Spin depolarization decay rates in α-symmetric stable fields on cubic lattices

We study the asymptotic, long-time behavior of the energy function where {X_s : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice Z^d, 1 < α ≤ 2, and f:R⁺ → R⁺ is any nondecreasing concave function. In the special case f(x) = x, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, α-stable, i.i.d., random, longitudinal field {λV(x) : x ∈ Z^d} with common marginal distribution, the standard α-symmetric stable distribution; the parameter λ describes the intensity of the field. Using large-deviation techniques, we show that S_c(λ α f) = lim_t→∞ E(t; λ f) exists. Moreover, we obtain a variational formula for this decay rate S_c. Finally, we analyze the behavior S_c(λ α f) as λ → 0 when f(x) = x^β for all 1 ≥ β > 0. Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting β = 1 in our results. We show that S_c(λ, α, 1) ≈ λ^α for d ≥ 3, λ^agr;(ln 1/λ)^α−1 in d = 2, and in d = 1. © 1996 John Wiley & Sons, Inc.     

Edge Close Ball Packings

For a packing of unit balls in R ³ , an edge-line is the intersection of the planes of two intersecting faces of some Dirichlet—Voronoi cell, like for example the line of an edge. Now the edge closeness of the arrangement is the maximal distance between an edge-line for a Dirichlet—Voronoi cell and the centre of the corresponding unit ball. We show that the unique packing with minimal edge closeness is generated by the space centred cubic lattice. Received June 22, 1999, and in revised form July 24, 2000. Online publication May 4, 2001.     

Hyperuniform and rigid stable matchings

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on R^d with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψ^τ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψ^τ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process Ψ, whose so‐called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Furthermore, if Ψ is a Poisson process then Ψ^τ has an exponentially decreasing truncated pair correlation function.     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

Ein Null-Zwei-Gesetz auf lokalkompakten abelschen Halbgruppen

Let λ, μ be regular probability measures on a locally compact abelian semigroup S, λ * μ the convolution of λ and μ, λⁿ the n^th iterated convolution of λ, δ_x the point measure of x?S. We study the totalvariation of λⁿ–δ_x * λⁿ for n → ∞. We shall see that for a certain class of semigroups the limit of this sequence is either 0 or 2.     

Critical phenomena in deep inelastic scattering

Saturation in deep inelastic scattering (DIS) and deeply virtual Compton scattering is associated with a phase transition between the partonic gas, typical of moderate x and Q ², and a partonic fluid (liquid), appearing at increasing Q ² and decreasing Bjorken x. In the statistical interpretation of DIS, the large-x, (1 − x) ⁿ factor in the structure functions is associated with a statistical distribution (perfect gas), while the low-x, Regge behaved factor x ^{b (Q 2}) introduces deviation from the perfect gas and ultimately leads to a gas-liquid phase transition.     

Phase Transition and Eigenvalue Degeneracy of a One Dimensional Anharmonic Oscillator

For a one dimensional lattice model with long range exponential interaction the problem of deriving the thermodynamic properties of the system reduces to the problem of determining the spectrum of a differential equation for a one dimensional anharmonic oscillator with potential αx² + βγx⁴ where γ is the inverse of the range of the interaction and α is a function of temperature which is positive at high temperatures and negative at low temperatures. It is proved that at low temperatures, α < 0, and in the limit γ → 0, (corresponding to an infinitely long range interaction) the spectrum becomes asymptotically degenerate. The relation of degeneracy to the occurrence of a phase transition is discussed and the generalization to higher dimensions is mentioned briefly.     

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ? ?³ with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H¹(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.     

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in d‐dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 262–286, 2017     

Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.

Consider the random graph on n vertices 1,…,n. Each vertex i is assigned a type x_i with x₁,…,x_n being independent identically distributed as a nonnegative random variable X. We assume that EX³< ∞. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability \begin{align*}\min \{1, \frac{x_ix_j}{n}\left(1+\frac{a}{n^{1/3}} \right) \}\end{align*}. We study the critical phase, which is known to take place when EX² = 1. We prove that normalized by n^‐2/3the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient \begin{align*}\sqrt{{\textbf{ E}}X{\textbf{ E}}X^3}\end{align*}and drift \begin{align*}a-\frac{{\textbf{ E}}X^3}{{\textbf{ E}}X}s\end{align*}. In particular, we conclude that the size of the largest connected component is of order n^2/3. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486–539, 2013     

Percolation in Voronoi tilings

We consider a percolation process on a random tiling of ℝ^d into Voronoi cells based on points of a realization of a Poisson process. We prove the existence of a phase transition as the proportion p of open cells is varied and provide estimates for the critical probability p_c. Specifically, we prove that for large d, 2^−d(9d log d)⁻¹ ≤ p_c(d) ≤ C2^−d log d. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

The decomposability of simple orthogonal arrays on 3 symbols having t + 1 rows and strength t

It is well‐known that all orthogonal arrays of the form OA(N, t + 1, 2, t) are decomposable into λ orthogonal arrays of strength t and index 1. While the same is not generally true when s = 3, we will show that all simple orthogonal arrays of the form OA(N, t + 1, 3, t) are also decomposable into orthogonal arrays of strength t and index 1. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 442–458, 2000     

Universality at the edge of the spectrum for unitary,orthogonal, and symplectic ensembles of random matrices

We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e^?V(x) where V is a polynomial, V(x) = κ_2mx^2m + · · ·, κ_2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [12], and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials found in [12] plays a central role. The formulae in [46] express the correlation kernels for β = 1, 4 as a sum of a Christoffel‐Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [9], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [49]. © 2006 Wiley Periodicals, Inc.     

Decomposition of multivariate infinitely divisible characteristic functions with absolutely continuous Poisson spectral measures

We shall consider the decomposition problem of multivariate infinitely divisible characteristic functions which have no Gaussian component and have absolutely continuous Poisson spectral measures. Under the condition that A = {x;f(x) > 0} is open, where f is the density of spectral measure, we shall show that a known sufficient condition for the membership of the class I_0m (i.e., infinitely divisible characteristic functions having only infinitely divisible factors) is also necessary.     

Stationary distributions of the simplest dynamical systems in Rn perturbed by generalized Poisson process

Let z(t) ∈ Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) ∈ Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

Random walks in random environments on metric groups

Random walks in random environments on countable metric groups with bounded jumps of the walking particle are considered. The transition probabilities of such a random walk from a pointx εG (whereG is the group in question) are described by a vectorp(x) ε ℝ^|W| (whereW ⊏G is fixed and |W|<∞). The set {p(x),x εG} is assumed to consist of independent identically distributed random vectors. A sufficient condition for this random walk to be transient is found. As an example, the groups ℤ ^d , free groups, and the free product of finitely many cyclic groups of second order are considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 129–135, January, 2000.     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

On a random graph with immigrating vertices: Emergence of the giant component

A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1/n, is studied. The detailed picture of emergence of giant components with O(n^2/3) vertices is shown to be the same as in the Erdős–Rényi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t=π/2, rather than t=1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum‐of‐squares and sum‐of‐cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky‐type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812–854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 79–102, 2000     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs

Let ∑ = (V,E) be a finite, d‐regular bipartite graph. For any λ > 0 let π_λ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ^|I| (π_λ is the hard‐core measure with activity λ on ∑). We study the Glauber dynamics, or single‐site update Markov chain, whose stationary distribution is π_λ. We show that when λ is large enough (as a function of d and the expansion of subsets of single‐parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d‐dimensional hypercube {0,1}^d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006