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Nina C. Snaith

Publications and source records attributed to Nina C. Snaith.

13 recordsLinked to original sources
arXiv2025

Thinned COE random matrix models for DNA replication

This paper details an observation that for more primitive organisms, such as some yeasts, the statistical distribution of the origins of replication sometimes looks remarkably like the distribution of eigenvalues from the Circular Orthogonal Ensemble (COE) of random matrices. This does not hold for more complex organisms, but a uniform thinning of the COE eigenvalues (which interpolates between the COE and uncorrelated, Poisson statistics) gives a platform to investigate characteristics of replication origin distribution in other species where data is available.

q-bio.OT↗
arXiv2025

Moments of the derivative of the characteristic polynomial of unitary matrices

Let $Λ_X(s)=\det(I-sX^{\dagger})$ be the characteristic polynomial of a Haar distributed unitary matrix $X$. It is believed that the distribution of values of $Λ_X(s)$ model the distribution of values of the Riemann zeta-function $ζ(s)$. This principle motivates many avenues of study. Of particular interest is the behavior of $Λ_X'(s)$ and the distribution of its zeros (all of which lie inside or on the unit circle). In this article we present several identities for the moments of $Λ_X'(s)$ averaged over $U(N)$, for $s \in \mathbb{C}$ as well as specialized to $|s|=1$. Additionally, we prove, for positive integer $k$, that the polynomial $\int_{U(N)} |Λ_X(1)|^{2k} \mathrm{dX}$ of degree $k^2$ in $N$ divides the polynomial $\int_{U(N)} |Λ_X'(1)|^{2k} \mathrm{dX}$ which is of degree $k^2+2k$ in $N$ and that the ratio, $f(N,k)$, of these moments factors into linear factors modulo $4k-1$ if $4k-1$ is prime. We also discuss the relationship of these moments to a solution of a second order non-linear Painléve differential equation. Finally we give some formulas in terms of the $_3F_2$ hypergeometric series for the moments in the simplest case when $N=2$, and also study the radial distribution of the zeros of $Λ_X'(s)$ in that case.

math-ph↗
arXiv2020

Moments of the logarithmic derivative of characteristic polynomials from $SO(N)$ and $USp(2N)$

We study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices. In particular, we compute the asymptotics for large matrix size, $N$, of these moments evaluated at points which are approaching 1. This follows work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith where they compute these asymptotics in the case of unitary random matrices.

math-ph↗
arXiv2019

Mixed moments of characteristic polynomials of random unitary matrices

Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N) (also known as the CUE) and relate the moment to the solution of a Painleve differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.

math-ph↗
arXiv2016

Symplectic $n$-level densities with restricted support

In this paper we demonstrate that the alternative form, derived by us in an earlier paper, of the $n$-level densities for eigenvalues of matrices from the classical compact group $USp(2N)$ is far better suited for comparison with derivations of the $n$-level densities of zeros in the family of Dirichlet $L$-functions associated with real quadratic characters than the traditional determinantal random matrix formula. Previous authors have found ingenious proofs that the leading order term of the $n$-level density of the zeros agrees with the determinantal random matrix result under certain conditions, but here we show that comparison is more straightforward if the more suitable form of the random matrix result is used. For the support of the test function in $[1,-1]$ and in $[-2,2]$ we compare with existing number theoretical results. For support in $[-3,3]$ no rigorous number theoretical result is known for the $n$-level densities, but we derive the densities here using random matrix theory in the hope that this may make the path to a rigorous number theoretical result clearer.

math.NT↗
arXiv2015

Beyond the excised ensemble: modelling elliptic curve L-functions with random matrices

The `excised ensemble', a random matrix model for the zeros of quadratic twist families of elliptic curve $L$-functions, was introduced by Dueñez, Huynh, Keating, Miller and Snaith. The excised model is motivated by a formula for central values of these $L$-functions in a paper by Kohnen and Zagier. This formula indicates that for a finite set of $L$-functions from a family of quadratic twists, the central values are all either zero or are greater than some positive cutoff. The excised model imposes this same condition on the central values of characteristic polynomials of matrices from $SO(2N)$. Strangely, the cutoff on the characteristic polynomials that results in a convincing model for the $L$-function zeros is significantly smaller than that which we would obtain by naively transferring Kohnen and Zagier's cutoff to the $SO(2N)$ ensemble. In this current paper we investigate a modification to the excised model. It lacks the simplicity of the original excised ensemble, but it serves to explain the reason for the unexpectedly low cutoff in the original excised model. Additionally, the distribution of central $L$-values is `choppier' than the distribution of characteristic polynomials, in the sense that it is a superposition of a series of peaks: the characteristic polynomial distribution is a smooth approximation to this. The excised model didn't attempt to incorporate these successive peaks, only the initial cutoff. Here we experiment with including some of the structure of the $L$-value distribution. The conclusion is that a critical feature of a good model is to associate the correct mass to the first peak of the $L$-value distribution.

math-ph↗
arXiv2011

A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor

We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in 2006; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling).Our purpose here is to provide a random matrix model for Miller's surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this sub-ensemble of SO(2N) the excised orthogonal ensemble. The sieving-off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of L-functions implied by the formula of Waldspurger and Kohnen-Zagier.The cut-off scale appropriate to modeling elliptic curve L-functions is exponentially small relative to the matrix size N. The one-level density of the excised ensemble can be expressed in terms of that of the well-known Jacobi ensemble, enabling the former to be explicitly calculated. It exhibits an exponentially small (on the scale of the mean spacing) hard gap determined by the cut-off value, followed by soft repulsion on a much larger scale. Neither of these features is present in the one-level density of SO(2N). When N tends to infinity we recover the limiting orthogonal behaviour. Our results agree qualitatively with Miller's discrepancy. Choosing the cut-off appropriately gives a model in good quantitative agreement with the number-theoretical data.

math.NT↗
arXiv2010

The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI

We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painleve VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painleve VI function. Our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve L-function families of finite conductor and of conjecturally orthogonal symmetry.

math.CA↗
arXiv2008

Lower order terms in the full moment conjecture for the Riemann zeta function

We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these coefficients. We also provide some numerical evidence in favour of the conjecture.

math.NT↗
arXiv2006

Random Matrix Theory and the Fourier Coefficients of Half-Integral Weight Forms

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of conjectures concerning the value-distribution of the Fourier coefficients of half-integral weight modular forms related to these L-functions. Our conjectures may be viewed as being analogous to the Sato-Tate conjecture for integral weight modular forms. Numerical evidence is presented in support of them.

math.NT↗
arXiv2006

Moments of the derivative of the Riemann zeta-function and of characteristic polynomials

We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann zeta-function on the critical line. We do the same for the analogue of Hardy's Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.

math.NT↗
arXiv2005

Discretisation for odd quadratic twists

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.

math.NT↗