Extremal statistics of curved growing interfaces in 1+1 dimensions

We study the joint probability distribution function (pdf) of the maximum M of the height and its position X_M of a curved growing interface belonging to the universality class described by the Kardar-Parisi-Zhang equation in 1+1 dimensions. We obtain exact results for the closely related problem of p non-intersecting Brownian bridges where we compute the joint pdf P_p(M,\tau_M) where \tau_M is there the time at which the maximal height M is reached. Our analytical results, in the limit p \to \infty, become exact for the interface problem in the growth regime. We show that our results, for moderate values of p \sim 10 describe accurately our numerical data of a prototype of these systems, the polynuclear growth model in droplet geometry. We also discuss applications of our results to the ground state configuration of the directed polymer in a random potential with one fixed endpoint.Comment: 6 pages, 4 figures. Published version, to appear in Europhysics Letters. New results added for non-intersecting excursion

Distribution of the time at which N vicious walkers reach their maximal height

We study the extreme statistics of N non-intersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum M and of the time \tau_M at which this maximum is reached. We focus in particular on non-intersecting Brownian bridges ("watermelons without wall") and non-intersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelons configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium (DPRM) with one free end-point on the other hand. We also check our results using numerical simulations of Dyson's Brownian motion and confront them with numerical simulations of the Polynuclear Growth Model (PNG) and of a model of DPRM on a discrete lattice. Some of the results presented here were announced in a recent letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].Comment: 30 pages, 12 figure

Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials

We study the distribution of the maximal height of the outermost path in the model of $N$ nonintersecting Brownian motions on the half-line as $N\to \infty$, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the $\textrm{Airy}_2$ process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.Comment: 39 pages, 4 figures; The title has been changed from "The limiting distribution of the maximal height of nonintersecting Brownian excursions and discrete Gaussian orthogonal polynomials." This is a reflection of the fact that the analysis has been adapted to include nonintersecting Brownian motions with either reflecting of absorbing boundaries at zero. To appear in J. Stat. Phy

Osteoarticular Infections in Pediatric Hospitals in Europe: A Prospective Cohort Study From the EUCLIDS Consortium

BACKGROUND: Pediatric osteoarticular infections (POAIs) are serious diseases requiring early diagnosis and treatment. METHODS: In this prospective multicenter cohort study, children with POAIs were selected from the European Union Childhood Life-threatening Infectious Diseases Study (EUCLIDS) database to analyze their demographic, clinical, and microbiological data. RESULTS: A cohort of 380 patients with POAIs, 203 with osteomyelitis (OM), 158 with septic arthritis (SA), and 19 with both OM and SA, was analyzed. Thirty-five patients were admitted to the Pediatric Intensive Care Unit; out of these, six suffered from shock, one needed an amputation of the right foot and of four left toes, and two had skin transplantation. According to the Pediatric Overall Performance Score, 36 (10.5%) showed a mild overall disability, 3 (0.8%) a moderate, and 1 (0.2%) a severe overall disability at discharge. A causative organism was detected in 65% (247/380) of patients. Staphylococcus aureus (S. aureus) was identified in 57.1% (141/247) of microbiological confirmed cases, including 1 (0.7%) methicillin-resistant S. aureus (MRSA) and 6 (4.2%) Panton-Valentine leukocidin (PVL)-producing S. aureus, followed by Group A Streptococcus (18.2%) and Kingella kingae (8.9%). K. kingae and PVL production in S. aureus were less frequently reported than expected from the literature. CONCLUSION: POAIs are associated with a substantial morbidity in European children, with S. aureus being the major detected pathogen. In one-third of patients, no causative organism is identified. Our observations show an urgent need for the development of a vaccine against S. aureus and for the development of new microbiologic diagnostic guidelines for POAIs in European pediatric hospitals