arXiv subjects

Christopher D. Sinclair

Publications and source records attributed to Christopher D. Sinclair.

At least 19 recordsLinked to original sources

arXiv2026

The Hirota Identity for Hyperpfaffian $\tau$-Functions in Charge-$L$ Ensembles

We study log-gas ensembles with inverse temperature $\beta = L^2$ using a confluent Vandermonde representation that admits a formulation in the exterior algebra of a finite-dimensional vector space. By interpreting the system as consisting of finitely many particles with integer charge $L$, partition functions can be expressed exactly as hyperpfaffians. In this formulation, the system is governed by a natural momentum grading arising from the confluent Vandermonde structure, and its statistical observables are determined entirely by the corresponding bigraded commutative subalgebra. The geometric identity that a particle's $L$-blade wedges with itself to zero produces momentum Pl\"ucker relations within this algebra. These relations generate momentum transport identities between sectors of different particle number. Upon introducing dynamic time variables, the partition functions become $\tau$-functions, and these transport identities are transformed into Hirota bilinear equations. This provides an explicit algebraic origin for the integrable hierarchy structure of the $\beta = L^2$ ensembles, which may be viewed as a finite-dimensional analogue of the Sato Grassmannian formulation of integrable systems.

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arXiv2026

Exact Solvability via the KP Hierarchy for $\beta=L^2$ Random Matrix Ensembles

Random matrix ensembles with Dyson index $\beta=L^{2}$ describe systems of $M$ charge-$L$ particles interacting logarithmically in the presence of an external potential, yet exact formulas for their physical observables have remained elusive for $L\neq 1,2$. We show that, for $L$ even, $\beta=L^{2}$ ensembles are governed by the KP hierarchy at finite particle number--paralleling the KP solvability of classical $\beta=1,2,4$ ensembles. The partition function is a hyperpfaffian $\tau$-function satisfying the Hirota bilinear identity, and correlation functions are generated by finite-order differential operators acting on this $\tau$-function. The key mechanism is an emergent quantized momentum that stratifies the system into discrete sectors, enforcing momentum conservation as a selection rule. This produces a dramatic dimensional reduction from ${LM\choose L}$ to $O(L^{2}M)$, enabling explicit computation of physical observables.

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arXiv2025

Hyperpfaffian Correlations for Beta-Ensembles: Beta an Even Square Integer

We give a hyperpfaffian formulation for correlation functions in $\beta$-ensembles arising in random matrix theory and statistical mechanics when $\beta = L^2$ is an even square integer. More specifically, for ensembles of $M$ points in a space $W \subset \mathbb C$ (typically $W=\mathbb R$ or $W=\mathbb T$), arising either as eigenvalues of a random matrix or as a system of charged particles with log interaction, to the $m$th correlation function $R_m : W^m \rightarrow [0, \infty)$ we associate the $L$-vector valued function $\gamma_m : W^m \rightarrow \Lambda^L \mathbb C^{L(M-m)}$ such that $R_m(\mathbf y)$ is given by the Vandermonde determinant in $y_1, \ldots, y_m$ times the hyperpfaffian of $\gamma_m(\mathbf y).$ The partition function of the ensemble was previously shown to be the hyperpfaffian of a {\it Gram} $L$-form $\gamma$ in $\Lambda^L \mathbb C^{LM},$ and we demonstrate the relationship between $\gamma_m(\mathbf y)$ and $\gamma$, both having coefficients built from integrals of Wronskians of monic polynomials. Assuming the existence of families of polynomials sympathetic with the weight of the ensemble, we may construct $\gamma(\mathbf y)$ so it is very sparse (relative to the expected ${L(M-m) \choose L}$ coefficients of a general $L$-vector). These generalize skew-orthogonal polynomials arising in the well-understood $\beta = 4$ situation. Finally we explore the situation in the circular $\beta = L^2$ ensembles. Here the monomials give a prototype, and we give explicit formulas for $\gamma$ and $\gamma_m$ in this setting. We use our hyperpfaffian framework to produce exact formulas for the two point function when $\beta = 16$ for small values $M.$ Along the way we will record hyperpfaffian evaluations using known values of partition functions of $\beta$-ensembles.

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arXiv2024

Electrostatics on Branching Processes

We introduce a random probability measure on the profinite completion of the random tree of a branching process and introduce the canonical and grand canonical ensembles of random repelling particles on this random profinite completion at inverse temperature $\beta > 0$. We think of this as a random spatial process of particles in a random tree, and we introduce the notion of the {\em mean} canonical and grand canonical partition functions where in this context `mean' means averaged over the random environment. We give a recursion for these mean partition functions and demonstrate that in certain instances, determined by the law for the branching process, these partition functions as a function of $\beta$ have algebraic properties which generalize those that appear in the non-random and $p$-adic environments.

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arXiv2020

Non-Archimedean Electrostatics

We introduce ensembles of repelling charged particles restricted to a ball in a non-archimedean field (such as the $p$-adic numbers) with interaction energy between pairs of particles proportional to the logarithm of the ($p$-adic) distance between them. In the {\em canonical ensemble}, a system of $N$ particles is put in contact with a heat bath at fixed inverse temperature $β$ and energy is allowed to flow between the system and the heat bath. Using standard axioms of statistical physics, the relative density of states is given by the $β$ power of the ($p$-adic) absolute value of the Vandermonde determinant in the locations of the particles. The partition function is the normalizing constant (as a function of $β$) of this ensemble, and we identify a recursion that allows this to be computed explicitly in finite time. Probabilities of interest, including the probabilities that specified subsets will have a prescribed occupation number of particles, and the conditional distribution of particles within a subset given a prescribed occupation number, are given explicitly in terms of the partition function. We then turn to the {\em grand canonical ensemble} where both the energy and number of particles are variable. We compute similar probabilities to those in the canonical ensemble and show how these probabilities can be given in terms the canonical and grand canonical partition functions. Finally, we briefly consider the multi-component ensemble where particles are allowed to take different integer charges, and we connect basic properties of this ensemble to the canonical and grand canonical ensembles.

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arXiv2018

The reciprocal Mahler ensembles of random polynomials

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval $[-2,2]$ on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakly to the arcsine distribution supported on $[-2,2]$. Outside this region the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from $[-2,2]$. These kernels, as well as the scaling limits for the kernels in the bulk $(-2,2)$ and at the endpoints $\{-2,2\}$ are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.

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arXiv2014

Extremal laws for the real Ginibre ensemble

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\rightarrow\infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.

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arXiv2014

A Solvable Two-Charge Ensemble on the Circle

We introduce an ensemble consisting of logarithmically repelling charge one and charge two particles on the unit circle constrained so that the total charge of all particles equals $N$, but the proportion of each species of particle is allowed to vary according to a fugacity parameter. We identify the proper scaling of the fugacity with $N$ so that the proportion of each particle stays positive in the $N \rightarrow \infty$ limit. This ensemble forms a Pfaffian point process on the unit circle, and we derive the scaling limits of the matrix kernel(s) as a function of the interpolating parameter. This provides a solvable interpolation between the circular unitary and symplectic ensembles.

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arXiv2014

Equidistribution of Elements of Norm 1 in Cyclic Extensions

Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1 + r_2 - 1$ where $r_1$ and $r_2$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in a finite cover of this torus with respect to a natural partial ordering induced by Hilbert's Theorem 90.

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arXiv2013

Root Statistics of Random Polynomials with Bounded Mahler Measure

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree $N$ polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yields a Pfaffian point process on the complex plane. When $N$ is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e., scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of $\pm 1$ which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of $\C$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.

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arXiv2012

Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary

We investigate a two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to β= 2. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a determinantal point process on the complex plane. We investigate the scaling limit, as N \to \infty, of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth. We find that the limiting kernel depends on the limiting value of N/s, and prove universality for these kernels. That is, we show that, the scaled kernel in a neighborhood of a point ζ\in \partial K can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement K to the complement of the closed unit disk. When N / s \to 0 we recover the universal kernel discovered by Doron Lubinsky in Universality type limits for Bergman orthogonal polynomials, Comput. Methods Funct. Theory, 10:135-154, 2010.

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arXiv2011

The Partition Function of Multicomponent Log-Gases

We give an expression for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charge at inverse temperature β = 1 (restricted to the line in the presence of a neutralizing field) in terms of the Berezin integral of an associated non- homogeneous alternating tensor. This is the analog of the de Bruijn integral identities [3] (for β = 1 and β = 4) ensembles extended to multicomponent ensembles.

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arXiv2010

A generalized plasma and interpolation between classical random matrix ensembles

The eigenvalue probability density functions of the classical random matrix ensembles have a well known analogy with the one component log-gas at the special couplings β= 1,2 and 4. It has been known for some time that there is an exactly solvable two-component log-potential plasma which interpolates between the β=1 and 4 circular ensemble, and an exactly solvable two-component generalized plasma which interpolates between β= 2 and 4 circular ensemble. We extend known exact results relating to the latter --- for the free energy and one and two-point correlations --- by giving the general (k_1+k_2)-point correlation function in a Pfaffian form. Crucial to our working is an identity which expresses the Vandermonde determinant in terms of a Pfaffian. The exact evaluation of the general correlation is used to exhibit a perfect screening sum rule.

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arXiv2010

Ensemble Averages when βis a Square Integer

We give a hyperpfaffian formulation of partition functions and ensemble averages for Hermitian and circular ensembles when L is an arbitrary integer and β=L^2 and when L is an odd integer and β=L^2 +1.

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arXiv2010

A Solvable Mixed Charge Ensemble on the Line: Global Results

We consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio 2 : 1 in the presence of the harmonic oscillator potential. The system is assumed to be at temperature corresponding to β= 1 and the sum of the charges is fixed. We investigate the distribution of the number as well as the spatial density of each species of particle in the limit as the total charge increases to \infty. These results will follow from the fact that the system of particles forms a Pfaffian point process. We produce the skew-orthogonal polynomials necessary to simplify the related matrix kernels.

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arXiv2010

Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in N can be written as α/\barα for some α\in O_K, which yields another ordering of \mathcal N given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that N is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that N is equidistributed with respect to norm, under the map β\mapsto \log| β| \bmod{\log | ε^2 |} where εis a fundamental unit of O_K.

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arXiv2008

Correlation Functions of Asymmetric Real Matrices

We give a closed form for the correlation functions of ensembles of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a $2 \times 2$ matrix kernel associated to the ensemble. We also derive closed forms for the matrix kernel and correlation functions for Ginibre's real ensemble.

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arXiv2007

The range of multiplicative functions on C[x], R[x] and Z[x]

Mahler's measure is generalized to create the class of {\it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C[x] and \R[x]. These {\it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithmetic structure. For instance, we show that the moment function associated to Mahler's measure restricted to real reciprocal polynomials of degree 2N has an analytic continuation to rational functions with rational coefficients, simple poles at integers between -N and N, and a zero of multiplicity 2N at the origin. This discovery leads to asymptotic estimates for the number of reciprocal integer polynomials of fixed degree with Mahler measure less than T as $T \to \infty$. To explain the structure of this moment functions we show that the real moment functions of a multiplicative distance function can be written as Pfaffians of antisymmetric matrices formed from a skew-symmetric bilinear form associated to the multiplicative distance function.

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