The Dirichlet Markov Ensemble

We equip the polytope of n×n Markov matrices with the normalized trace of the Lebesgue measure of Rⁿ². This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,…,1/n). We show that if is such a random matrix, then the empirical distribution built from the singular values of tends as n→∞ to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of tends as n→∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.     

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

We study the Epstein zeta function En(L,s) for and a random lattice L of large dimension n. For any fixed we determine the value distribution and moments of En(⋅,cn) (suitably normalized) as n→∞. We further discuss the random function c?En(⋅,cn) for c∈[A,B] with and determine its limit distribution as n→∞.     

Stable extendibility of normal bundles over lens spaces

We study the stable extendibility of R-vector bundles over the (2n+1)-dimensional standard lens space Ln(p) with odd prime p, focusing on the normal bundle to an immersion of Ln(p) in the Euclidean space R2n+1+t. We show several concrete cases in which is stably extendible to Lk(p) for any k with k?n, and in several cases we determine the exact value m for which is stably extendible to Lm(p) but not stably extendible to Lm+1(p).     

The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map acting on a von Neumann algebra—or more generally a dual operator system—we show that there is a unique reversible system (i.e., a complete order automorphism α of a dual operator system N) that captures all of the asymptotic behavior of L, called the asymptotic lift of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n×n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W^∗-dynamical system (N,Z), and we identify (N,Z) as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed algebra Nα. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense that

     

Asymptotic bounds for permutations containing many different patterns

We say that a permutation σ∈Sn contains a permutation π∈Sk as a pattern if some subsequence of σ has the same order relations among its entries as π. We improve on results of Wilf, Coleman, and Eriksson et al. that bound the asymptotic behavior of pat(n), the maximum number of distinct patterns of any length contained in a single permutation of length n. We prove that by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of k-superpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that Lk, the length of the shortest k-superpattern, is at most . This may lend evidence to a conjecture of Eriksson et al. that .     

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature

Let L=Δ−∇φ⋅∇ be a symmetric diffusion operator with an invariant measure on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry-Émery Ricci curvature satisfying Ricm,n(L)?−(n−1), and therefore generalize a Cheng's result on the Laplacian (S.-Y. Cheng (1975) [8]) to the case of the diffusion operator.     

Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ={Φ(m,n)}_(m,n)∈Δ the subspace . Supposing that X₁ is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on X₁, we obtain that the admissibility of the pair with p∈(1,∞] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.     

Transient random walks in random environment on a Galton–Watson tree

We consider a transient random walk (X _n ) in random environment on a Galton–Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that X _n is of order of magnitude n ^Λ, with ${\Lambda \in (0,1)}$ . We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio (Probab Theory Related 136(1):81–101, 2006).     

High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between and with q≤p, based on a sample of size of N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m=n−p→∞ and c=p/n→c₀∈[0,1), assuming that and have a joint (q+p)-variate normal distribution. An extended Fisher’s z-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,q, and n and the population canonical correlations.     

Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators Le₁,…,Len,Lf₁,…,Lfm subject to the unitary commutation relations of the form

     

Kolmogorov and linear widths of weighted Besov classes

We study the Kolmogorov m-widths and the linear m-widths of the weighted Besov classes on [−1,1], where Lq,μ, 1?q?∞, denotes the Lq space on [−1,1] with respect to the measure , μ>0. Optimal asymptotic orders of and as m→∞ are obtained for all 1?p,τ?∞. It turns out that in many cases, the orders of are significantly smaller than the corresponding orders of the best m-term approximation by ultraspherical polynomials, which is somewhat surprising.     

Square-free values of the Carmichael function

We obtain an asymptotic formula for the number of square-free values among p−1, for primes p?x, and we apply it to derive the following asymptotic formula for L(x), the number of square-free values of the Carmichael function λ(n) for 1?n?x,

     

Random maximal independent sets and the unfriendly theater seating arrangement problem

People arrive one at a time to a theater consisting of m rows of length n. Being unfriendly they choose seats at random so that no one is in front of them, behind them or to either side. What is the expected number of people in the theater when it becomes full, i.e., it cannot accommodate any more unfriendly people? This is equivalent to the random process of generating a maximal independent set of an m×n grid by randomly choosing a node, removing it and its neighbors, and repeating until there are no nodes remaining. The case of m=1 was posed by Freedman and Shepp [D. Freedman, L. Shepp, An unfriendly seating arrangement (problem 62-3), SIAM Rev. 4 (2) (1962) 150] and solved independently by Friedman, Rothman and MacKenzie [H.D. Friedman, D. Rothman, Solution to: An unfriendly seating arrangement (problem 62-3), SIAM Rev. 6 (2) (1964) 180-182; J.K. MacKenzie, Sequential filling of a line by intervals placed at random and its application to linear adsorption, J. Chem. Phys. 37 (4) (1962) 723-728] by proving the asymptotic limit . In this paper we solve the case m=2 and prove the asymptotic limit . In addition, we consider the more general case of m×n grids, m≥1, and prove the existence of asymptotic limits in this general setting. We also make several conjectures based upon Monte Carlo simulations.     

On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

In this paper, we use a combination of VDR theory and patchwork method to derive an efficient algorithm for generating uniform random points on a unit d-sphere. We first propose an algorithm to generate random vector with uniform distribution on a unit 2-sphere on the plane. Then we use VDR theory to reduce random vector X_d with uniform distribution on a unit d-sphere into , such that the random vector (X_d-1,X_d) is uniformly distributed on a unit 2-sphere and X_d-2 has conditional uniform distribution on a (d-2)-sphere of radius , given V=v with V having the p.d.f. . Finally, we arrive by induction at an algorithm for generating uniform random points on a unit d-sphere.     

Degenerate principal series representations and their holomorphic extensions

Let X=H/L be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain D=G/K. The intersection S of the Shilov boundary of D with X defines a distinguished subset of the topological boundary of X and is invariant under H. It can be realized as S=H/P for certain parabolic subgroup P of H. We study the spherical representations of H induced from P. We find formulas for the spherical functions in terms of the Macdonald hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces D. We consider a class of H-invariant integral intertwining operators from the representations on L²(S) to the holomorphic representations of G restricted to H. We construct a new class of complementary series for the groups H=SO(n,m), SU(n,m) (with n−m>2) and Sp(n,m) (with n−m>1). We realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of G=SU(n,m), SU(n,m)×SU(n,m) and SU(2n,2m) respectively.