arXiv subjects

Robert Hough

Publications and source records attributed to Robert Hough.

11 recordsLinked to original sources

arXiv2024

Lower order terms in the shape of cubic fields

We demonstrate equidistribution of the lattice shape of cubic fields when ordered by discriminant, giving an estimate in the Eisenstein series spectrum with a lower order main term. The analysis gives a separate discussion of the contributions of reducible and irreducible binary cubic forms, following a method of Shintani. Our work answers a question posed at the American Institute of Math by giving a precise geometric and spectral description of an evident barrier to equidistribution in the lattice shape.

math.NT↗

arXiv2022

Subconvexity of Shintani's zeta function

Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani $ζ$ function enumerating class numbers of binary cubic forms.

math.NT↗

arXiv2022

Subconvexity of twisted Shintani zeta functions

Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version.

math.NT↗

arXiv2021

Eisenstein series twisted Shintani zeta function

We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.

math.NT↗

arXiv2019

Solution of the 15 puzzle problem

A generalized `$15$ puzzle' consists of an $n \times n$ numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' `15 puzzle problem' by proving that the asymptotic total variation mixing time of the board is at least order $ n^4 $ when the board is given periodic boundary conditions and when random moves are made. We demonstrate that for any $f(n) \to \infty$ with $n$, the number of fixed points after $n^4 f(n)$ moves converges to a Poisson distribution of parameter 1. The order of total variation mixing time for this convergence is $n^4$ without cut-off. We also prove an upper bound of order $n^{4 }\log n$ for the total variation mixing time.

math.PR↗

arXiv2019

The spectrum of the abelian sandpile model

In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral factors which govern the asymptotic spectral gap and asymptotic mixing time. This article gives a general method of determining the spectral factor either computationally or asymptotically and performs the determination in specific examples.

math.PR↗

arXiv2019

Cut-off for sandpiles on tiling graphs

Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling either torus or open boundary conditions. A general method of obtaining the Green's function of the tiling is given, and a total variation cut-off phenomenon is demonstrated under general conditions. It is shown that the boundary condition does not affect the mixing time for planar tilings, nor does it change the asymptotic mixing time for the cubic lattice $\zed^d$ for all sufficiently large $d$. In a companion paper, computational methods are used to demonstrate that the mixing time is altered for the $\Dfour$ lattice in dimension 4.

math.PR↗

arXiv2019

The shape of cubic fields

We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the quantitative equidistribution of the shape of cubic fields when the fields are ordered by discriminant.

math.NT↗

arXiv2018

The local zeta function in enumerating quartic fields

An exact formula is obtained for the Fourier transform of the local condition of maximality modulo primes $p>3$ in the prehomogeneous vector space $2 \otimes \mathrm{Sym}^2(\mathbb{Z}_p^3)$ parametrizing quartic fields, thus solving the local `quartic case' in enumerating quartic fields.

math.NT↗

arXiv2018

The shape of quartic fields

We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant. Our estimate saves a small power in the corresponding Weyl sums.

math.NT↗

arXiv2018

The local limit theorem on nilpotent Lie groups

A local limit theorem is proven on connected, simply connected nilpotent Lie groups, for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization. The result extends an earlier local limit theorems of Alexopoulos which treated absolutely continuous measures with a continuous density of compact support, and also extends local limit theorems of Breuillard and Diaconis-Hough which treated general measures on the Heisenberg group.

math.PR↗