arXiv subjects

Emma Bailey

Publications and source records attributed to Emma Bailey.

7 recordsLinked to original sources

arXiv2026

Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function

Assuming the Riemann Hypothesis, we show that for $k>0$ $$ \frac{1}{T}\text{meas}\Big\{t\in [T,2T]:|\zeta(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}}, $$ where $C_k=\exp(e^{ck})$ for some absolute constant $c>0$. This implies that the $2k$-moments of $|\zeta|$ are bounded above by $C_k(\log T)^{k^2}$, recovering the bound of Harper. The proof relies on the recursive scheme of one of the authors with Bourgade and Radziwill (2020), and combines ideas of Soundararajan (2009) and Harper (2013).

math.NT↗

arXiv2026

Agent-based macroeconomics for the UK's Seventh Carbon Budget

In June 2026, the UK government will set its carbon budget for the period 2038 to 2042, the seventh such carbon budget (CB7) since the Climate Change Act became law in 2008. For the first time, this carbon budget will be accompanied by a macroeconomic assessment of its impact on growth, employment, inflation and inequality. Researchers from the Institute of New Economic Thinking (INET) Oxford are working in partnership with the Department for Energy Security and Net Zero to deliver this assessment using our data-driven macroeconomic agent-based model (ABM). This extended abstract presents the work in progress towards this pioneering policymaking using our data-driven macroeconomic ABM. We are conducting our work in three work packages. By the time of the workshop, we hope to be able to present preliminary findings from the first two work packages. In WP1, we adapt an existing macro-ABM prototype and build a UK macroeconomic baseline. The main task for this is initialising the model with suitable UK household microdata. We present the options considered and the approach settled upon. In WP2, we conduct preliminary modelling that represents UK decarbonisation as an external shock to financial flows and technical coefficients. In order to present results in time to influence the June 2026 policy decision, this second work package exogenously forces the ABM to follow the CB7 green investment and associated technological change projections provided by the Climate Change Committee. Finally, we will implement more sophisticated social and technological learning packages in WP3, building our own projections of likely decarbonisation pathways that may diverge from UK government plans. For the workshop, we will present the progress of WP1 and WP2.

econ.GN↗

arXiv2025

Chase-escape with conversion as a multiple sclerosis lesion model

We introduce conversion to the stochastic process known as chase-escape in an effort to model aspects of inflammatory damage from multiple sclerosis. We prove monotonicity results for aggregate damage for the model on the positive integers, trees, stars, and the complete graph. Additionally, we establish the existence and asymptotic order of a phase transition on bounded degree graphs with a non-trivial site percolation threshold.

math.PR↗

arXiv2024

Lower Bounds for the Large Deviations of Selberg's Central Limit Theorem

Let $\delta>0$ and $\sigma=\frac{1}{2}+\tfrac{\delta}{\log T}$. We prove that, for any $\alpha>0$ and $V\sim \alpha\log \log T$ as $T\to\infty$, $\frac{1}{T}\text{meas}\big\{t\in [T,2T]: \log|\zeta(\sigma+\rm{i} \tau)|>V\big\}\geq C_\alpha(\delta)\int_V^\infty \frac{e^{-y^2/\log\log T}}{\sqrt{\pi\log\log T}} \rm{d} y,$ where $\delta$ is large enough depending on $\alpha$. The result is unconditional on the Riemann hypothesis. As a consequence, we recover the sharp lower bound for the moments on the critical line proved by Heap & Soundararajan and Radziwi{\l}{\l} & Soundararajan. The constant $C_\alpha(\delta)$ is explicit and is compared to the one conjectured by Keating & Snaith for the moments.

math.NT↗

arXiv2023

Critical drift estimates for the frog model on trees

Place an active particle at the root of a $d$-ary tree and a single dormant particle at each non-root site. In discrete time, active particles move towards the root with probability $p$ and, otherwise, away from the root to a uniformly sampled child vertex. When an active particle moves to a site containing a dormant particle, the dormant particle becomes active. The critical drift $p_d$ is the infimum over all $p$ for which infinitely many particles visit the root almost surely. Guo, Tang, and Wei proved that $\sup_{d\geq 3} p_d \leq 1/3$. We improve this bound to $5/17$ with a shorter argument that generalizes to give bounds on $\sup_{d \geq m} p_d$. We additionally prove that $\limsup p_d \leq 1/6$ by finding the limiting critical drift for a non-backtracking variant.

math.PR↗

arXiv2022

Large Deviation Estimates of Selberg's Central Limit Theorem and Applications

For $V\sim \alpha \log\log T$ with $0<\alpha<2$, we prove \[ \frac{1}{T}\text{meas}\{t\in [T,2T]: \log|\zeta(1/2+ {\rm i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^2/\log\log T}. \] This improves prior results of Soundararajan and of Harper on the large deviations of Selberg's Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwi{\l}{\l} and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $(\log T)^\theta$, $0<\theta <3$, that is expected to be sharp for $\theta > 0$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwi{\l}{\l} and one of the authors to prove fine asymptotics for the maximum on intervals of length $1$.

math.NT↗

arXiv2021

Evidence of Random Matrix Corrections for the Large Deviations of Selberg's Central Limit Theorem

Selberg's central limit theorem states that the values of $\log|\zeta(1/2+i \tau)|$, where $\tau$ is a uniform random variable on $[T,2T]$, is distributed like a Gaussian random variable of mean $0$ and standard deviation $\sqrt{\frac{1}{2}\log \log T}$. It was conjectured by Radziwi{\l}{\l} that this breaks down for values of order $\log\log T$, where a multiplicative correction $C_k$ would be present at level $k\log\log T$, $k>0$. This constant should be equal to the leading asymptotic for the $2k^{th}$ moment of $\zeta$, as first conjectured by Keating and Snaith using random matrix theory. In this paper, we provide numerical and theoretical evidence for this conjecture. We propose that this correction has a significant effect on the distribution of the maximum of $\log|\zeta|$ in intervals of size $(\log T)^\theta$, $\theta>0$. The precision of the prediction enables the numerical detection of $C_k$ even for low $T$'s of order $T=10^8$. A similar correction appears in the large deviations of the Keating-Snaith central limit theorem for the logarithm of the characteristic polynomial of a random unitary matrix, as first proved by F\'eray, M\'eliot and Nikeghbali.

math.PR↗