Censored Glauber Dynamics for the mean field Ising Model

We study Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss Model. It is well known that at high temperature ($\beta < 1$) the mixing time is $\Theta(n\log n)$, whereas at low temperature ($\beta > 1$) it is $\exp(\Theta(n))$. Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed $\beta > 1$, the mixing-time of this model is $\Theta(n\log n)$, analogous to the high-temperature regime of the original dynamics. Furthermore, they showed \emph{cutoff} for the original dynamics for fixed $\beta<1$. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Currie-Weiss model. Namely, we found a scaling window of order $1/\sqrt{n}$ around the critical temperature $\beta_c=1$, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if $\beta = 1 + \delta$ for some $\delta > 0$ with $\delta^2 n \to \infty$, then the mixing-time has order $(n / \delta)\log(\delta^2 n)$. The cutoff constant is $(1/2+[2(\zeta^2 \beta / \delta - 1)]^{-1})$, where $\zeta$ is the unique positive root of $g(x)=\tanh(\beta x)-x$, and the cutoff window has order $n / \delta$.Comment: 55 pages, 4 figure

Introduction to Arithmetic Mirror Symmetry

We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics

Metal-insulator transition at B=0 in an ultra-low density (rs=23r_{s}=23) two dimensional GaAs/AlGaAs hole gas

We have observed a metal-insulator transition in an ultra-low density two dimensional hole gas formed in a high quality GaAs-AlGaAs heterostructure at B=0. At the highest carrier density studied ($p_{s}=2.2x10^{10} cm^{-2}, r_{s}=16$) the hole gas is strongly metallic, with an exceptional mobility of $425,000 cm^{2}V^{-1}s^{-1}$. The low disorder and strength of the many-body interactions in this sample are highlighted by the observation of re-entrant metal insulator transitions in both the fractional ($\nu < 1/3$) and integer ($2 > \nu > 1$) quantum Hall regimes. On reducing the carrier density the temperature and electric field dependence of the resistivity show that the sample is still metallic at $p_{s}=1.3x10^{10} cm^{-2}$ ($r_{s}=21$), becoming insulating at $p_{s}{\simeq}1x10^{10} cm^{-2}$. Our results indicate that electron-electron interactions are dominant at these low densities, pointing to the many body origins of this metal-insulator transition. We note that the value of $r_{s}$ at the transition ($r_{s}=23 +/- 2$) is large enough to allow the formation of a weakly pinned Wigner crystal, and is approaching the value calculated for the condensation of a pure Wigner crystal.Comment: 4 pages, latex, 4 postscript figures, submitted to EP2DS-12 on 21st August 1997, to appear in Physica

Horizontal and vertical movements of starry smooth-hound Mustelus asterias in the northeast Atlantic

Commercial landings of starry smooth-hound Mustelus asterias in northern European seas are increasing, whilst our knowledge of their ecology, behaviour and population structure remains limited. M. asterias is a widely distributed demersal shark, occupying the waters of the southern North Sea and Irish Sea in the north, to at least the southern Bay of Biscay in the south, and is seasonally abundant in UK waters. There are no species-specific management measures for the northeast Atlantic stock, and the complexity of its population structure is not yet fully understood. To address this issue, we deployed both mark-recapture and electronic tags on M. asterias to gain novel insights into its horizontal and vertical movements. Our data suggest that the habitat use of M. asterias changes on a seasonal basis, with associated changes in geographical distribution, depth utilisation and experienced temperature. We report the first direct evidence of philopatry for this species, and also provide initial evidence of sex-biased dispersal and potential metapopulation-like stock structuring either side of the UK continental shelf. Investigations of finer-scale vertical movements revealed clear diel variation in vertical activity. The illustrated patterns of seasonal space-use and behaviour will provide important information to support the stock assessment process and will help inform any future management options

Hypertension and albuminuria in chronic kidney disease mapped to a mouse chromosome 11 locus

Chronic kidney disease (CKD) is a key cause of hypertension and a potent independent risk for cardiovascular disease. Epidemiological studies suggest a strong genetic component determining susceptibility for renal disease and, by inference, the associated cardiovascular risk. With a subtotal nephrectomy model of kidney disease, we found the 129S6 mouse strain to be very susceptible to the development of hypertension, albuminuria, and kidney injury, whereas the C57BL/6 strain is relatively resistant. Accordingly, we set out to map quantitative trait loci conferring susceptibility to hypertension and albuminuria using this model with F2 mice. We found significant linkage of the blood pressure trait to two loci. At D11Mit143, mice homozygous for the 129S6 allele had significantly higher systolic blood pressure than mice heterozygous or homozygous for the C57BL/6 allele. Similarly, at D1Mit308, there was an excellent correlation between genotype and the blood pressure phenotype. The effect of the chromosome 11 locus was verified with a separate cohort of F2 mice. For the albuminuria trait, a significant locus was found at D11Mit143, which overlaps the blood pressure trait locus. Our studies have identified a region spanning ∼8 cM on mouse chromosome 11 that is associated with susceptibility to hypertension and albuminuria in CKD

Engineering the spin polarization of one-dimensional electrons

We present results of magneto-focusing on the controlled monitoring of spin polarization within a one-dimensional (1D) channel, and its subsequent effect on modulating the spin-orbit interaction (SOI) in a 2D GaAs electron gas. We demonstrate that electrons within a 1D channel can be partially spin polarized as the effective length of the 1D channel is varied in agreement with the theoretical prediction. Such polarized 1D electrons when injected into a 2D region result in a split in the odd-focusing peaks, whereas the even peaks remain unaffected (single peak). On the other hand, the unpolarized electrons, achieved by reducing the effective length of the 1D channel, do not affect the focusing spectrum and the odd and even peaks remain as single peaks, respectively. The split in odd-focusing peaks is evidence of direct measurement of spin polarization within a 1D channel, where each sub-peak represents the population of a particular spin state. Confirmation of the spin splitting is determined by a selective modulation of the focusing peaks due to the Zeeman energy in the presence of an in-plane magnetic field. We suggest that the SOI in the 2D regime is enhanced by a stream of polarized 1D electrons. The spatial control of spin states of injected 1D electrons and the possibility of tuning the SOI may open up a new regime of spin-engineering with application in future quantum information schemes

High-resolution error detection in the capture process of a single-electron pump

The dynamic capture of electrons in a semiconductor quantum dot (QD) by raising a potential barrier is a crucial stage in metrological quantized charge pumping. In this work, we use a quantum point contact (QPC) charge sensor to study errors in the electron capture process of a QD formed in a GaAs heterostructure. Using a two-step measurement protocol to compensate for 1/f noise in the QPC current, and repeating the protocol more than 106 times, we are able to resolve errors with probabilities of order 106. For the studied sample, one-electron capture is affected by errors in 30 out of every million cycles, while two-electron capture was performed more than 106 times with only one error. For errors in one-electron capture, we detect both failure to capture an electron and capture of two electrons. Electron counting measurements are a valuable tool for investigating non-equilibrium charge capture dynamics, and necessary for validating the metrological accuracy of semiconductor electron pumps

Enhanced indistinguishability of in-plane single photons by resonance fluorescence on an integrated quantum dot

Integrated quantum light sources in photonic circuits are envisaged as the building blocks of future on-chip architectures for quantum logic operations. While semiconductor quantum dots have been proven to be the highly efficient emitters of quantum light, their interaction with the host material induces spectral decoherence, which decreases the indistinguishability of the emitted photons and limits their functionality. Here, we show that the indistinguishability of in-plane photons can be greatly enhanced by performing resonance fluorescence on a quantum dot coupled to a photonic crystal waveguide. We find that the resonant optical excitation of an exciton state induces an increase in the emitted single-photon coherence by a factor of 15. Two-photon interference experiments reveal a visibility of 0.80 ± 0.03, which is in good agreement with our theoretical model. Combined with the high in-plane light-injection efficiency of photonic crystal waveguides, our results pave the way for the use of this system for the on-chip generation and transmission of highly indistinguishable photons

Photon Statistics; Nonlinear Spectroscopy of Single Quantum Systems

A unified description of multitime correlation functions, nonlinear response functions, and quantum measurements is developed using a common generating function which allows a direct comparison of their information content. A general formal expression for photon counting statistics from single quantum objects is derived in terms of Liouville space correlation functions of the material system by making a single assumption that spontaneous emission is described by a master equation

Some investigations into non passive listening

Our knowledge of the function of the auditory nervous system is based upon a wealth of data obtained, for the most part, in anaesthetised animals. More recently, it has been generally acknowledged that factors such as attention profoundly modulate the activity of sensory systems and this can take place at many levels of processing. Imaging studies, in particular, have revealed the greater activation of auditory areas and areas outside of sensory processing areas when attending to a stimulus. We present here a brief review of the consequences of such non-passive listening and go on to describe some of the experiments we are conducting to investigate them. In imaging studies, using fMRI, we can demonstrate the activation of attention networks that are non-specific to the sensory modality as well as greater and different activation of the areas of the supra-temporal plane that includes primary and secondary auditory areas. The profuse descending connections of the auditory system seem likely to be part of the mechanisms subserving attention to sound. These are generally thought to be largely inactivated by anaesthesia. However, we have been able to demonstrate that even in an anaesthetised preparation, removing the descending control from the cortex leads to quite profound changes in the temporal patterns of activation by sounds in thalamus and inferior colliculus. Some of these effects seem to be specific to the ear of stimulation and affect interaural processing. To bridge these observations we are developing an awake behaving preparation involving freely moving animals in which it will be possible to investigate the effects of consciousness (by contrasting awake and anaesthetized), passive and active listening