Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions

Consider a critical nearest neighbor branching random walk on the $d$-dimensional integer lattice initiated by a single particle at the origin. Let $G_{n}$ be the event that the branching random walk survives to generation $n$. We obtain limit theorems conditional on the event $G_{n}$ for a variety of occupation statistics: (1) Let $V_{n}$ be the maximal number of particles at a single site at time $n$. If the offspring distribution has finite $\alpha$th moment for some integer $\alpha\geq 2$, then in dimensions 3 and higher, $V_n=O_p(n^{1/\alpha})$; and if the offspring distribution has an exponentially decaying tail, then $V_n=O_p(\log n)$ in dimensions 3 and higher, and $V_n=O_p((\log n)^2)$ in dimension 2. Furthermore, if the offspring distribution is non-degenerate then $P(V_n\geq \delta \log n | G_{n})\to 1$ for some $\delta >0$. (2) Let $M_{n} (j)$ be the number of multiplicity-$j$ sites in the $n$th generation, that is, sites occupied by exactly $j$ particles. In dimensions 3 and higher, the random variables $M_{n} (j)/n$ converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a "typical" site (that is, at the location of a randomly chosen particle of the $n$th generation) is of order $O_p(\log n)$, and the number of occupied sites is $O_p(n/\log n)$

Interacting nuclei in distant galaxies

The N-galaxy 3C 390.3 has been monitored spectroscopically since 1974 (Osterbrock, Koski and Phillips 1975; Oke 1988). From various archives and literature, it is found that the Balmer lines change their intensities and profiles in a dramatic manner. The H alpha profile is very broad and peculiar, and the relative intensities of its two humps changes consistently with time, possibly periodically. Before 1980, the blue hump was significantly stronger than the one in the red. From 1980 to 1983 the blue hump became stronger (see Oke 1988). After 1983 the H alpha profile has returned to its early shape and seems to have completed a full circle. Unlike the rapid (on the order of a month or even less) and aperiodic variation in the continuum and integrated line intensities, the change in broad profile seems slow and consistent. Taking the analogy of cataclysmic variables, the double-horn profiles have been observed in cases of interacting stars. For example, the emission lines, both in He II and hydrogen Balmer lines in GD 552 (Stover 1985) show double-horn profiles and periodical changes in their line profiles, including the change in ratios of two humps. It is understood that the D-wave components (Smak 1976) are the signature of an emitting disk and the S-wave component is from the emission at a hot spot which rotates and results in a moving component in the velocity space. The mass flow from the nearby interacting star provides the stream toward the core of a neutron star or white dwarf. Therefore, it is proposed that the variation of broad line profiles observed in 3C 390.3 may be the result of a pair of interacting massive cores. The rotational velocity dominates and produces a variable double-horn profile. However, the line widths observed in broad line radio galaxies are one order larger than that in interacting stars. The Balmer decrements imply a much smaller density (10(exp 10-12) cm(exp-3)) than that in the cataclysmic variables. The much larger velocity and much thinner density make it unlikely that the broad line emission is simply formed in an accretion disk. The authors postulate that a significant rotational motion is involved. If the observed squared profiles are indeed due to the rotational velocity field, one can naturally explain their rare occurrence

A new view of nonlinear water waves: the Hilbert spectrum

We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes

A phase transition for measure-valued SIR epidemic processes

We consider measure-valued processes $X=(X_t)$ that solve the following martingale problem: for a given initial measure $X_0$, and for all smooth, compactly supported test functions $\varphi$, \begin{eqnarray*}X_t(\varphi )=X_0(\varphi)+\frac{1}{2}\int _0^tX_s(\Delta \varphi )\,ds+\theta \int_0^tX_s(\varphi )\,ds\\{}-\int_0^tX_s(L_s\varphi )\,ds+M_t(\varphi ).\end{eqnarray*} Here $L_s(x)$ is the local time density process associated with $X$, and $M_t(\varphi )$ is a martingale with quadratic variation $[M(\varphi )]_t=\int_0^tX_s(\varphi ^2)\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $\theta_c(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $\theta>\theta_c(d)$, then the solution survives forever with positive probability, but if $\theta<\theta_c(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $\theta$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $\theta$; that is, for any compact set $K$, the process $X_t(K)=0$ eventually.Comment: Published in at http://dx.doi.org/10.1214/13-AOP846 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Differential geometric regularization for supervised learning of classifiers

We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to an estimator of the class probability P(y|\vec x). The regularization term measures the volume of this submanifold, based on the intuition that overfitting produces rapid local oscillations and hence large volume of the estimator. This technique can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. In experiments, we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification.http://proceedings.mlr.press/v48/baia16.pdfPublished versio