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Bob Hough

Publications and source records attributed to Bob Hough.

12 recordsLinked to original sources
arXiv2018

Random walk on unipotent matrix groups

We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the $n\times n$ uni-upper triangular group with entries taken modulo $p$. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order $p^2$ steps for randomness, coordinates on the second diagonal require order $p$ steps; coordinates on the $k$th diagonal require order $p^{\frac{2}{k}}$ steps.

math.PR↗
arXiv2018

Sandpiles on the square lattice

We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $\mathbb{Z}^2$. We also determine the asymptotic spectral gap, asymptotic mixing time and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus $\left( \mathbb{Z} / m\mathbb{Z} \right)^2$. The techniques use analysis of the space of functions on $\mathbb{Z}^2$ which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in $\ell^p(\mathbb{Z}^2)$ as linear combinations of certain discrete derivatives of Green's functions, extending a result of Schmidt and Verbitskiy.

math.PR↗
arXiv2018

Mixing and cut-off in cycle walks

Given a sequence $(\mathfrak{X}_i, \mathscr{K}_i)_{i=1}^\infty$ of Markov chains, the cut-off phenomenon describes a period of transition to stationarity which is asymptotically lower order than the mixing time. We study mixing times and the cut-off phenomenon in the total variation metric in the case of random walk on the groups $\mathbb{Z}/p\mathbb{Z}$, $p$ prime, with driving measure uniform on a symmetric generating set $A_p \subset \mathbb{Z}/p\mathbb{Z}$.

math.NT↗
arXiv2016

Maass form twisted Shintani $\mathscr{L}$-functions

The Maass-form twisted Shintani $\mathscr{L}$-functions are introduced, and some of their analytic properties are studied. These functions contain data regarding the distribution of shapes of cubic rings.

math.NT↗
arXiv2016

Equidistribution of bounded torsion CM points

Averaging over imaginary quadratic fields, we prove, quantitatively, the equidistribution of CM points associated to 3-torsion classes in the class group. We conjecture that this equidistribution holds for points associated to ideals of any fixed odd order. We prove a partial equidistribution result in this direction and present empirical evidence.

math.NT↗
arXiv2016

The random $k$ cycle walk on the symmetric group

We study the random walk on the symmetric group $S_n$ generated by the conjugacy class of cycles of length $k$. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after $\frac{n}{k} log n$ steps, uniformly in $k = o(n)$ as $n \to \infty$. The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles.

math.PR↗
arXiv2015

Cut-off phenomenon in the uniform plane Kac walk

We consider an analogue of the Kac random walk on the special orthogonal group $SO(N)$, in which at each step a random rotation is performed in a randomly chosen 2-plane of $\bR^N$. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large $N$ limit. In the special case where the angle of rotation is deterministic this confirms a conjecture of Rosenthal \cite{Rosenthal}. Under mild conditions we also establish a cut-off for convergence of the walk to stationarity under the $L^2$ norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and $L^2$ norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.

math.PR↗
arXiv2014

Zero density estimate for modular form $L$-functions in weight aspect

Considering the family of $L$-functions $\{L(s,f)\}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp forms for $SL_2(\mathbb{Z})$, we prove a zero density estimate near the central point, valid as the weight $k \to \infty$. This is an ingredient in the author's related paper, which gives an unconditional upper bound on the distribution of the central values.

math.NT↗
arXiv2014

The distribution of the logarithm in an orthogonal and a symplectic family of $L$-functions

We consider the logarithm of the central value $\log L(1/2)$ in the orthogonal family ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp form for $SL_2(\mathbb{Z})$, and in the symplectic family ${L(s,χ_{8d})}_{d \asymp D}$ where $χ_{8d}$ is the real character associated to fundamental discriminant $8d$. Unconditionally, we prove that the two distributions are asymptotically bounded above by Gaussian distributions, in the first case of mean $-\frac{1}{2} \log \log k$ and variance $\log \log k$, and in the second case of mean $\frac{1}{2}\log \log D$ and variance $\log \log D$. Assuming both the Riemann and Zero Density Hypotheses in these families we obtain the full normal law in both families, confirming a conjecture of Keating and Snaith.

math.NT↗
arXiv2011

The resonance method for large character sums

We consider the size of large character sums, proving new lower bounds for the quantity $Δ(N,q) = \sup_{χ\neq χ_0 mod q} |\sum_{n < N} χ(n)|$ for almost all ranges of $N$. The results are proven using the resonance method and saddle point analysis.

math.NT↗