Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

We consider the double scaling limit for a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n → ∞ that intersects the hard edge at x = 0 at a critical time t = t ^*. In a previous paper, the scaling limits for the positions of the paths at time t ≠ t ^* were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as n → ∞ of the correlation kernel at critical time t ^* and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix.     

An Infinite Family of Adsorption Models and Restricted Lukasiewicz Paths

We define (k,?)-restricted Lukasiewicz paths, k??????₀, and use these paths as models of polymer adsorption. We write down a polynomial expression satisfied by the generating function for arbitrary values of (k,?). The resulting polynomial is of degree ?+1 and hence cannot be solved explicitly for sufficiently large ?. We provide two different approaches to obtain the phase diagram. In addition to a more conventional analysis, we also develop a new mathematical characterisation of the phase diagram in terms of the discriminant of the polynomial and a zero of its highest degree coefficient. We then give a bijection between (k,?)-restricted Lukasiewicz paths and ??rise??-restricted Dyck paths, identifying another family of path models which share the same critical behaviour. For (k,?)=(1,??) we provide a new bijection to Motzkin paths. We also consider the area-weighted generating function and show that it is a q-deformed algebraic function. We determine the generating function explicitly in particular cases of (k,?)-restricted Lukasiewicz paths, and for (k,?)=(0,??) we provide a bijection to Dyck paths.     

Symmetrized Models of Last Passage Percolation and Non-Intersecting Lattice Paths

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.     

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function h_t(x) with corner initialization. We prove, with one exception, that the limiting distribution function of h_t(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin–Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppäläinen–Johansson model. Hence some of our results are not new, but the proofs are.     

Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions

The ground state energy of an atom of nuclear chargeZe in a magnetic fieldB is exactly evaluated to leading order asZ in the following three regions:BZ ^4/3,BZ ^4/3 andZ ^4/3BZ³. In each case this is accomplished by a modified Thomas-Fermi (TF) type theory. We also analyze these TF theories in detail, one of their consequences being the nonintuitive fact that atoms are spherical (to leading order) despite the leading order change in energy due to theB field. This paper complements and completes our earlier analysis [1], which was primarily devoted to the regionsBZ ³ andBZ³ in which a semiclassical TF analysis is numerically and conceptually wrong. There are two main mathematical results in this paper, needed for the proof of the exactitude of the TF theories. One is a generalization of the Lieb-Thirring inequality for sums of eigenvalues to include magnetic fields. The second is a semiclassical asymptotic formula for sums of eigenvalues that isuniform in the fieldB.Work partially supported by U.S. National Science Foundation grant PHY90-19433 A02Work partially supported by U.S. National Science Foundation grant DMS 92-03829Work partially supported by the Heraeus Stiftung and the Research Fund of the University of Iceland.     

Weighted Exponential Random Graph Models: Scope and Large Network Limits

We study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. Analogous to fundamental results derived for standard (unweighted) exponential random graph models in the work of Chatterjee and Diaconis, we derive limiting results for the structure of these models as the number of nodes goes to infinity. Our results are applicable for a wide variety of base measures including measures with unbounded support. We also derive sufficient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates. Finally we include a number of open problems to spur further understanding of this model especially in the context of applications.     

Phenomenological Models of Nucleation and Growth of Metal on a Semiconductor

Physics of the Solid State - Four modes of metal growth on a semiconductor substrate have been detected and distinguished on the basis of experimental data obtained under similar conditions using...     

Effect of Time Delay on Stochastic Tumor Growth

We study the dynamics of tumor cell growth with time-delayed feedback driven by multiplicative noise in an asymmetrical bistable potential well. For a small delay time, the analytical solutions of the probability distribution and the first passage time show that, with the increasing delay time, the peak of the probability distribution in a lower population state would increase, but in a higher population state it decreases. It is shown that the multiplicative noise and the time delay play opposite roles in the tumor cell growth.     

Large Numbers and the Time Variation of Physical Constants

We consider a cosmological model consistent withobservation which not only explains the well-knownlarge-number coincidences, but also deduces the valuesof the mass, radius, and age of the universe, the Hubble constant and the cosmological constant,a relation between the pion mass and the Hubble constantknown so far only as a mysterious empirical coincidence,and other features. This model predicts an ever-expanding universe, as indeed latestastrophysical data indicate.     

Equilibration of Small and Large Subsystems in Field Theories and Matrix Models

Integrable quantum field theories in 1+1 dimensions have recently become amenable to a rigorous construction, but many questions about the structure of their local observables remain open. Our goal is to characterize these local observables in terms of their expansion coefficients in a series expansion by interacting annihilators and creators, similar to form factors. We establish a rigorous one-to-one characterization, where locality of an observable is reflected in analyticity properties of its expansion coefficients; this includes detailed information about the high-energy behaviour of the observable and the growth properties of the analytic functions. Our results hold for generic observables, not only smeared pointlike fields, and the characterizing conditions depend only on the localization region—we consider wedges and double cones—and on the permissible high energy behaviour.     

大型水平极化电磁脉冲有界波模拟器的并行模拟分析

采用将大地设置为均匀有耗介质,并用单轴完全匹配层（UPML）吸收边界截断的方式,对离散电阻加载的地面上方大型水平极化电磁脉冲（EMP）有界波模拟器的时域辐射场进行并行时域有限差分模拟.给出大地电导率和相对介电常数及模拟器圆锥的锥半径不同时模拟器辐射场的时域波形,分析三者对辐射场的影响,并给出模拟器内有10 m长的效应物时耦合场的时域波形.并行模拟时,计算网格总数达18亿.研究表明：大地相对介电常数和电导率越大,近地面测点接收到的地面反射作用越大;测点场的峰值受锥半径的影响最大,且随着锥半径的增大,同一水平面内的场也越不均匀;对地面上方1 m处的几个测点,其脉冲峰值及半高宽受地面反射及地面损耗的影响较大,而地面上方5 m处的几个测点则受地面影响较小;当效应物开孔位置位于模拟器场泄露一侧时耦合进入圆筒内的电磁波能量较多.     

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If , and is a finite (nonabelian) group, then is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of . We characterize when MCA are group endomorphisms of , and show that MCA on inherit a natural structure theory from the structure of . We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.     

微晶硅薄膜在粗糙表面的大面积均匀生长机制在

用氩气稀释硅烷作为反应气体,利用感应耦合等离子体在廉价衬底上大面积均匀沉积微晶硅薄膜．X射线衍射和拉曼分析表明衬底能显著地影响晶向,且微晶硅薄膜由小晶粒组成．台阶仪测试表明,微晶硅薄膜具有大范围均匀的特点．探针分析表明衬底附近区域的离子密度及电子温度分布均匀．基于以上结果可知：粗糙表面在晶体结构形成具有重要的作用,电感耦合等离子体反应器可以大面积均匀沉积薄膜．此外,氩气能影响反应过程并提高微晶硅薄膜特性．     

Effect of Grating Spacing on Response Time and Sensitivity in Photorefractive Materials at Large Modulation Depth

Photorefractive response time and photorefractive sensitivity are two important parameters characterizing photorefractive materials. We study theoretically the effect of the grating spacing on these parameters at large modulation depth in the absence and presence of an applied electric field for some of the most promising photorefractive materials, namely, LiNbO₃, KNSBN, SPS, BGO, and GaAs. We find that the response time increases with increasing grating spacing for LiNbO₃ and KNSBN, whereas the response time decreases with increasing grating spacing for SPS, BGO, and GaAs. The photorefractive sensitivity is mainly affected by the mobility–lifetime product in the presence of an applied electric field.     

Growth of the Eigensolutions of Laplacians on Riemannian Manifolds II: Positivity of the Initial Energy

In this paper, the energy function is used to investigate the eigen-solutions of ?Δu + V u = λu on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions.