Exact Asymptotic Results Research Articles

Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates

Abstract This article investigates the long-wave anti-plane shear motion in a symmetric three-layered laminate composed of pre-stressed compressible elastic layers. The layers of the laminate are perfectly bonded, while traction-free and fixed boundary conditions are considered on the outer faces of the laminate. In both cases, the dispersion relation is obtained in terms of symmetric and anti-symmetric decompositions. Numerical results and an asymptotic long-wave analysis are presented, corresponding to the three possible vibration modes. It is revealed that a low-frequency mode only exists in respect of symmetric motion with free-faces, while all other cases pose a series of non-zero cut-off frequencies. Comparisons between the exact and approximate asymptotic results are presented, and excellent agreement is observed.

Long-Wave Anti-Plane Motion in a Pre-Stressed Compressible Elastic Laminate with One Fixed and One Free Face

In this paper, long-wave anti-plane shear motion in a multilayered laminate composed of pre-stressed compressible elastic layers is investigated. The layers of the laminate are perfectly bonded, while a fixed-free boundary condition is prescribed on the outer faces of the laminate. The solution of the model is determined analytically via the propagator matrix and numerically through the asymptotic approach. Moreover, the numerical results featuring harmonic curves are presented graphically, together with an asymptotic long-wave analysis of the vibration modes. As a special case of materials, linear isotropic with one shear modulus is considered. A polynomial long-wave low-frequency approximation of the related dispersion relation is also studied. It governs dispersion curves including the lowest harmonic. It is revealed that a low-frequency mode exists in both the two- and three-layered laminates, which are taken as prototypical structures. Lastly, comparisons between the exact and approximate asymptotic results are presented, and excellent agreement is observed.

Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model

We investigate the non-equilibrium dynamics of a one-dimensional spin-1/2 XXZ model at zero-temperature in the regime |\Delta|< 1|Δ|<1, initially prepared in a product state with two domain walls i.e, |\downarrow\dots\downarrow\uparrow\dots\uparrow\downarrow\dots\downarrow \rangle|↓…↓↑…↑↓…↓⟩. At early times, the two domain walls evolve independently and only after a calculable time a non-trivial interplay between the two emerges and results in the occurrence of a split Fermi sea. For \Delta=0Δ=0, we derive exact asymptotic results for the magnetization and the spin current by using a semi-classical Wigner function approach, and we exactly determine the spreading of entanglement entropy exploiting the recently developed tools of quantum fluctuating hydrodynamics. In the interacting case, we analytically solve the Generalized Hydrodynamics equation providing exact expressions for the conserved quantities. We display some numerical results for the entanglement entropy also in the interacting case and we propose a conjecture for its asymptotic value.

Towards Degree Distribution of a Duplication-Divergence Graph Model

We present a rigorous and precise analysis of degree distribution in a dynamic graph model introduced by Solé, Pastor-Satorras et al. in which nodes are added according to a duplication-divergence mechanism. This model is discussed in numerous publications with only very few recent rigorous results, especially for the degree distribution. In this paper we focus on two related problems: the expected value and variance of the degree of a given node over the evolution of the graph and the expected value and variance of the average degree over all nodes. We present exact and precise asymptotic results showing that both quantities may decrease or increase over time depending on the model parameters. Our findings are a step towards a better understanding of the graph behaviors such as degree distributions, symmetry, power law, and structural compression.

Height of a liquid drop on a wetting stripe.

Adsorption of liquid on a planar wall decorated by a hydrophilic stripe of width L is considered. Under the condition that the wall is only partially wet (or dry) while the stripe tends to be wet completely, a liquid drop is formed above the stripe. The maximum height ℓ_{m}(δμ) of the drop depends on the stripe width L and the chemical potential departure from saturation δμ where it adopts the value ℓ_{0}=ℓ_{m}(0). Assuming a long-range potential of van der Waals type exerted by the stripe, the interfacial Hamiltonian model is used to show that ℓ_{0} is approached linearly with δμ with a slope which scales as L^{2} over the region satisfying L≲ξ_{∥}, where ξ_{∥} is the parallel correlation function pertinent to the stripe. This suggests that near the saturation there exists a universal curve ℓ_{m}(δμ) to which the adsorption isotherms corresponding to different values of L all collapse when appropriately rescaled. Although the series expansion based on the interfacial Hamiltonian model can be formed by considering higher order terms, a more appropriate approximation in the form of a rational function based on scaling arguments is proposed. The approximation is based on exact asymptotic results, namely, that ℓ_{m}∼δμ^{-1/3} for L→∞ and that ℓ_{m} obeys the correct δμ→0 behavior in line with the results of the interfacial Hamiltonian model. All the predictions are verified by the comparison with a microscopic density functional theory (DFT) and, in particular, the rational function approximation-even in its simplest form-is shown to be in a very reasonable agreement with DFT for a broad range of both δμ and L.

Enhanced Permutation Tests via Multiple Pruning.

Big multi-omics data in bioinformatics often consists of a huge number of features and relatively small numbers of samples. In addition, features from multi-omics data have their own specific characteristics depending on whether they are from genomics, proteomics, metabolomics, etc. Due to these distinct characteristics, standard statistical analyses using parametric-based assumptions may sometimes fail to provide exact asymptotic results. To resolve this issue, permutation tests can be a way to exactly analyze multi-omics data because they are distribution-free and flexible to use. In permutation tests, p-values are evaluated by estimating the locations of test statistics in an empirical null distribution generated by random shuffling. However, the permutation approach can be infeasible when the number of features increases, because more stringent control of type I error is needed for multiple hypothesis testing, and consequently, much larger numbers of permutations are required to reach significance. To address this problem, we propose a well-organized strategy, “ENhanced Permutation tests via multiple Pruning (ENPP).” ENPP prunes the features in every permutation round if they are determined to be non-significant. In other words, if the feature statistics from the permuted datasets exceed the feature statistics from the original dataset, beyond a predetermined threshold, the feature is determined to be non-significant. If so, ENPP removes the feature and iterates the process without the feature in the next permutation round. Our simulation study showed that the ENPP method could remove about 50% of the features at the first permutation round, and, by the 100th permutation round, 98% of the features had been removed and only 7.4% of the computation time with the original unpruned permutation approach had elapsed. In addition, we applied this approach to a real data set (Korea Association REsource: KARE) of 327,872 SNPs to find association with a non-normally distributed phenotype (fasting plasma glucose), interpreted the results, and discussed the feasibility and advantages of the approach.

Some Results on the Optimal Matching Problem for the Jacobi Model

We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let $X_1, \ldots, X_n$ be independent random variables with common distribution the symmetric Jacobi measure $d\mu (x) = C_d (1-x^2)^{\frac d2 -1} dx$ with dimension $ d \geq 1$ on $[-1, 1]$, and let $\mu_n = \frac{1}{n} \sum_{i = 1}^{n} \delta_{X_i}$ be the associated empirical measure. We show that $\lim_{n \to \infty} n\E \left[ W_2^2( \mu^n, \mu ) \right] = \sum_{k = 1}^{\infty} \frac{1}{k(k+d-1)}$, where $W_2$ is the quadratic Kantorovich distance with respect to the intrinsic cost $\rho(x, y) = |\arccos(x) - \arccos (y)|$, $(x, y) \in [-1, 1]^2$, associated to the model. When $\mu$ is the product measure of two Jacobi measures with dimensions $d$ and $d'$ respectively, then $\E \left[ W_2^2( \mu^n, \mu ) \right] \approx \frac{\log n}{n}$. In the particular case $d = d' = 1$ (corresponding to the product of arcsine laws), $\lim_{n \to \infty} \frac{n}{\log n} \E \left[ W_2^2( \mu^n, \mu ) \right] = \frac{\pi}{4}$. Similar results do hold for non-symmetric Jacobi distributions. The proofs are based on the recent PDE and mass transportation approach developed by L.~Ambrosio, F.~Stra and D.~Trevisan.

Tail asymptotics for a state-dependent bulk matching queueing system with impatient customers

We study a state-dependent bulk matching queueing system with impatient customers, where customers and servers visit the system from both sides. Servers provide services in a batch with a maximum size and take matching customers away instantly. To characterize such a queueing system, the corresponding Markov process is constructed by the number of complete batches of customers and the number of remaining customers in the incomplete batch. By analyzing this system, we find it difficult to obtain the joint stationary distribution of the Markov process. Therefore, we consider the tail asymptotics for the joint probabilities. Using the matrix analytic method and censoring technique, we obtain the one-term and general expansions for the nonzero elements of the rate matrices, where the coefficients of the expansions are presented in the closed form. On the basis of these expansion formulae, the exact tail asymptotic result for the joint stationary probabilities is derived.

Transport in Hamiltonian systems with slowly changing phase space structure

Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of one-dimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion of the phase space structures: chaotic, weakly chaotic and regular sets change with time. We show that theoretical predictions of transport rates can be made in the limit where the variations of the stochastic parameters are very slow compared to the Hamiltonian dynamics. Exact asymptotic results can be obtained in the one-dimensional case where the Hamiltonian dynamics is integrable for fixed values of the parameters. For the more interesting chaotic Hamiltonian dynamics case, we show that two mechanisms contribute to the transport. For some range of the parameter variations, one mechanism -called ”transport by migration with the mixing regions” - is dominant. We are then able to model transport in phase space by a Markov model, the local diffusion model, and to give reasonably good transport estimates.

Generalizations of TASEP in Discrete and Continuous Inhomogeneous Space

We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams. For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation results which put the systems into the Kardar-Parisi-Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy-Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions. A homogeneous version of a discrete space system we consider is a one-parameter deformation of the geometric last-passage percolation, and we obtain extensions of the limit shape parabola and the corresponding asymptotic fluctuation results. The exact solvability and asymptotic behavior results are powered by a new nontrivial connection to Schur measures and processes.

Surface van der Waals forces in a nutshell.

Most often in chemical physics, long range van der Waals surface interactions are approximated by the exact asymptotic result at vanishing distance, the well known additive approximation of London dispersion forces due to Hamaker. However, the description of retardation effects that is known since the time of Casimir is completely neglected for the lack of a tractable expression. Here we show that it is possible to describe surface van der Waals forces at arbitrary distances in one single simple equation. The result captures the long sought crossover from non-retarded (London) to retarded (Casimir) interactions, the effect of polarization in condensed media, and the full suppression of retarded interactions at large distance. This is achieved with similar accuracy and the same material properties that are used to approximate the Hamaker constant in conventional applications. The results show that at ambient temperature, retardation effects significantly change the power law exponent of the conventional Hamaker result for distances of just a few nanometers.

The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework

AbstractWe consider a modeling setup where the volatility index (VIX) dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed‐form expansions and sharp error bounds for VIX futures, options, and implied volatilities. In particular, we derive exact asymptotic results for VIX‐implied volatilities, and their sensitivities, in the joint limit of short time‐to‐maturity and small log‐moneyness. The expansions obtained are explicit based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol‐of‐vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has previously been adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.

The Short-Time Behaviour of VIX Implied Volatilities in a Multifactor Stochastic Volatility Framework

We consider a modeling setup where the volatility index (VIX) dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed‐form expansions and sharp error bounds for VIX futures, options, and implied volatilities. In particular, we derive exact asymptotic results for VIX‐implied volatilities, and their sensitivities, in the joint limit of short time‐to‐maturity and small log‐moneyness. The expansions obtained are explicit based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol‐of‐vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has previously been adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.

Linear algebra of the permutation invariant Crow–Kimura model of prebiotic evolution

A particular case of the famous quasispecies model — the Crow–Kimura model with a permutation invariant fitness landscape — is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special tridiagonal form, a change of the basis is suggested such that in the new coordinates a number of analytical results can be obtained. In particular, using the eigenvectors of the mutation matrix as the new basis, we show that the quasispecies distribution approaches a binomial one and give simple estimates for the speed of convergence. Another consequence of the suggested approach is a parametric solution to the system of equations determining the quasispecies. Using this parametric solution we show that our approach leads to exact asymptotic results in some cases, which are not covered by the existing methods. In particular, we are able to present not only the limit behavior of the leading eigenvalue (mean population fitness), but also the exact formulas for the limit quasispecies eigenvector for special cases. For instance, this eigenvector has a geometric distribution in the case of the classical single peaked fitness landscape. On the biological side, we propose a mathematical definition, based on the closeness of the quasispecies to the binomial distribution, which can be used as an operational definition of the notorious error threshold. Using this definition, we suggest two approximate formulas to estimate the critical mutation rate after which the quasispecies delocalization occurs.

Weighted Kolmogorov-Smirnov test: Accounting for the tails

Accurate goodness-of-fit tests for the extreme tails of empirical distributions is a very important issue, relevant in many contexts, including geophysics, insurance, and finance. We have derived exact asymptotic results for a generalization of the large-sample Kolmogorov-Smirnov test, well suited to testing these extreme tails. In passing, we have rederived and made more precise the approximate limit solutions found originally in unrelated fields, first in [L. Turban, J. Phys. A 25, 127 (1992)] and later in [P. L. Krapivsky and S. Redner, Am. J. Phys. 64, 546 (1996)].

The Poisson-Boltzmann theory for the two-plates problem: Some exact results

The general solution to the nonlinear Poisson-Boltzmann equation for two parallel charged plates, either inside a symmetric electrolyte, or inside a 2q:-q asymmetric electrolyte, is found in terms of Weierstrass elliptic functions. From this we derive some exact asymptotic results for the interaction between charged plates, as well as the exact form of the renormalized surface charge density.

PRICES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS IN LÉVY-DRIVEN MODELS, NEAR BARRIER

We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Lévy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carr's randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Lévy processes commonly used in option pricing.

Poisson-Boltzmann theory for two parallel uniformly charged plates

We solve the nonlinear Poisson-Boltzmann equation for two parallel and like-charged plates both inside a symmetric electrolyte, and inside a 2:1 asymmetric electrolyte, in terms of Weierstrass elliptic functions. From these solutions we derive the functional relation between the surface charge density, the plate separation, and the pressure between plates. For the one plate problem, we obtain exact expressions for the electrostatic potential and for the renormalized surface charge density, both in symmetric and in asymmetric electrolytes. For the two plate problems, we obtain new exact asymptotic results in various regimes.

Tail asymptotics for M/M/c retrial queues with non-persistent customers

In this paper, we consider a variant of the classical M/M/c retrial queue, in which we allow non-persistent customers. When c > 1, this system does not have an explicit closed form solution for the joint stationary distribution of the number of retrial customers in the orbit and the number of busy servers. Our main focus is on the tail asymptotics for the joint probabilities. We first present a matrix-product solution for the joint stationary probability vectors, which is further simplified to a scalar-product form, according to matrix-analytic theory. We then apply the censoring technique, which has been proven an efficient approach for analyzing queueing systems including retrial queues, to obtain the censored equations and the Key Lemma. In terms of these results, we finally prove an exact tail asymptotic result for the stationary probabilities.

Sample‐Path Large Deviations in Credit Risk

The event of large losses plays an important role in credit risk. As these large losses are typically rare, and portfolios usually consist of a large number of positions, large deviation theory is the natural tool to analyze the tail asymptotics of the probabilities involved. We first derive a sample‐path large deviation principle (LDP) for the portfolio′s loss process, which enables the computation of the logarithmic decay rate of the probabilities of interest. In addition, we derive exact asymptotic results for a number of specific rare‐event probabilities, such as the probability of the loss process exceeding some given function.