Inhomogeneous condensation in the Gross-Neveu model in non-integer spatial dimensions 1≤d<31 \leq d < 3

The Gross-Neveu model in the $N \to \infty$ approximation in $d=1$ spatial dimensions exhibits a chiral inhomogeneous phase (IP), where the chiral condensate has a spatial dependence that spontaneously breaks translational invariance and the $\mathbb{Z}_2$ chiral symmetry. This phase is absent in $d=2$, while in $d=3$ its existence and extent strongly depends on the regularization and the value of the finite regulator. This work connects these three results smoothly by extending the analysis to non-integer spatial dimensions $1 \leq d <3$, where the model is fully renormalizable. To this end, we adapt the stability analysis, which probes the stability of the homogeneous ground state under inhomogeneous perturbations, to non-integer spatial dimensions. We find that the IP is present for all $d<2$ and vanishes exactly at $d=2$. Moreover, we find no instability towards an IP for $2\leq d<3$, which suggests that the IP in $d=3$ is solely generated by the presence of a regulator.Comment: 14 pages, 6 figure

Improving spatial nitrogen dioxide prediction using diffusion tubes: a case study in West Central Scotland

It has been well documented that air pollution adversely affects health, and epidemiological pollutionhealth studies utilise pollution data from automatic monitors. However, these automatic monitors are small in number and hence spatially sparse, which does not allow an accurate representation of the spatial variation in pollution concentrations required for these epidemiological health studies. Nitrogen dioxide (NO2) diffusion tubes are also used to measure concentrations, and due to their lower cost compared to automatic monitors are much more prevalent. However, even combining both data sets still does not provide sufficient spatial coverage of NO2 for epidemiological studies, and modelled concentrations on a regular grid from atmospheric dispersion models are also available. This paper proposes the first modelling approach to using all three sources of NO2 data to make fine scale spatial predictions for use in epidemiological health studies. We propose a geostatistical fusion model that regresses combined NO2 concentrations from both automatic monitors and diffusion tubes against modelled NO2 concentrations from an atmospheric dispersion model in order to predict fine scale NO2 concentrations across our West Central Scotland study region. Our model exhibits a 47% improvement in fine scale spatial prediction of NO2 compared to using the automatic monitors alone, and we use it to predict NO2 concentrations across West Central Scotland in 2006

How robust are the estimated effects of air pollution on health? Accounting for model uncertainty using Bayesian model averaging

The long-term impact of air pollution on human health can be estimated from small-area ecological studies in which the health outcome is regressed against air pollution concentrations and other covariates, such as socio-economic deprivation. Socio-economic deprivation is multi-factorial and difficult to measure, and includes aspects of income, education, and housing as well as others. However, these variables are potentially highly correlated, meaning one can either create an overall deprivation index, or use the individual characteristics, which can result in a variety of pollution-health effects. Other aspects of model choice may affect the pollution-health estimate, such as the estimation of pollution, and spatial autocorrelation model. Therefore, we propose a Bayesian model averaging approach to combine the results from multiple statistical models to produce a more robust representation of the overall pollution-health effect. We investigate the relationship between nitrogen dioxide concentrations and cardio-respiratory mortality in West Central Scotland between 2006 and 2012

Stability of homogeneous chiral phases against inhomogeneous perturbations in 2+1 dimensions

In this work, inhomogeneous chiral phases are studied in a variety of Four-Fermion and Yukawa models in $2+1$ dimensions at zero and non-zero temperature and chemical potentials. Employing the mean-field approximation, we do not find indications for an inhomogeneous phase in any of the studied models. We show that the homogeneous phases are stable against inhomogeneous perturbations. At zero temperature, full analytic results are presented.Comment: 10 pages, 1 figure, contains ancillary files with plot data; talk given at the 39th International Symposium on Lattice Field theory (LATTICE 2022) in Bonn; August 8-13 202

Phase diagram of the 2+1-dimensional Gross-Neveu model with chiral imbalance

In this work, the phase diagram of the $2+1$-dimensional Gross-Neveu model is investigated with baryon chemical potential as well as chiral chemical potential in the mean-field approximation. We study the theory using two lattice discretizations, which are both based on naive fermions. An inhomogeneous chiral phase is observed only for one of the two discretizations. Our results suggest that this phase disappears in the continuum limit.Comment: 9 pages, 2 figures, contains ancillary files with plot data; talk given at the 38th International Symposium on Lattice Field theory (LATTICE 2021); July 26-30 202

Spatial modelling of air pollution, deprivation and mortality in Scotland

Air pollution is not only a major risk to the environment, but also a major environmental risk to the health of the population in developed and developing countries. The health impact of both short-term and long-term exposure to air pollution has been the focus of much research in the past few decades, which has investigated the relationship between specific air pollutants, such as carbon monoxide (CO), nitrogen dioxide (NO_2), particulate matter (PM_2.5 and PM_10), and sulphur dioxide (SO_2), to cardiovascular and respiratory diseases. The health impact of short-term exposure is conducted through time series studies, whereas long-term exposure is investigated through cohort studies. Cohort studies are considered the gold-standard research design since inference is made at the individual level and can directly assess cause and effect. However, cohort studies are costly and require a long follow-up period meaning they take a long time to conduct. To counteract these limitations, spatial ecological studies are used instead, which make use of routinely available disease data and air pollutant concentrations at a small areal level, such as census tracts or postcodes. This is to ensure the population under study is relatively homogeneous within the areal unit in terms of socio-demographic characteristics, and thus complements inference from a cohort study. These studies quantify the health impact of exposure to air pollution by relating geographical contrasts between air pollutant concentrations and disease risk across the chosen spatial resolution. The disease data are counts of the numbers of disease cases occurring in each areal unit, and Poisson log-linear models are used to assess the pollutant-health relationship. Other covariate information, such as socio-economic deprivation, is also included to help explain the spatial pattern in disease risk. However, the residual disease risk after the covariate effects have been accounted for tends to contain spatial autocorrelation, which has to be modelled in order to make sound inferences. Residual spatial autocorrelation is typically modelled by a set of random effects that utilise a neighbourhood matrix in order to induce spatial autocorrelation into the model. There are a number of specifications to model this, but this thesis makes use of the Leroux specification due to its flexibility in being able to model both strong and weak spatial autocorrelation. An important issue with using a spatial ecological study design is the estimation of spatially representative pollutant concentrations that are available in each areal unit. Studies can typically use measured data from fixed-location monitors that are spatially sparse and do not provide a pollutant concentration for each areal unit; or they make use of modelled concentrations available at a fine grid square resolution, which are known to contain biases and no measure of uncertainty. There have been numerous statistical approaches to combine both sets of information in order to estimate accurate and spatially representative concentrations. This thesis will develop previous methodology that utilises extra data sources in order to improve the prediction performance of the model for use in a Scottish context. The overarching aim of this thesis is to investigate the cardio-respiratory health effects of long-term exposure to air pollution in West Central Scotland, UK. As the majority of air pollution in this region results from vehicle emissions, nitrogen dioxide (NO_2), a traffic-related gaseous pollutant, will be used to measure air pollution. Models investigating its health effect will incorporate predicted measures of NO_2 developed in this thesis. The sensitivity of the pollutant-health effect to the choice of NO_2 concentrations, indicator of deprivation, and choice of spatial model will be investigated. Changing these factors has been shown to modify estimated pollutant-health effects.\\ Findings in this thesis demonstrated that improvements in the accuracy of fine scale spatial prediction of NO_2 concentrations can be made by utilising extra sources of data in addition to the commonly-used monitoring stations. In addition, the estimated pollutant-health effect is not robust to the choice of the aforementioned factors and the choice of these factors can have a major impact on the resulting pollutant-health effects. This justified the combination of all statistical models into a single effect size, which estimated a small, but positive effect of NO_2 concentrations on cardio-respiratory ill health. However, the estimated NO_2-health relationship was not substantial, possibly due to the NO_2 concentrations in West Central Scotland being too low. Greater variation in the exposure would be needed to observe substantial health impacts