Interface growth in two dimensions: A Loewner-equation approach

The problem of Laplacian growth in two dimensions is considered within the Loewner-equation framework. Initially the problem of fingered growth recently discussed by Gubiec and Szymczak [T. Gubiec and P. Szymczak, Phys. Rev. E 77, 041602 (2008)] is revisited and a new exact solution for a three-finger configuration is reported. Then a general class of growth models for an interface growing in the upper-half plane is introduced and the corresponding Loewner equation for the problem is derived. Several examples are given including interfaces with one or more tips as well as multiple growing interfaces. A generalization of our interface growth model in terms of ``Loewner domains,'' where the growth rule is specified by a time evolving measure, is briefly discussed.Comment: To appear in Physical Review

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work. For the shape consisting of $n=\omega_d r^d$ sites, where $\omega_d$ is the volume of the unit ball in $\R^d$, we show that the inradius of the set of occupied sites is at least $r-O(\log r)$, while the outradius is at most $r+O(r^\alpha)$ for any $\alpha > 1-1/d$. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with $n=\pi r^2$ particles, we show that the inradius is at least $r/\sqrt{3}$, and the outradius is at most $(r+o(r))/\sqrt{2}$. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions.Comment: [v3] Added Theorem 4.1, which generalizes Theorem 1.4 for the abelian sandpile. [v4] Added references and improved exposition in sections 2 and 4. [v5] Final version, to appear in Potential Analysi

Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all \b>0 there exists \rho (d,\b)N(d,\b), any $x \in \Z^d$, and any $A\subset \{1,..., n\}^d$ | \{y\in\Z^d\colon \nu_{A,x}(y) \geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as \b \to \infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne

Families of Vicious Walkers

We consider a generalisation of the vicious walker problem in which N random walkers in R^d are grouped into p families. Using field-theoretic renormalisation group methods we calculate the asymptotic behaviour of the probability that no pairs of walkers from different families have met up to time t. For d>2, this is constant, but for d<2 it decays as a power t^(-alpha), which we compute to O(epsilon^2) in an expansion in epsilon=2-d. The second order term depends on the ratios of the diffusivities of the different families. In two dimensions, we find a logarithmic decay (ln t)^(-alpha'), and compute alpha' exactly.Comment: 20 pages, 5 figures. v.2: minor additions and correction

Projected climate-induced faunal change in the western hemisphere

Climate change is predicted to be one of the greatest drivers of ecological change in the coming century. Increases in temperature over the last century have clearly been linked to shifts in species distributions. Given the magnitude of projected future climatic changes, we can expect even larger range shifts in the coming century. These changes will, in turn, alter ecological communities and the functioning of ecosystems. Despite the seriousness of predicted climate change, the uncertainty in climate-change projections makes it difficult for conservation managers and planners to proactively respond to climate stresses. To address one aspect of this uncertainty, we identified predictions of faunal change for which a high level of consensus was exhibited by different climate models. Specifically, we assessed the potential effects of 30 coupled atmosphere–ocean general circulation model (AOGCM) future-climate simulations on the geographic ranges of 2954 species of birds, mammals, and amphibians in the Western Hemisphere. Eighty percent of the climate projections based on a relatively low greenhouse-gas emissions scenario result in the local loss of at least 10% of the vertebrate fauna over much of North and South America. The largest changes in fauna are predicted for the tundra, Central America, and the Andes Mountains where, assuming no dispersal constraints, specific areas are likely to experience over 90% turnover, so that faunal distributions in the future will bear little resemblance to those of today

Analysis of a fully packed loop model arising in a magnetic Coulomb phase

The Coulomb phase of spin ice, and indeed the Ic phase of water ice, naturally realise a fully-packed two-colour loop model in three dimensions. We present a detailed analysis of the statistics of these loops, which avoid themselves and other loops of the same colour, and contrast their behaviour to an analogous two-dimensional model. The properties of another extended degree of freedom are also addressed, flux lines of the emergent gauge field of the Coulomb phase, which appear as "Dirac strings" in spin ice. We mention implications of these results for related models, and experiments.Comment: 5 pages, 4 figure

Determinantal Correlations of Brownian Paths in the Plane with Nonintersection Condition on their Loop-Erased Parts

As an image of the many-to-one map of loop-erasing operation \LE of random walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk (LERW) model is the statistical ensemble of SAWs such that the weight of each SAW $\zeta$ is given by the total weight of all random walks $\pi$ which are inverse images of $\zeta$, \{\pi: \LE(\pi)=\zeta \}. We regard the Brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the LERW model. Following the theory of Fomin on nonintersecting LERWs, we introduce a nonintersecting system of $N$-tuples of LEBPs in a domain $D$ in the complex plane, where the total weight of nonintersecting LEBPs is given by Fomin's determinant of an $N \times N$ matrix whose entries are boundary Poisson kernels in $D$. We set a sequence of chambers in a planar domain and observe the first passage points at which $N$ Brownian paths $(\gamma_1,..., \gamma_N)$ first enter each chamber, under the condition that the loop-erased parts (\LE(\gamma_1),..., \LE(\gamma_N)) make a system of nonintersecting LEBPs in the domain in the sense of Fomin. We prove that the correlation functions of first passage points of the Brownian paths of the present system are generally given by determinants specified by a continuous function called the correlation kernel. The correlation kernel is of Eynard-Mehta type, which has appeared in two-matrix models and time-dependent matrix models studied in random matrix theory. Conformal covariance of correlation functions is demonstrated.Comment: v3: REVTeX4, 27 pages, 10 figures, corrections made for publication in Phys.Rev.

The Hamburg/ESO R-process Enhanced Star survey (HERES). V. Detailed abundance analysis of the r-process enhanced star HE 2327-5642

We report on a detailed abundance analysis of the strongly r-process enhanced giant star, HE 2327-5642 ([Fe/H] = -2.78, [r/Fe] = +0.99). Determination of stellar parameters and element abundances was based on analysis of high-quality VLT/UVES spectra. The surface gravity was calculated from the NLTE ionization balance between Fe I and Fe II, and Ca I and Ca II. Accurate abundances for a total of 40 elements and for 23 neutron-capture elements beyond Sr and up to Th were determined. The heavy element abundance pattern of HE 2327-5642 is in excellent agreement with those previously derived for other strongly r-process enhanced stars. Elements in the range from Ba to Hf match the scaled Solar r-process pattern very well. No firm conclusion can be drawn with respect to a relationship between the fisrt neutron-capture peak elements, Sr to Pd, in HE 2327-5642 and the Solar r-process, due to the uncertainty of the latter. A clear distinction in Sr/Eu abundance ratios was found between the halo stars with different europium enhancement. The strongly r-process enhanced stars reveal a low Sr/Eu abundance ratio at [Sr/Eu] = -0.92+-0.13, while the stars with 0 < [Eu/Fe] < 1 and [Eu/Fe] < 0 have 0.36 dex and 0.93 dex larger Sr/Eu values, respectively. Radioactive dating for HE 2327-5642 with the observed thorium and rare-earth element abundance pairs results in an average age of 13.3 Gyr, when based on the high-entropy wind calculations, and 5.9 Gyr, when using the Solar r-residuals. HE 2327-5642 is suspected to be radial-velocity variable based on our high-resolution spectra, covering ~4.3 years.Comment: 16 pages, 12 figures, accepted to A&

On harmonic measure of critical curves

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge $c\leqslant 1$, scaling exponents of harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. {\bf 84}, 1363 (2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument that allows us to connect harmonic measure of critical curves to operators obtained by fusion of primary fields, and compute characteristics of fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with $c\leqslant 1$.Comment: Some more correction

Massage therapy: attitudes and use among General Practitioners and Patients in Auckland

Aim To begin to explore beliefs, attitudes and use of massage therapy and complementary and alternative medicine (CAM) among GPs and GP patients in the Auckland region. Methods Two hundred GPs were randomly selected from local directories and sent a self-administered, anonymous questionnaire. Patients were recruited from four clinics in the Auckland region. Questions addressed referral, reasons for referral and use of massage therapy and CAM in general. Items assessing attitudes and beliefs about the effectiveness and safety of massage therapy were also included. Results Of the 200 GPs, 86 (43%) returned questionnaires; of those, 74 were useable. The majority of GP respondents (55, 74%) had recommended that a patient see a massage therapist in the last year, most commonly for musculoskeletal problems, stress and pain. Most GP respondents (63, 85%) felt they had some knowledge of massage therapy and agreed that massage reduces stress (67, 91%) and anxiety (62, 84%); only six (8%) GP respondents believed that massage effects were due to a placebo effect. Of the 53 patients approached at the clinics 92% agreed to complete the anonymous questionnaire. Of these patients, 30 (61%) had received a massage over the last year, and 12 (24%) reported receiving a doctor’s recommendation for massage. Reasons for receiving massage included relaxation, stress, and sprains and strains. Only 15 (39%) of the patients who had received a massage reported checking massage therapists’ qualifications. Most patient respondents agreed that massage therapy reduces stress and anxiety, stimulates the body’s natural healing ability and improves circulation. Most GP and patient respondents were unsure about whether massage therapy enhanced immune function. Most GP respondents (63, 85%) want to see regulatory standards established for CAM, with the same scientific investigation of complementary and alternative therapies as there is for conventional therapies. Only nine (18%) patient respondents reported discussing massage therapy as a treatment with their GPs. Importantly, only five (10%) patients reported feeling reluctant to tell their GPs about their use of CAM. Conclusions This is the first study to specifically assess massage therapy attitudes and use in samples of GPs and noncancer patients in New Zealand. Although the samples involve only a small proportion of GPs and patients in the Auckland area and the GP response rate was low, the study provides preliminary evidence of favourable attitudes and use of massage therapy among GPs and patients in this region. GP and patient respondents in this study held positive views about the effects of massage therapy, particularly for treating stress and anxiety, and had high referral and use rates. A larger, more representative study is needed to determine the generalisability of the observed levels of interest and use in massage therapy and other forms of CAM. In contrast to trends noted in the United States, patient respondents reported that they feel comfortable telling their GPs about their use of CAM