Finding blocks and other patterns in a random coloring of ℤ

Let ξ = (ξ_k)_k∈? be i.i.d. with P(ξ_k = 0) = P(ξ_k = 1) = 1/2, and let S: = (S_k) be a symmetric random walk with holding on ?, independent of ξ. We consider the scenery ξ observed along the random walk path S, namely, the process (χ_k := ξ). With high probability, we reconstruct the color and the length of blockⁿ, a block in ξ of length ≥ n close to the origin, given only the observations (χ_k). We find stopping times that stop the random walker with high probability at particular places of the scenery, namely on blockⁿ and in the interval [?3ⁿ,3ⁿ]. Moreover, we reconstruct with high probability a piece of ξ of length of the order 3 around blockⁿ, given only 3 observations collected by the random walker starting on the boundary of blockⁿ. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Total relative displacement of vertex permutations of K

Let α denote a permutation of the n vertices of a connected graph G. Define δ_α(G) to be the number , where the sum is over all the unordered pairs of distinct vertices of G. The number δ_α(G) is called the total relative displacement of α (in G). So, permutation α is an automorphism of G if and only if δ_α(G) = 0. Let π(G) denote the smallest positive value of δ_α(G) among the n! permutations α of the vertices of G. A permutation α for which π(G) = δ_α(G) has been called a near‐automorphism of G [ 2 ]. We determine π(K) and describe permutations α of K for which π(K) = δ_α(K). This is done by transforming the problem into the combinatorial optimization problem of maximizing the sums of the squares of the entries in certain t by t matrices with non–negative integer entries in which the sum of the entries in the ith row and the sum of the entries in the ith column each equal to n_i,1≤i≤t. We prove that for positive integers, n₁≤n₂≤…≤n_t, where t≥2 and n_t≥2, where k₀ is the smallest index for which n = n+1. As a special case, we correct the value of π(K_m,n), for all m and n at least 2, given by Chartrand, Gavlas, and VanderJagt [ 2 ]. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 85–100, 2002     

Word maps and spectra of random graph lifts

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ₁ and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19–35), in almost every n‐lift H of G, all “new” eigenvalues of H are ≤ O(λ ρ^1/2). Here we improve this bound to O(λ ρ^2/3). It is conjectured in (Friedman, Graphs Duke Math J 118 (2003) 19–35) that the statement holds with the bound ρ + o(1) which, if true, is tight by (Greenberg, PhD thesis, 1995). For G a bouquet with d/2 loops, our arguments yield a simple proof that almost every d‐regular graph has second eigenvalue O(d^2/3). For the bouquet, Friedman (2008). has famously proved the (nearly?) optimal bound of . Central to our work is a new analysis of formal words. Let w be a formal word in letters g,…,g. The word map associated with w maps the permutations σ₁,…,σ_k ∈ S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by σ_i. We investigate the random variable X that counts the fixed points in this permutation when the σ_i are selected uniformly at random. The analysis of the expectation ??(X) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica (Nica, Random Struct Algorithms 5 (1994), 703–730), which determines, for every fixed w, the limit distribution (as n →∞) of X. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = u^d for some word u. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

On smallest triangles

Pick n points independently at random in ?², according to a prescribed probability measure μ, and let Δ ≤ Δ ≤ … be the areas of the () triangles thus formed, in nondecreasing order. If μ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n³Δ : i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(μ). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that if μ is the uniform probability measure on the region S, then κ(μ) ≤ 2/|S|, where |S| denotes the area of S. Equality holds in that κ(μ) = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitányi. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 206–223, 2003     

Robust characterizations of k‐wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non‐uniform) k ‐wise independent if for any subset of k indices {i₁,…,i_k} and for any z₁∈Σ,…,z_k∈Σ, Pr_X～D[X···X = z₁···z_k] = Pr_X～D[X = z₁]···Pr_X～D[X = z_k]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

m‐ary Search trees when m ≥ 27: A strong asymptotics for the space requirements

It is known that the joint distribution of the number of nodes of each type of an m‐ary search tree is asymptotically multivariate normal when m ≤ 26. When m ≥ 27, we show the following strong asymptotics of the random vector X_n = ^t(X, … , X), where X denotes the number of nodes containing i ? 1 keys after having introduced n ? 1 keys in the tree: There exist (nonrandom) vectors X, C, and S and random variables ρ and φ such that (X_n ? nX)/n ? ρ(C cos(τ₂log n + φ) + S sin(τ₂log n + φ)) →_n→∞ 0 almost surely and in L²; σ₂ and τ₂ denote the real and imaginary parts of one of the eigenvalues of the transition matrix, having the second greatest real part. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Packing perfect matchings in random hypergraphs

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer , the binomial ‐uniform random hypergraph contains edge‐disjoint perfect matchings, provided , where is an integer depending only on . Our result for is asymptotically optimal and for is optimal up to the factor. This significantly improves a result of Frieze and Krivelevich.     

The thresholds for diameter 2 in random Cayley graphs

Given a group G, the model denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any and any family of groups G_k of order n_k for which , a graph with high probability has diameter at most 2 if and with high probability has diameter greater than 2 if . We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erd?s‐Renyi random graphs. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 218–235, 2014     

A new construction of perfect nonlinear functions using Galois rings

For a prime p, we give a construction of perfect nonlinear functions from ? to ? when either of the following conditions holds: (1) n≥p; (2) n<p, and n is a composite number or is the sum of positive composite numbers. It follows that when n≥12, there is a perfect nonlinear function from ? to ? for any prime p. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 229‐239, 2009     

Tauberian theorem for m‐spherical transforms on the Heisenberg group

In this paper we prove a Tauberian type theorem for the space L ( H _n ). This theorem gives sufficient conditions for a L ( H _n ) submodule J ? L ( H _n ) to make up all of L ( H _n ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L^∞( H _n ). In connection with the Pompeiu problem, given the vanishing of integrals ∫ z ^m L _g f ( z , 0) dσ ( z ) = 0 for all g ∈ H _n and i = 1, 2 for appropriate radii r₁ and r₂, we now have the (improved) conclusion f ≡ 0, where = · · · and form the standard basis for T^(0,1)( H _n ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase transitions for random walk asymptotics on free products of groups

Suppose we are given finitely generated groups Γ₁,…,Γ_m equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ₁* … *Γ_m and give a complete classification of the possible asymptotic behaviour of the corresponding n‐step return probabilities. They either inherit a law of the form ?^nδn log n from one of the free factors Γ_i or obey a ?^nδn^?3/2‐law, where ? < 1 is the corresponding spectral radius and δ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form $\mathbb{Z}^{d_1}\ast \ldots \ast \mathbb{Z}^{d_m}Suppose we are given finitely generated groups Γ₁,…,Γ_m equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ₁* … *Γ_m and give a complete classification of the possible asymptotic behaviour of the corresponding n‐step return probabilities. They either inherit a law of the form ?^nδn log n from one of the free factors Γ_i or obey a ?^nδn^?3/2‐law, where ? < 1 is the corresponding spectral radius and δ is the period of the random walk. In addition, we determine the full range of the asymptotic behaviour in the case of nearest neighbour random walks on free products of the form $\mathbb{Z}^{d_1}\ast \ldots \ast \mathbb{Z}^{d_m}$. Moreover, we characterize the possible phase transitions of the non‐exponential types n log n in the case Γ₁ * Γ₂. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Covering arrays with mixed alphabet sizes

Covering arrays with mixed alphabet sizes, or simply mixed covering arrays, are natural generalizations of covering arrays that are motivated by applications in software and network testing. A (mixed) covering array A of type is a k × N array with the cells of row i filled with elements from ? and having the property that for every two rows i and j and every ordered pair of elements (e,f) ∈ ? × ?, there exists at least one column c, 1 ≤ c ≤ N, such that A_i,c = e and A_j,c = f. The (mixed) covering array number, denoted by , is the minimum N for which a covering array of type with N columns exists. In this paper, several constructions for mixed covering arrays are presented, and the mixed covering array numbers are determined for nearly all cases with k = 4 and for a number of cases with k = 5. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 413–432, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10059     

The random connection model: Connectivity,edge lengths,and degree distributions

Consider the random graph whose vertex set is a Poisson point process of intensity n on . Any two vertices are connected by an edge with probability , independently of all other edges, and independent of the other points of . d is the toroidal metric, r > 0 and is non‐increasing and . Under suitable conditions on g, almost surely, the critical parameter M_n for which does not have any isolated nodes satisfies . Let , and θ be the volume of the unit ball in . Then for all , is connected with probability approaching one as . The bound can be seen to be tight for the usual random geometric graph obtained by setting . We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy .     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Interpolating between random walk and rotor walk

We introduce a family of stochastic processes on the integers, depending on a parameter and interpolating between the deterministic rotor walk () and the simple random walk (). This p‐rotor walk is not a Markov chain but it has a local Markov property: for each the sequence of successive exits from is a Markov chain. The main result of this paper identifies the scaling limit of the p‐rotor walk with two‐sided i.i.d. initial rotors. The limiting process takes the form , where is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation (1) for all . Here is a standard Brownian motion and are constants depending on the marginals of the initial rotors on and respectively. Chaumont and Doney have shown that Equation 1 has a pathwise unique solution , and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, and . This last result, together with the main result of this paper, implies that the p‐rotor walk is recurrent for any two‐sided i.i.d. initial rotors and any .     

On Decomposition in Exponential Orlicz Spaces

Various characterizations are given of the exponential Orlicz space L and the Orlicz‐Lorentz space L. By way of application we give a simple proof of the celebrated theorem of Brézis and Wainger concerning a limiting case of a Sobolev imbedding theorem.     

Phase transition for Parking blocks,Brownian excursion and coalescence

In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n < m), and ? = m ? n empty places. For a noncomputer science‐minded reader, we shall use the metaphore of n cars parking on m places: each car c_i chooses a place p_i at random, and if p_i is occupied, c_i tries successively p_i + 1, p_i + 2, until it finds an empty place. Pittel [42] proves that when ?/m goes to some positive limit β < 1, the size B of the largest block of consecutive cars satisfies 2(β ? 1 ? log β)B = 2 log m ? 3 log log m + Ξ_m, where Ξ_m converges weakly to an extreme‐value distribution. In this paper we examine at which level for n a phase transition occurs between B = o(m) and m ? B = o(m). The intermediate case reveals an interesting behavior of sizes of blocks, related to the standard additive coalescent in the same way as the sizes of connected components of the random graph are related to the multiplicative coalescent. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 76–119, 2002     

The Calderón approach to an elliptic boundary problem

We consider a boundary problem for an elliptic system in a bounded region Ω ? ?ⁿ and where the spectral parameter is multiplied by a discontinuous weight function ω (x) = diag(ω₁(x), …, ω_N (x)). The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Recently, this problem was studied under the assumption that the ω_j (x)^–1 are essentially bounded in Ω. In this paper we suppose that ω (x) vanishes identically in a proper subregion Ω of Ω and that the ω_j (x)^–1 are essentially bounded in . Then by using methods which are a variant of those used in constructing the Calderón projectors for the boundary Γ of Ω, we shall derive results here which will enable us in a subsequent work to apply the ideas of Calderón to develop the spectral theory associated with the problem under consideration here (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Star factorizations of graph products

A k‐star is the graph K_1,k. We prove a general theorem about k‐star factorizations of Cayley graphs. This is used to give necessary and sufficient conditions for the existence of k‐star factorizations of any power (K_q)^s of a complete graph with prime power order q, products C × C ×··· × C of k cycles of arbitrary lengths, and any power (C_r)^s of a cycle of arbitrary length. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 59–66, 2001