahler measure, Advances in Mathematics, Vol.272, No.:124-199, 2015.  link, DOI, arXiv
• Brian Rider and Christopher Sinclair, Extremal laws for the real Ginibre ensemble, Annals of Applied Probability, Vol.24, No.4:1621-1651, 2014.  link, DOI, arXiv
• Brian Rider, Christopher D. Sinclair and Yuan Xu, A solvable mixed charge ensemble on the line, Probability Theory and Related Fields, Vol.55, No.1-2:127-164, 2013.  link, DOI, arXiv
• Christopher D. Sinclair and Maxim Yattselev, Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary, Journal of Approximation Theory, Vol.164, No.5:682-708, 2012.  link, DOI, arXiv
• Christopher D. Sinclair, The partition function of multicomponent log-gases, Journal of Physics A: Mathematical and Theoretical, Vol.45, No.16: 165002, 2012.  link, DOI, arXiv
• Christopher D. Sinclair, Ensemble averages when β is a square integer, Monatshefte für Mathematik, Vol.166, No.1:121-144, 2012.  link, DOI, arXiv
• Peter Forrester and Christopher D. Sinclair, A generalized plasma and interpolation between classical random matrix ensembles, Journal of Statistical Physics, Vol.143, No.2:326-345, 2011.  link, DOI, arXiv
• Kathleen Petersen and Christopher D. Sinclair, Equidistribution of algebraic numbers of norm one in quadratic number fields, International Journal of Number Theory, Vol.7, No.7:1841-1861, 2011.  link, DOI, arXiv
• Kevin Hare, David McKinnon and Christopher D. Sinclair, Patterns and periodicity in a family of resultants, Journal de Théorie des Nombres de Bordeaux, Vol.21, No.:215-234, 2009.  link, DOI, arXiv
• Christopher D. Sinclair, Correlation functions for β=1 ensembles of matrices of odd size, Journal of Statistical Physics, Vol.136, No.1:17-33, 2009.  link, DOI, arXiv
• Alexei Borodin and Christopher D. Sinclair, The Ginibre ensemble of real random matrices and its scaling limits, Communications in Mathematical Physics, Vol.291, No.1:177-224, 2009.  link, DOI, arXiv
• Christopher D. Sinclair, The range of multiplicative functions on ℂ[x],ℝ[x] and ℤ[x], Proceedings of the London Mathematical Society, Vol.96, No.3:697-737, 2008.  link, DOI, arXiv
• Christopher D. Sinclair and Jeffrey Vaaler, Self-inversive polynomials with all zeros on the unit circle, London Mathematical Society Lecture Note Series: Number Theory and Polynomials, Vol.352, No.:312-321, 2008.  link, DOI, arXiv
• Kathleen Petersen and Christopher D. Sinclair, Conjugate reciprocal polynomials with all roots on the unit circle, Canadian Journal of Mathematics, Vol.60, No.5:1149-1167, 2008.  link, DOI, arXiv
• Christopher D. Sinclair, Averages over Ginibre's ensemble of random real matrices, International Research Mathematics Notices, Vol.2007, No.:1-15, 2007.  link, DOI, arXiv
• Christopher D. Sinclair, The distribution of Mahler's measures of reciprocal polynomials, International Journal of Mathematics and Mathematical Sciences, Vol.52, No.:2773-2786, 2004.  link, DOI, arXiv

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