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.

Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model

We present a mathematical proof of theoretical predictions made by Arguin and Saint-Aubin, as well as by Bauer, Bernard, and Kytola, about certain non-local observables for the two-dimensional Ising model at criticality by combining Smirnov's recent proof of the fact that the scaling limit of critical Ising interfaces can be described by chordal SLE(3) with Kozdron and Lawler's configurational measure on mutually avoiding chordal SLE paths. As an extension of this result, we also compute the probability that an SLE(k) path, k in (0,4], and a Brownian motion excursion do not intersect.Comment: v1: 17 pages, 4 figures, to appear in J. Phys. A: Math. Theor

Moderate exercise may attenuate some aspects of immunosenescence

BACKGROUND: Immunosenescence is related to the deterioration of many immune functions, which may be manifested in increased susceptibility to infection, cancer, and autoimmunity. Lifestyle factors, such as diet or physical activity, may influence the senescence of the immune system. It is widely accepted that moderate physical activity may cause beneficial effects for physical and psychological health as well as for the immune system activity in aged people. METHODS: Thirty elderly women aged 62 to 86 were subjected to a two-years authorized physical activity program. Peripheral blood lymphocytes distribution and the production of cytokines involved in the immune response development and regulation (IL-2, IL-4 and IFN-γ) were investigated. The same parameters were evaluated in two control groups of women: a sedentary group of 12 elderly women selected for the second round of the physical activity program and in a group of 20 sedentary young women. Flow cytometry methods were used for the examination of surface markers on peripheral blood lymphocytes and intracellular cytokines expression. RESULTS: The distribution of the main lymphocytes subpopulations in the peripheral blood of elderly women did not show changes after long-term moderate physical training. The percentage of lymphocytes expressing intracellular IL-2 was higher in the group of women attending 2-years physical activity program than in the control group of elderly sedentary women, and it was similar to the value estimated in the group of young sedentary women. There was no difference in the intracellular expression of IL-4 and IFN-γ between the active and elderly sedentary women. CONCLUSIONS: Our results suggest that moderate, long-term physical activity in elderly women may increase the production of IL-2, an important regulator of the immune response. This may help ameliorate immunosenescence in these women

The Difference Between a Discrete and Continuous Harmonic Measure

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius of $h$. For a simply connected domain $D$ in the plane, let $\omega_h(0,\cdot;D)$ be the discrete harmonic measure at $0\in D$ associated with this random walk, and $\omega(0,\cdot;D)$ be the (continuous) harmonic measure at $0$. For domains $D$ with analytic boundary, we prove there is a bounded continuous function $\sigma_D(z)$ on $\partial D$ such that for functions $g$ which are in $C^{2+\alpha}(\partial D)$ for some $\alpha>0$ $$\lim_{h\downarrow 0} \frac{\int_{\partial D} g(\xi) \omega_h(0,|d\xi|;D) -\int_{\partial D} g(\xi)\omega(0,|d\xi|;D)}{h} = \int_{\partial D}g(z) \sigma_D(z) |dz|.$$ We give an explicit formula for $\sigma_D$ in terms of the conformal map from $D$ to the unit disc. The proof relies on some fine approximations of the potential kernel and Green's function of the random walk by their continuous counterparts, which may be of independent interest.Comment: 16 pages, revision after the referee's report, to appear in Journal of Theoretical Probabilit