A local limit theorem for the critical random graph

We consider the limit distribution of the orders of the k largest components in the Erd¿os-Rényi random graph inside the critical window for arbitrary k. We prove a local limit theorem for this joint distribution and derive an exact expression for the joint probability density function

Transforming fixed-length self-avoiding walks into radial SLE_8/3

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE with kappa=8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and then apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values

Dehydroepiandrosterone sulfate levels associated with decreased malaria parasite density and increased hemoglobin concentration in pubertal girls from western Kenya

In areas where Plasmodium falciparum malaria is endemic, parasite density, morbidity, and mortality decrease with increasing age, which supports the view that years of cumulative exposure are necessary for the expression of maximal protective immunity. Developmental changes in the host also have been implicated in the expression of maximal resistance. To further evaluate the contribution of host developmental factors in malaria resistance, we examined the relationship between P. falciparum parasitemia and pubertal development in a cross-sectional sample of 12 - 18-year-old schoolgirls from an area of intense transmission in western Kenya. Among pubertal girls, dehydroepiandrosterone sulfate (DHEAS) levels were significantly associated with decreased parasite density, even after adjustment for age. DHEAS levels also were related to increased hemoglobin levels, even after accounting for age and other determinants of hemoglobin level. These findings support the hypothesis that host pubertal development, independent of age and, by proxy, cumulative exposure, is necessary for maximal expression of resistance to malarial infection and morbidity, as assessed by hemoglobin leve

Osteosarcoma of the mobile spine†

Background The aims of this analysis were to investigate features and outcome of high-grade osteosarcomas of the mobile spine. Patients and methods Since 1977, 20 Cooperative Osteosarcoma Study Group patients had a diagnosis of high-grade osteosarcomas of the mobile spine and were included in this retrospective analysis of patient-, tumor- and treatment-related variables and outcome. Results The median age was 29 years (range 5-58). Most frequent tumor sites were thoracic and lumbar spine. All but three patients had nonmetastatic disease at diagnosis. Treatment included surgery and chemotherapy for all patients, 13 were also irradiated. Eight patients failed to achieve a macroscopically complete surgical remission (five local, one primary metastases, two both), six died, two are alive, both with radiotherapy. Of 12 patients with complete remission at all sites, three had a recurrence (two local, one metastases) and died. The median follow-up of the 11 survivors was 8.7 years (range 3.1-22.3), 5-year overall and event-free survival rates were 60% and 43%. Age <40 years, nonmetastatic disease at diagnosis and complete remission predicted for better overall survival (OS, P < 0.05). Conclusions Osteosarcomas of the mobile spine are rare. With complete resection (and potentially radiotherapy) and chemotherapy, prognosis may be comparable with that of appendicular osteosarcoma

Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.Comment: 18 pages, 4 figures. Some sections rewritten (including title and abstract), numerical results added, references added. Accepted for publication in J. Stat. Phy

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Comment: 45 pages, 12 figures. This is a revised version of math.PR/0504036 without the appendice

Geometric Exponents, SLE and Logarithmic Minimal Models

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.Comment: Added reference