Precise Asymptotics Research Articles

Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere

In our previous work (Angelopoulos 2018 Adv. Math. 323 529–621) we showed that the coefficient in the precise leading-order late-time asymptotics for solutions to the wave equation with smooth, compactly supported initial data on Schwarzschild backgrounds is proportional to the time-inverted Newman–Penrose constant (TINP), that is the Newman–Penrose constant of the associated time integral. The time integral (and hence the TINP constant) is canonically defined in the domain of dependence of any Cauchy hypersurface along which the stationary Killing field is non-vanishing. As a result, an explicit expression of the late-time polynomial tails was obtained in terms of initial data on Cauchy hypersurfaces intersecting the future event horizon to the future of the bifurcation sphere.In this paper, we extend the above result to Cauchy hypersurfaces intersecting the bifurcation sphere via a novel geometric interpretation of the TINP constant in terms of a modified gradient flux on Cauchy hypersurfaces. We show, without appealing to the time integral construction, that a general conservation law holds for these gradient fluxes. This allows us to express the TINP constant in terms of initial data on Cauchy hypersurfaces for which the time integral construction breaks down.

Precise Large Deviations for Subexponential Distributions in a Multi Risk Model

The precise large deviations asymptotics for the sums of independent identical random variables when the distribution of the summand belongs to the class S ∗ of heavy tailed distributions is studied. Under mild conditions, we extend the previous results from the paper Denisov et al. (2010) to asymptotics that are valid uniformly over some time interval. Finally, we apply the main result on the multi-risk model introduced by Wang and Wang (2007).

Strong law of large numbers for the capacity of the Wiener sausage in dimension four

We prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four, where a logarithmic correction appears in the scaling. The main step of the proof is to obtain precise asymptotics for the expected value of the capacity. This requires a delicate analysis of intersection probabilities between two independent Wiener sausages.

On the phase transition curve in a directed exponential random graph model

Abstract We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the important statistical content of such models is given by the normalization constant of the models, and, in particular, an appropriately scaled limit of the normalization, which is called the free energy. We derive precise asymptotics for the normalization constant for finite graphs. We use this to derive a formula for the free energy. The limit is analytic everywhere except along a curve corresponding to a first-order phase transition. We examine unusual behavior of the model along the phase transition curve.

Christoffel functions with power type weights

Precise asymptotics for Christoffel functions are established for power type weights on unions of Jordan curves and arcs. The asymptotics involve the equilibrium measure of the support of the measure. The result at the endpoints of arc components is obtained from the corresponding asymptotics for internal points with respect to a different power weight. On curve components the asymptotic formula is proved via a sharp form of Hilbert's lemniscate theorem while taking polynomial inverse images. The situation is completely different on the arc components, where the local asymptotics is obtained via a discretization of the equilibrium measure with respect to the zeros of an associated Bessel function. The proofs are potential theoretical, and fast decreasing polynomials play an essential role in them.

Extinction times in the subcritical stochastic SIS logistic epidemic.

Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given sizeN. We study the behaviour of the process as the population size N tends to infinity. Our results cover the entire subcritical regime, including the "barely subcritical" regime, where the recovery rate exceeds the infection rate by an amount that tends to0 as [Formula: see text] but more slowly than [Formula: see text]. We derive precise asymptotics for the distribution of the extinction time and the total number of cases throughout the subcritical regime, give a detailed description of the course of the epidemic, and compare to numerical results for a range of parameter values. We hypothesise that features of the course of the epidemic will be seen in a wide class of other epidemic models, and we use real data to provide some tentative and preliminary support for this theory.

High degrees in random recursive trees

AbstractFor , let Tn be a random recursive tree (RRT) on the vertex set . Let be the degree of vertex v in Tn, that is, the number of children of v in Tn. Devroye and Lu showed that the maximum degree Δn of Tn satisfies almost surely; Goh and Schmutz showed distributional convergence of along suitable subsequences. In this work we show how a version of Kingman's coalescent can be used to access much finer properties of the degree distribution in Tn. For any , let . Also, let be a Poisson point process on with rate function . We show that, up to lattice effects, the vectors converge weakly in distribution to . We also prove asymptotic normality of when slowly, and obtain precise asymptotics for when and is not too large. Our results recover and extends the previous distributional convergence results on maximal and near‐maximal degrees in RRT.

Exact solution and precise asymptotics of a Fisher–KPP type front

The present work concerns a version of the Fisher–KPP equation where the nonlinear term is replaced by a saturation mechanism, yielding a free boundary problem with mixed conditions. Following an idea proposed in Brunet and Derrida (2015 J. Stat. Phys. 161 801), we show that the Laplace transform of the initial condition is directly related to some functional of the front position . We then obtain precise asymptotics of the front position by means of singularity analysis. In particular, we recover the so-called Ebert and van Saarloos correction (Ebert and van Saarloos 2000 Physica D 146 1), we obtain an additional term of order in this expansion, and we give precise conditions on the initial condition for those terms to be present.

Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior

Precise asymptotics known for the Green function of the Laplacian have found their analogs for bounded below periodic elliptic operators of the second-order below and at the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. In a previous work, two of the authors considered the case of a spectral edge. The main result of this article is finding such asymptotics near a gap edge, for “generic” periodic elliptic operators of second-order with real coefficients in dimension d ≥ 2 , when the gap edge occurs at a symmetry point of the Brillouin zone.

Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes

We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner–Nordström families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman–Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's polynomial law τ−3 as a lower bound. Our analysis relies on physical space techniques and uses the vector field approach for almost-sharp decay estimates introduced in our companion paper. In the black hole case, our estimates hold in the domain of outer communications up to and including the event horizon. Our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors.

Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times τu=inf{n:Yn>u} for the perpetuity sequence Yn=B1+A1B2+⋯+(A1…An−1)Bn,where (An,Bn) are i.i.d. random variables with values in R+×R. Recently, a number of limit theorems related to τu were proved including the law of large numbers, the central limit theorem and large deviations theorems (see Buraczewski et al., in press). We obtain a precise asymptotics of the sequence P[τu=logu∕ρ], ρ>0, u→∞ which considerably improves the previous results of Buraczewski et al. (in press). There, probabilities P[τu∈Iu] were identified, for some large intervals Iu around ku, with lengths growing at least as loglogu. Remarkable analogies and differences to random walks (Buraczewski and Maślanka, in press; Lalley, 1984) are discussed.

Uniform Asymptotic Expansions for the Fundamental Solution of Infinite Harmonic Chains

We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by 1/t uniformly in space. In particlar we give precise asymptotics for the cross-over from the t^{-1/2} decay of nondegenerate wave numbers to the degenerate t^{-1/3} decay of degenerate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.

Spectral Geometry of the Steklov Problem on Orbifolds

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two-dimensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components.

Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime

Nous considerons l’equation d’Anderson parabolique conduite par un bruit gaussien, fractionnaire en temps, et blanc ou fractionnaire en espace, qu’on resout dans un sens faible defini par une integrale de Skorokhod. Notre objectif est de donner l’exposant de Lyapounov pour les moments, et les asymptotiques des grands moments. Pour les asymptotiques en temps long, nos resultats mettent en evidence des phenomenes differents de ceux observes pour le regime Stratonovich, et dans le cas d’un bruit blanc en temps. Ces differences s’effacent neanmoins lorsque l’on considere les asymptotiques des grands moments. Nos resultats sont obtenus en introduisant une nouvelle inegalite variationnelle, et a l’aide d’outils nouveaux tels qu’un principe de grandes deviations de type Feynman–Kac, la linearisation par des approximations tangentes, et des techniques inspirees des probabilites dans les espaces de Banach.

A pointwise limit theorem for counting processes of perturbed random walks with an application to repeated significance tests

ABSTRACTHsu and Robbins (1947) introduced the concept of complete convergence as a complement to the Kolmogorov strong law in that they proved that provided the mean of the summands is zero and that the variance is finite. Later, Erdős proved the necessity (1949, 1950). Heyde (1975) proved that, under the same conditions, , thereby opening an area of research that has been called precise asymptotics. Both results above have been extended and generalized in various directions. Kao (1978) proved a pointwise version of Heyde’s result, viz. for the counting process , he showed that , where W(⋅) is the standard Wiener process. In this article, we prove an analog for perturbed random walks and illustrate how they enter naturally within the theory of repeated significance tests in exponential families.

Convergence rates in precise asymptotics for a kind of complete moment convergence

Liu and Lin (Statist. Probab. Lett. 2006) introduced a kind of complete moment convergence which includes complete convergence as a special case. In this paper, we study the convergence rates of the precise asymptotics for complete moment convergence introduced by Liu and Lin (2006) and get the corresponding convergence rates.

Exit times densities of the Bessel process

We examine the density functions of the first exit times of the Bessel process from the intervals [ 0 , 1 ) [0,1) and ( 0 , 1 ) (0,1) . First, we express them by means of the transition density function of the killed process. Using that relationship we provide precise estimates and asymptotics of the exit time densities. In particular, the results hold for the first exit time of the n n -dimensional Brownian motion from a ball.

NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS

We consider the spectral behavior and noncommutative geometry of commutators$[P,f]$, where$P$is an operator of order 0 with geometric origin and$f$a multiplication operator by a function. When$f$is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions$f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

Spectrum of Neumann--Poincaré Operator on Annuli and Cloaking by Anomalous Localized Resonance for Linear Elasticity

We investigate anomalous localized resonance on the circular coated structure and cloaking due to it in the context of elastostatic systems. The structure consists of the circular core with constant Lame parameters and the circular shell with negative Lame parameters proportional to those of the core. We show that there is a nonzero number $k_0$ determined by Lame parameters such that two nonempty eigenvalue sequences of the Neumann--Poincare operator associated with the structure converge to $k_0$ and $-k_0$, respectively, and derive precise asymptotics of the convergence. We then show by qualitative estimates based on asymptotics of eigenvalues that cloaking by anomalous localized resonance takes place if and only if the dipole-type source lies inside critical radii determined by radii of the core and the shell. The critical radii corresponding to $k_0$ and $-k_0$ are different.

Profiles of PATRICIA Tries

A PATRICIA trie is a trie in which non-branching paths are compressed. The external profile $$B_{n,k}$$ , defined to be the number of leaves at level k of a PATRICIA trie on n nodes, is an important “summarizing” parameter, in terms of which several other parameters of interest can be formulated. Here we derive precise asymptotics for the expected value and variance of $$B_{n,k}$$ , as well as a central limit theorem with error bound on the characteristic function, for PATRICIA tries on n infinite binary strings generated by a memoryless source with bias $$p > 1/2$$ for $$k\sim \alpha \log n$$ with $$\alpha \in (1/\log (1/q) + \varepsilon , 1/\log (1/p) - \varepsilon )$$ for any fixed $$\varepsilon > 0$$ . In this range, $$ {\mathbb {E}}[B_{n,k}] = \varTheta ( {\mathrm {Var}}[B_{n,k}])$$ , and both are of the form $$\varTheta (n^{\beta (\alpha )}/\sqrt{\log n})$$ , where the $$\varTheta $$ hides bounded, periodic functions in $$\log n$$ whose Fourier series we explicitly determine. The compression property leads to extra terms in the Poisson functional equations for the profile which are not seen in tries or digital search trees, resulting in Mellin transforms which are only implicitly given in terms of the moments of $$B_{m,j}$$ for various m and j. Thus, the proofs require information about the profile outside the main range of interest. Our derivations rely on analytic techniques, including Mellin transforms, analytic de-Poissonization, the saddle point method, and careful bounding of complex functions.