Connectedness of certain random graphs

L. Dubins conjectured in 1984 that the graph on vertices {1, 2, 3, ...} where an edge is drawn between verticesi andj with probabilityp _ij=λ/max(i, j) independently for each pairi andj is a.s. connected forλ=1. S. Kalikow and B. Weiss proved that the graph is a.s. connected for anyλ>1. We prove Dubin’s conjecture and show that the graph is a.s. connected for anyλ>1/4. We give a proof based on a recent combinatorial result that forλ≦1/4 the graph is a.s. disconnected. This was already proved forλ<1/4 by Kalikow and Weiss. Thusλ=1/4 is the critical value for connectedness, which is surprising since it was believed that the critical value is atλ=1.     

On samples of independent random vectors in spaces of infinitely increasing dimension

We study the asymptotic behavior of a set of random vectors ξ_i, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate the asymptotics of the distribution of the random vectors, the consistency of the sets M _m⁽ⁿ⁾ = ξ₁,..., ξ_m and X _n^λ = x ∈ X _n: ρ(x) ≤ λn, and the mutual location of pairs of vectors. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1706–1711, December, 1998.     

On the replications of certain multiplicative designs

Bounds on the number of row sums in ann×n, non-singular (0,1)-matrixA sarisfyingA ^tA=diag (k ₁-λ₁,…,k _n-λ_n),k _j>λ_j>0,λ₁=…=λ_e,λ_e+1=…=λ_n are obtained which extend previous results for such matrices.     

Arithmetical properties of the number of t-core partitions

Let Λ={λ ₁≥⋅⋅⋅≥λ _s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ _i nodes in the ith row. Let λ _j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ _i +λ _j ′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 ^t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.      

Moments of an exponential functional of random walks and permutations with given descent sets

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ₁ + ξ₁ξ₂ + ξ₁ξ₂ξ₃ + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μ _k = E(ξ ^k ) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle. This revised version was published online in August 2006 with corrections to the Cover Date.     

Large Deviation Principles for Moving Average Processes of Real Stationary Sequences

Let X _k =∑ _i=−∞^∞ a _i ξ _k−i ,k≥1, be the moving average processes, where (ξ _i ) _i∈ℤ is a sequence of real stationary random variables. Under the assumptions that the large deviation principle (LDP) for real stationary sequence holds, LDP for the moving average processes of real stationary sequence is established.      

Transient phenomena for random walks in the absence of the expected value of jumps

Let ξ, ξ₁, ξ₂, ... be independent identically distributed random variables, and S _n :=Σ _j=1 ⁿ ,ξ _j , $ \bar S $ \bar S := sup _n≥0 S _n . If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $ \bar S $ \bar S as a → 0 and $ \bar S $ \bar S tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:

On Spectral Minimal Partitions: A Survey

Given a bounded open set Ω in \mathbbRⁿ{\mathbb{R}^n} (or a Riemannian manifold) and a partition of Ω by k open sets D _j , we can consider the quantity max _j λ(D _j ) where λ(D _j ) is the groundstate energy of the Dirichlet realization of the Laplacian in D _j . If we denote by \mathfrakL_k(W){\mathfrak{L}_k(\Omega)} the infimum over all the k-partitions of max _j λ(D _j ), a minimal (spectral) k-partition is then a partition which realizes the infimum. Although the analysis is rather standard when k = 2 (we find the nodal domains of a second eigenfunction), the analysis of higher k’s becomes non trivial and quite interesting.     

Estimates for the closeness of successive convolutions of multidimensional symmetric distributions

Summary Let ξ₁, ξ₂,... be i.i.d random vectors in ℝ ^k with a common distribution ℒ(ξ_i),... = F, i = 1, 2,.... Let S _n = ξ₁+...+ξ _n . We investigate how small is the difference between ℒ(S _n ) and ℒ(S _{n+ m} ) in the case when ξ _i have symmetric distributions.     

Inequalities and sphere-packing inl p

If 1<p<∞, there is a constantr _p <1/2 so that ifr>r _p only a bounded number of balls inl _p of radiusr can be packed into the unit ball ofl _p . We obtain the exact value of this bound for eachp andr as a consequence of several new inequalities relating the expressions Σλ _i λ _j ‖x _i –x _j ‖ ^p , Σλ _i ‖x _i ‖ ^p and Σλ _i ^/2 for sequences (x _i ) ₁ ⁿ ⊂l _p and (λ _i ) ₁ ⁿ ⊂R.     

A sieve result for Farey fractions

This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ₁, λ₂). An asymptotic formula for the sifting function is derived under the assumption that (λ₁, λ₂)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p ₁ p ₂...p _s ,p ₁≥p ₂>-..., ξk)_is defined by ξ(k)=(logp ₁/logk, logp ₂/logk,..., logp _s /logk, 0, ...); allp _i are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel. Supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 108–127, January–March, 1999. Translated by V. Stakénas     

DNA labelled graphs with DNA computing

Let k≥2, 1≤i≤k andα≥1 be three integers. For any multiset which consists of some k-long oligonucleotides, a DNA labelled graph is defined as follows: each oligonucleotide from the multiset becomes a point; two points are connected by an arc from the first point to the second one if the i rightmost uucleotides of the first point overlap with the i leftmost nucleotides of the second one. We say that a directed graph D can be(k, i;α)-labelled if it is possible to assign a label(l_1(x),..., l_k(x))to each point x of D such that l_j(x)∈{0,...,a-1}for any j∈{1,...,k}and(x,y)∈E(D)if and only if(l_k-i 1(x),..., l_k(x))=(l_1(y),..., l_i(y)). By the biological background, a directed graph is a DNA labelled graph if there exist two integers k, i such that it is(k, i; 4)-labelled. In this paper, a detailed discussion of DNA labelled graphs is given. Firstly, we study the relationship between DNA labelled graphs and some existing directed graph classes. Secondly, it is shown that for any DNA labelled graph, there exists a positive integer i such that it is(2i, i; 4)-labelled. Furthermore, the smallest i is determined, and a polynomial-time algorithm is introduced to give a(2i, i; 4)-labelling for a given DNA labelled graph. Finally, a DNA algorithm is given to find all paths from one given point to another in a(2i, i; 4)-labelled directed graph.     

Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA ₁, ...,A _k are distinct subsets of a set of cardinalityn, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, andk>n, then some two ofA ₁, ...,A _k have empty intersection. We prove a strengthening, that at leastk /n ofA ₁, ...,A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440     

The second centralizer of a Bernoulli shift is just its powers

If ℐ is a collection of measure preserving transformations of a probability space, byC(ℐ), the centralizer of ℐ, we mean the group of all measure preserving transformationsS such thatTS=ST for allT ∈ ℐ. We show here that ifT is a Bernoulli shift, thenC(C(T))={T ⁱ |i ∈ Z}. The proof is carried out by constructing an action of Z², {T ₁ ⁱ °T ₂ ⁱ |i, j ∈ Z}, whereT ₁ is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T ₁,T ₂ ^i} ={T ₁ ⁱ °T ₂ ^k l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation.     

The degree distribution of the random multigraphs

In this paper, as a generalization of the binomial random graph model, we define the model of multigraphs as follows: let G(n; {p _k }) be the probability space of all the labelled loopless multigraphs with vertex set V = {υ ₁, υ ₂, …, υ _n }, in which the distribution of t_{vi ,vj} t_{v_i ,v_j } , the number of the edges between any two vertices υ _i and υ _j is

Forest-Like Permutations

Given a permutation , construct a graph G _π on the vertex set {1, 2,..., n} by joining i to j if (i) i < j and π(i) < π(j) and (ii) there is no k such that i < k < j and π(i) < π(k) < π(j). We say that π is forest-like if G _π is a forest. We first characterize forest-like permutations in terms of pattern avoidance, and then by a certain linear map being onto. Thanks to recent results of Woo and Yong, these show that forest-like permutations characterize Schubert varieties which are locally factorial. Thus forest-like permutations generalize smooth permutations (corresponding to smooth Schubert varieties). We compute the generating function of forest-like permutations. As in the smooth case, it turns out to be algebraic. We then adapt our method to count permutations for which G _π is a tree, or a path, and recover the known generating function of smooth permutations. Received March 27, 2006     

Sphere and Dot Product Representations of Graphs

A graph G is a k-sphere graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the distance between v _i and v _j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the dot product of v _i and v _j is at least 1.     

Small deviations of the maximal element of a sequence of weighted independent random variables

Consider a sequence {X _i } of independent copies of a nonnegative random variable X and let M = sup _{j ≥ 1}λ _j X _j , where {λ _j } is a nonincreasing sequence of positive numbers for which P(M < ∞) = 1. The asymptotic behavior of -logP(M < r) as r → 0 is studied.     

Orthogonal random vectors and the Hurwitz-Radon-Eckmann theorem

In several different aspects of algebra and topology the following problem is of interest: find the maximal number of unitary antisymmetric operatorsU _i inH = ℝ ⁿ with the propertyU _i U _j = −U _j U _i (i≠j). The solution of this problem is given by the Hurwitz-Radon-Eckmann formula. We generalize this formula in two directions: all the operatorsU _i must commute with a given arbitrary self-adjoint operator andH can be infinite-dimensional. Our second main result deals with the conditions for almost sure orthogonality of two random vectors taking values in a finite or infinite-dimensional Hilbert spaceH. Finally, both results are used to get the formula for the maximal number of pairwise almost surely orthogonal random vectors inH with the same covariance operator and each pair having a linear support inH⊕H. The paper is based on the results obtained jointly with N.P. Kandelaki (see [1,2,3]).     

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000