Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

The birth of the giant component

Limiting distributions are derived for the sparse connected components that are present when a random graph on n vertices has approximately 1/2n edges. In particular, we show that such a graph consists entirely of trees, unicyclic components, and bicyclic components with probability approaching √2/3 cosh √5/18 ≈ 0.9325 as n→∞. The limiting probability that it is consists of trees, unicyclic components, and at most one another component is approximately 0.9957; the limiting probability that it is planar lies between 0.987 and 0.9998. When a random graph evolves and the number of edges passes 1/2n, its components grow in cyclic complexity according to an interesting Markov process whose asymptotic structure is derived. The probability that there never is more than a single component with more edges than vertices, throughout the veolution, approaches 5 π/18 ≈ 0.8727. A “uniform” model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substanitially simpler than the analogous formulas for the classical random graphs of Erdõs and Rényi. The notions of “excess” and “deficiency,” which are significant characteristics of the generating function as well as of the graphs themselves, lead to a mathematically attractive structural theory for the uniform model. A general approach to the study of stopping configurations makes it possible to sharpen previously obtained estimates in a uniform manner and often to obtain closed forms for the constants of interest. Empirical results are presented to complement the analysis, indicating the typical behavior when n is near 2oooO. © 1993 John Wiley & Sons, Inc.     

Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees

We consider conditioned Galton–Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 57–101, 2016     

Formation of precipitates in heavily boron doped 4H-SiC

Secondary ion mass spectrometry (SIMS) and transmission electron microscopy (TEM) are utilized to study precipitation and the solubility of B in 4H-SiC epitaxial layers super saturated with B. Heat treatments are performed in Ar atmosphere in an rf-heated furnace at temperatures between 1700 and 2000 °C. SIMS ion images, and TEM micrographs reveal the formation of two types of precipitates where the larger, more thermally stable one is suggested to be B₄C. The boron solubility is determined from SIMS depth profiles and is shown to follow the Arrhenius expression: 7.1 × 10²² exp(−1.4 eV/k_BT) cm⁻³ over the studied temperature range.     

On String Graph Limits and the Structure of a Typical String Graph

We study limits of convergent sequences of string graphs, that is graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.     

The Greedy Independent Set in a Random Graph with Given Degrees

We analyse the size of an independent set in a random graph on n vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as (the jamming constant), expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges in probability to the jamming constant as . The results hold under weaker assumptions in a random multigraph with given degrees constructed via the configuration model. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 565–586, 2017     

On the variance of the random sphere of influence graph

We show that the variance of the number of edges in the random sphere of influence graph built on n i.i.d. sites which are uniformly distributed over the unit cube in R ^d, grows linearly with n. This is then used to establish a central limit theorem for the number of edges in the random sphere of influence graph built on a Poisson number of sites. Some related proximity graphs are discussed as well. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 139–152, 1999     

On moment conditions for normed sums of independent variables and martingale differences

Let X₁, X₂,… be a sequence of i.i.d. random variables and S_n their partial sums. Necessary and sufficient conditions are given for $\text{{n}{}^{\mathrm{<mtext>?1</mtext><mtext>q</mtext>}}\text{S}{}_{\mathrm{n}}\text{}}{}^{\mathrm{\propto }}{}_1$ to have uniformly bounded pth moments, 0<p<q?2.Some of the results are generalized to martingle differences.     

On the limiting distribution of the number of ‘near-matches’

When (R₁,...,R_n) is a random permutation of the numbers (1,...,n), a ‘near-match’ at the ith place is defined to have occured if |R_i ? i| < k, for some fixed integer k. This note studies the asymptotic distribution of the number of ‘near-matches’ when k is fixed and when k is allowed to go to infinity with n.