Multidimensional sticky Brownian motions as limits of exclusion processes

We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and the entire particle system is slowed down until the ``collision'' is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and reflects at the boundary of the wedge after spending an instant of time there. In particular, this leads to a natural multidimensional generalization of sticky Brownian motion on the half-line, which is of interest in both queuing theory and stochastic portfolio theory. For instance, this can model a market, which experiences a slowdown due to a major event (such as a court trial between some of the largest firms in the market) deciding about the new market leader.Comment: Published at http://dx.doi.org/10.1214/14-AAP1019 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counterintuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erd\H{o}s-R\'enyi random graphs $G(n,p)$ with constant edge density $p \in (0,1)$, the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of $G(n,p)$, which might be of independent interest.Comment: Version 2, minor change

Testing for high-dimensional geometry in random graphs

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erd\H{o}s-R\'enyi random graph $G(n,p)$. Under the alternative, the graph is generated from the $G(n,p,d)$ model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere $\mathbb{S}^{d-1}$, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., $p$ is a constant), we propose a near-optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in $G(n,p,d)$.Comment: 28 pages; v2 contains minor change

A glükokortikoidok hatását és metabolizmusát befolyásoló gén-polimorfizmusok patofiziológiai szerepének vizsgálata = Pathophysiological significance of gene polymorphisms affecting glucocorticoid action and metabolism

A HSD11B1 génen in silico kutatásainkkal azonosított 65 polimorfizmus közül - transzkripciós faktorok kötőhelyein elhelyezkedésük alapján - 12 polimorfizmust részletesen vizsgáltunk és meghatároztuk populáció-szintű gyakoriságukat. A vizsgált 12 polimorfizmus közül a rs4844880 polimorf allél bizonyult kiemelten jelentősnek; a polimorfizmus hordozása kedvezőbb csont ásványianyag tartalommal (BMC) társult és jelenléte esetén Cushing-szindrómás betegekben magasabb szérum kortizol és osteokalcin koncentrációkat detektáltunk. Kimutattuk, hogy a polimorf allélt tartalmazó vektorral transzfektált Hela sejtek csökkent luciferáz aktivitást mutattak, ami magyarázza a rs4844880 allél hordozás csontanyagcserére kifejtett kedvező hatását. A GR gén 4 polimorfizmusának (Bcl1, N363S, ER22/23EK és A3669G) vizsgálatával kimutattuk, hogy a Bcl1 polimorfizmus jelenléte csökkent BMC értékkel társul Cushing-szindrómás betegekben. Kimutattuk továbbá, hogy várandós nőkben a Bcl1 polimorfizmus hordozása kockázati tényező a HELLP szindróma kialakulásában, és az ER22/23EK polimorfizmus jelenléte várandós nőkben csökkenti a graviditás alatti testsúly növekedés kockázatát. Mindezek az eredmények új adatokkal egészítik ki a glükokortikoidok iránti érzékenység pathofiziológiai jelentőségét, melyet nagyrészt genetikai tényezők, köztük elsősorban a HSD11B1 és GR gének variánsai határoznak meg. | Of the 65 polymorphisms identified in the HSD11B1 gene using our in silico analysis, 12 were further investigated based on their critical locations in sites which bind transcription factors, and their allelic frequencies were determined in healthy subjects. Of these 12 polymorphic alleles, the rs4844880 proved to be particularly important, because this polymorphic allele was associated with better bone mineral content (BMC) and, in patients with Cushing’s syndrome, with higher serum cortisol and osteocalcin concentrations. Hela cells transfected with this polymorphic variant showed decreased luciferase activity compared to those transfected with the wild-type variant, thus explaining the beneficial effect of the rs4844880 polymorphism on bone metabolism. Studies on the 4 polymorphisms of the GR gene (Bcl1, N363S, ER22/23EK and A3669G) indicated that the polymorphic Bcl1 allele was associated with a lower bone mineral content in patients with Cushing’s syndrome. In addition, we showed that the presence of this allele is a risk factor for HELLP syndrome in pregnant women, and that carriers of the ER22/23/EK polymorphism have a lower risk for weight gain during pregnancy. All these finding provide novel data on the pathophysiologic significance of glucocorticoid sensitivity, determined by genetic factors mainly including the HSD11B1 and GR gene variants

Can one hear the shape of a population history?

Reconstructing past population size from present day genetic data is a major goal of population genetics. Recent empirical studies infer population size history using coalescent-based models applied to a small number of individuals. Here we provide tight bounds on the amount of exact coalescence time data needed to recover the population size history of a single, panmictic population at a certain level of accuracy. In practice, coalescence times are estimated from sequence data and so our lower bounds should be taken as rather conservative.Comment: 22 pages, 7 figures; v2 is significantly revised from v