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

Genomic and geographical structure of human cytomegalovirus

Human cytomegalovirus (CMV) has infected humans since the origin of our species and currently infects most of the world’s population. Variability between CMV genomes is the highest of any human herpesvirus, yet large portions of the genome are conserved. Here, we show that the genome encodes 74 regions of relatively high variability each with 2 to 8 alleles. We then identified two patterns in the CMV genome. Conserved parts of the genome and a minority (32) of variable regions show geographic population structure with evidence for African or European clustering, although hybrid strains are present. We find no evidence that geographic segregation has been driven by host immune pressure affecting known antigenic sites. Forty-two variable regions show no geographical structure, with similar allele distributions across different continental populations. These “nongeographical” regions are significantly enriched for genes encoding immunomodulatory functions suggesting a core functional importance. We hypothesize that at least two CMV founder populations account for the geographical differences that are largely seen in the conserved portions of the genome, although the timing of separation and direction of spread between the two are not clear. In contrast, the similar allele frequencies among 42 variable regions of the genome, irrespective of geographical origin, are indicative of a second evolutionary process, namely balancing selection that may preserve properties critical to CMV biological function. Given that genetic differences between CMVs are postulated to alter immunogenicity and potentially function, understanding these two evolutionary processes could contribute important information for the development of globally effective vaccines and the identification of novel drug targets

Decay Properties of the Connectivity for Mixed Long Range Percolation Models on Zd\Z^d

In this short note we consider mixed short-long range independent bond percolation models on $\Z^{k+d}$. Let $p_{uv}$ be the probability that the edge $(u,v)$ will be open. Allowing a $x,y$-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity $\tau_{xy}$ is governed by the probability $p_{xy}$. The result holds up to the critical point.Comment: 6 page

Long Cycles in a Perturbed Mean Field Model of a Boson Gas

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density $\rho=\rho_{{\rm short}}+\rho_{{\rm long}}$ into the number density of particles belonging to cycles of finite length ($\rho_{{\rm short}}$) and to infinitely long cycles ($\rho_{{\rm long}}$) in the thermodynamic limit. For this model we prove that when there is Bose condensation, $\rho_{{\rm long}}$ is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of $\rho_{{\rm long}}\neq 0$ with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas.Comment: 10 page

Improving professional psychological practice through an increased repertoire of research methodologies : illustrated by the development of MOL.

Mental health problems present an increasing global disease burden making the development of effective and efficient psychological treatments an urgent public health priority. Despite the continued proliferation of treatments and large numbers of randomized controlled trials (RCTs), evidence suggests that pre-post effect sizes have been decreasing over time not increasing. Promoting RCTs as a gold standard of evidence has not been a useful strategy for advancing progress in the development of increasingly effective and efficient psychological treatments and has, in fact, created a divide between research and practice in professional psychology. To close this divide, other methodologies are needed that can assist in the rigorous development and evaluation of treatments in routine clinical practice. We outline some of the problems with using RCTs as the sole means of generating evidence for treatment effectiveness and efficiency and we use the development and evaluation of a transdiagnostic cognitive therapy to illustrate an alternative way of accumulating evidence through a much closer connection between research and practice. Ultimately, including other methodologies alongside RCTs that combine research and practice more seamlessly, will produce treatments of greater effectiveness and efficiency and help to reduce the global burden of mental health problems. (PsycINFO Database Record (c) 2017 APA, all rights reserved

Deep mining of oxysterols and cholestenoic acids in human plasma and cerebrospinal fluid: Quantification using isotope dilution mass spectrometry

Both plasma and cerebrospinal fluid (CSF) are rich in cholesterol and its metabolites. Here we describe in detail a methodology for the identification and quantification of multiple sterols including oxysterols and sterol-acids found in these fluids. The method is translatable to any laboratory with access to liquid chromatography – tandem mass spectrometry. The method exploits isotope-dilution mass spectrometry for absolute quantification of target metabolites. The method is applicable for semi-quantification of other sterols for which isotope labelled surrogates are not available and approximate quantification of partially identified sterols. Values are reported for non-esterified sterols in the absence of saponification and total sterols following saponification. In this way absolute quantification data is reported for 17 sterols in the NIST SRM 1950 plasma along with semi-quantitative data for 8 additional sterols and approximate quantification for one further sterol. In a pooled (CSF) sample used for internal quality control, absolute quantification was performed on 10 sterols, semi-quantification on 9 sterols and approximate quantification on a further three partially identified sterols. The value of the method is illustrated by confirming the sterol phenotype of a patient suffering from ACOX2 deficiency, a rare disorder of bile acid biosynthesis, and in a plasma sample from a patient suffering from cerebrotendinous xanthomatosis, where cholesterol 27-hydroxylase is deficient

Twistors and Black Holes

Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex coordinates adapted to the Heisenberg symmetries. The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. We explicitly construct the ``covariant c-map'' and the ``twistor map'', which relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates on Z (resp. S). As applications, we solve for the general BPS geodesic motion on M, and provide explicit integral formulae for the quaternionic Penrose transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated by first or second order differential operators. Finally, we compute the exact radial wave function (in the supergravity approximation) for BPS black holes with fixed electric and magnetic charges.Comment: 47 pages, v2: typos corrected, reference added, v3: minor change

On Ruelle's construction of the thermodynamic limit for the classical microcanonical entropy

In this note we make a very elementary technical observation to the effect that Ruelle's construction of the thermodynamic limit of the classical entropy density defined with a regularized microcanonical measure actually establishes the thermodynamic limit for the entropy density defined with the proper microcanonical measure. At this stage a key formula is still derived from the regularized measures. We also show that with only minor changes in the proof the regularization of the microcanonical measure is actually not needed at all.Comment: Short communication (7p), accepted for publication in J.Stat.Phy

'Education, education, education' : legal, moral and clinical

This article brings together Professor Donald Nicolson's intellectual interest in professional legal ethics and his long-standing involvement with law clinics both as an advisor at the University of Cape Town and Director of the University of Bristol Law Clinic and the University of Strathclyde Law Clinic. In this article he looks at how legal education may help start this process of character development, arguing that the best means is through student involvement in voluntary law clinics. And here he builds upon his recent article which argues for voluntary, community service oriented law clinics over those which emphasise the education of students

The phase transition of the quantum Ising model is sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called `random-parity' representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.Comment: Small changes. To appear in the Journal of Statistical Physic