Discrete Polynuclear Growth and Determinantal Processes

We consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F ₁ GOE Tracy- Widom distribution in terms of the Airy process. We also show some results, and give a conjecture, about the transversal fluctuations in a point to line last passage percolation problem. Furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions.     

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single time (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y ^–2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.     

Dynamics of a Tagged Particle in the Asymmetric Exclusion Process with the Step Initial Condition

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in the TASEP with the step initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of the TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.     

The Pearcey Process

The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fixed time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel.     

关于Airy光束衍射及自加速性质的研究

对Airy光束的特性做进一步探讨, 一方面对无限宽Airy光束的重心问题给出新的理论说明, 另一方面着重对有限宽情形下的Airy光束的奇特性质进行探讨. 文中采用反证法给出无衍射的讨论, 同时结合数值模拟给出高斯函数及矩形函数限定下的有限宽Airy光束的场分布, 并由此得到其重心位置的轨迹: 重心位置是不变的, 不可能整体自由加速. 最终得到有限宽Airy光束既不可能在自由空间加速, 也不可能是无衍射光束. 关键词： Airy光束 无衍射 自加速 数值模拟     

Random Skew Plane Partitions and the Pearcey Process

We study random skew 3D partitions weighted by q ^vol and, specifically, the q → 1 asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.     

Generation and tight-focusing properties of cylindrical vector circular Airy beams

Tight-focusing properties of cylindrical vector circular Airy beams [i.e., azimuthally polarized (AP) circular Airy beam and radially polarized (RP) circular Airy beam] passing through a high numerical aperture thin lens are investigated in detail. It is found that a super long subwavelength dark channel with full width at half maximum about 0.49λ and depth of focus (DOF) about 52λ can be achieved near the focal region for the case of tight focusing of an AP circular Airy beam, and a super long needle with DOF about 27.5λ of strong longitudinally polarized field can be obtained near the focal region for the case of tight focusing of a RP circular Airy beam. Furthermore, we report experimental generation of an AP circular Airy. Our results will be useful for atom guiding and trapping, particle acceleration and fluorescent imaging.     

Airy plasmons: non‐diffracting optical surface waves

Airy beams represent an important class of non‐diffracting waves which can be realized on a flat surface. Being generated in the form of surface‐plasmon polaritons, such Airy plasmons demonstrate many remarkable properties: they do not diffract while propagating along parabolic trajectories, and they recover their shape after passing through obstacles. This paper reviews the basic physics of Airy plasmons in both paraxial and non‐paraxial cases, and describes the experimental methods for generation of Airy surface waves on metal surfaces, including a control of their trajectories, as well as the interference of Airy plasmons and hot‐spot generation. Many unusual properties of Airy plasmons can be utilized for useful applications, including plasmonic circuitry and surface tweezers. Picture: Observation of two colliding Airy plasmons.     

Quantitative comparison of self-healing ability between Bessel–Gaussian beam and Airy beam

The self-healing ability during propagation process is one of the most important properties of non-diffracting beams. This ability has crucial advantages to light sheet-based microscopy to reduce scattering artefacts, increase the quality of the image and enhance the resolution of microscopy. Based on similarity between two infinite-dimensional complex vectors in Hilbert space, the ability to a Bessel–Gaussian beam and an Airy beam have been studied and compared. Comparing the evolution of the similarity of Bessel–Gaussian beam with Airy beam under the same conditions, we find that Bessel–Gaussian beam has stronger self-healing ability and is more stable than that of Airy beam. To confirm this result, the intensity profiles of Bessel–Gaussian beam and Airy beam with different similarities are numerically calculated and compared.     

Tracy–Widom at High Temperature

We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature \(\beta \) tends to \(0\) . We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy–Widom \(\beta \) law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index \(k\) . Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in Dumaz and Virág (Ann Inst H Poincaré Probab Statist 49(4):915–933, 2013). We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when \(\beta \rightarrow 0\) . As an application, we investigate the maximal eigenvalues statistics of \(\beta _N\) -ensembles when the repulsion parameter \(\beta _N\rightarrow 0\) when \(N\rightarrow +\infty \) . We study the double scaling limit \(N\rightarrow +\infty , \beta _N \rightarrow 0\) and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of Edelman and Sutton (J Stat Phys 127(6):1121–1165, 2007) and Ramírez et al. (J Am Math Soc 24:919–944, 2011) from our later study of the stochastic Airy operator.     

Airy Functions Over Local Fields

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated with compact p-adic Lie groups also have these properties.     

The Interpolating Airy Kernels for the \beta =1 and \beta =4 Elliptic Ginibre Ensembles

We consider two families of non-Hermitian Gaussian random matrices, namely the elliptic Ginibre ensembles of asymmetric $N$ -by- $N$ matrices with Dyson index $\beta =1$ (real elements) and with $\beta =4$ (quaternion-real elements). Both ensembles have already been solved for finite $N$ using the method of skew-orthogonal polynomials, given for these particular ensembles in terms of Hermite polynomials in the complex plane. In this paper we investigate the microscopic weakly non-Hermitian large- $N$ limit of each ensemble in the vicinity of the largest or smallest real eigenvalue. Specifically, we derive the limiting matrix-kernels for each case, from which all the eigenvalue correlation functions can be determined. We call these new kernels the “interpolating” Airy kernels, since we can recover—as opposing limiting cases—not only the well-known Airy kernels for the Hermitian ensembles, but also the complementary error function and Poisson kernels for the maximally non-Hermitian ensembles at the edge of the spectrum. Together with the known interpolating Airy kernel for $\beta =2$ , which we rederive here as well, this completes the analysis of all three elliptic Ginibre ensembles in the microscopic scaling limit at the spectral edge.     

面向近原子尺度制造的光学测量精度极限分析

纳米级乃至更高精度的测量是原子及近原子尺度制造技术发展的基础和保障.光学测量具有精度高、测量范围广、测量直观等优点,其对单个成像光斑中心的定位可达远超衍射极限的精度.但由于光本身散粒噪声、探测器暗电流噪声等随机性的存在,光学测量存在精度极限.本文基于克拉美罗下界理论发展了可适用于任意强度分布像斑的精度极限计算方法,并以典型艾里斑为例,分析了成像过程中反映信噪比、能量集中度、计算方式的参数对定位精度极限的影响规律并给出提高精度的建议和结论.对实验所得像斑进行了精度极限计算,验证了所得结论对类似艾里斑的像斑的适用性.研究为原子及近原子尺度制造过程中光学测量的应用和优化提供了分析方法和理论指导.     

Airy光纤:基于阵列波导耦合的光场调控方法

近年来,Airy光束作为一种无衍射光束,其特性引起了研究者的广泛关注,人们对它的理论研究、实验验证、实际应用多个方面都取得了长足的进步.而Airy光纤作为一种可生成Airy光束的波导器件,结合其光纤自身优点可适用于多种应用领域,因此开展新型Airy光纤的探索研究、拓展Airy光束的应用范围具有重要现实意义.本文从Airy光束的原理、光纤结构设计、光纤内部光束生成机理、生成光束波长响应特性以及Airy光纤研究现状和应用五个方面展开了较系统的讨论.     

Airy Functions in the Thermodynamic Bethe Ansatz

Thermodynamic Bethe ansatz equations are coupled nonlinear integral equations which appear frequently when solving integrable models. Those associated with models with N=2 supersymmetry can be related to differential equations, among them Painlevé III and the Toda hierarchy. In the simplest such case, the massless limit of these nonlinear integral equations can be solved in terms of the Airy function. This is the only known closed-form solution of thermodynamic Bethe ansatz equations, outside of free or classical models. This turns out to give the spectral determinant of the Schrödinger equation in a linear potential.     

Universality of the topological string at large radius and NS-brane resurgence

We show that there is a natural universal limit of the topological string free energies at the large radius point. The new free energies keep a nonholomorphic dependence on the complex structure moduli space and their functional form is the same for all Calabi–Yau geometries, compact and noncompact alike. The asymptotic nature of the free energy expansion changes in this limit due to a milder factorial growth of its coefficients, and this implies a transseries extension with instanton effects in \(\exp (- 1/g_s^2)\), of NS-brane type, rather than \(\exp (-1/g_s)\), of D-brane type. We show a relation between the instanton action of NS-brane type and the volume of the Calabi–Yau manifold which points to a possible interpretation in terms of NS5-branes. A similar rescaling limit has been considered recently leading to an Airy equation for the partition function which is here used to explain the resurgent properties of the rescaled transseries.     

Nuclear-rainbow scattering and nucleus-nucleus potentials at short distances

The elastic scattering of strongly bound nuclei at energies of 10 to 70 MeV per nucleon shows the phenomenon of “rainbow scattering.” A nuclear rainbow appears because of deflection to negative angles. This process involves a strong overlap of nuclear densities, with values of up to twice the saturation density of nuclear matter. The ¹⁶O+¹⁶O system is studied with a high precision over a wide energy range from 7 to 70 MeV per nucleon in several laboratories. Primary Airy maxima and higher order Airy structures are observed. At all energies, excellent fits are obtained with deep potentials as deduced from the double-folding model involving a nucleon-nucleon interaction weakly dependent on the density. It is shown that Pauli blocking expected at low energies is strongly reduced if the local momenta are calculated self-consistently. Systematics confirms a refractive origin of large-angle scattering, at low energies inclusive. Thus, nuclear-rainbow scattering yields unique information about the properties of cold nuclear matter at higher densities.     

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.     

Propagation dynamics and optical trapping of a radial Airy array beam

Analytical propagation expression of a radial Airy array beam in coherent and incoherent combination passing through paraxial ABCD system is derived, and used to investigate the effect of combination scheme, array orientation and initial phase of Airy beamlet on propagation dynamics of the resulting beam in free space, where optical spot array and vortex array with different shapes are also found, respectively. And then taking four-beamlet Airy array beam in same array orientation as an example, square optical spot array obtained in focal field can be used for simultaneous trapping multiple Rayleigh particles with relative refractive index larger than 1. The transverse gradient forces serving as restore forces tend to push particles at different initial positions to their individual optical spot center. The analysis of trapping stability indicates that larger input peak intensity of Airy beamlet and smaller particle size are benefit to trapping particle owing to many deeper potential wells. Vortex array produced by coherent combined Airy array beam in this paper is expected to be useful for simultaneous trapping microparticles with relative refractive index smaller than 1.     

Edge Scaling of the β-Jacobi Ensemble

We study the scaling limit of the spectrum of the ??-Jacobi ensemble at the soft edge and hard edge for general values of ??. We show that the limiting point processes correspond respectively to the stochastic Airy and Bessel point processes introduced in Ramírez et al. (J. Am. Math. Soc. 24(4):919?C944, 2011) and Ramírez and Rider (Commun. Math. Phys. 288(3):887?C906, 2009).