The ζ(2) limit in the random assignment problem

The random assignment (or bipartite matching) problem asks about A_n=min_π ∑c(i, π(i)), where (c(i, j)) is a n×n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. Mézard and Parisi (1987) used the replica method from statistical physics to argue nonrigorously that EA_n→ζ(2)=π²/6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ(2) limit and of the conjectured limit distribution of edge‐costs and their rank‐orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almost‐optimal matching coincides with the optimal matching except on a small proportion of edges. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 381–418, 2001     

On increasing subsequences of minimal Erdös-Szekeres permutations

Let π be a minimal Erdös-Szekeres permutation of 1, 2, ..., n ², and let l _n,k be the length of the longest increasing subsequence in the segment (π(1), ..., π(k)). Under uniform measure we establish an exponentially decaying bound of the upper tail probability for l _n,k , and as a consequence we obtain a complete convergence, which is an improvement of Romik’s recent result. We also give a precise lower exponential tail for l _n,k .     

Largest planar matching in random bipartite graphs

For a distribution ?? over labeled bipartite (multi) graphs G = (W, M, E), |W| = |M| = n, let L(n) denote the size of the largest planar matching of G (here W and M are posets drawn on the plane as two ordered rows of nodes and edges are drawn as straight lines). We study the asymptotic (in n) behavior of L(n) for different distributions ??. Two interesting instances of this problem are Ulam's longest increasing subsequence problem and the longest common subsequence problem. We focus on the case where ?? is the uniform distribution over the k‐regular bipartite graphs on W and M. For k = o(n^1/4), we establish that $L(n) \slash \sqrt{kn}$ tends to 2 in probability when n → ∞. Convergence in mean is also studied. Furthermore, we show that if each of the n² possible edges between W and M are chosen independently with probability 0 < p < 1, then L(n)/n tends to a constant γ_p in probability and in mean when n → ∞. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 162–181, 2002     

Packing of graphic n‐tuples

An n‐tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v₁, …, v_n} in which the degree of v_i is the ith entry of π. Graphic n‐tuples (d, …, d) and (d, …, d) pack if there are edge‐disjoint n‐vertex graphs G₁ and G₂ such that d(v_i) = d and d(v_i) = d for all i. We prove that graphic n‐tuples π₁ and π₂ pack if , where Δand δdenote the largest and smallest entries in π₁ + π₂ (strict inequality when δ = 1); also, the bound is sharp. Kundu and Lovász independently proved that a graphic n‐tuple π is realized by a graph with a k‐factor if the n‐tuple obtained by subtracting k from each entry of π is graphic; for even n we conjecture that in fact some realization has k edge‐disjoint 1‐factors. We prove the conjecture in the case where the largest entry of π is at most n/2 + 1 and also when k?3. © 2011 Wiley Periodicals, Inc. J Graph Theory     

The Length of the Longest Increasing Subsequence of a Random Mallows Permutation

The Mallows measure on the symmetric group S _n is the probability measure such that each permutation has probability proportional to q raised to the power of the number of inversions, where q is a positive parameter and the number of inversions of π is equal to the number of pairs i<j such that π _i >π _j . We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that n→∞ and q→1 in such a way that n(1?q) has a limit in R.     

A power law for connectedness of some random graphs at the critical point

We consider P(G is connected) when G is a graph with vertex set Z⁺ = {1,2, …}, and the edge between i and j is present with probability p(i, j) = min(λ h(i, j), 1) for certain functions h(i, j) homogeneous of degree -1. It is known that there is a critical value λ_c of λ such that . We show that the probability, at the critical point λ_c, that n₁, and n₂ are connected satisfies a power law, in the sense that for n₂ ≧ n_t ≧ 1 for any δ > 0 and certain constants c₁ and c₂.     

A local compactness theorem for Maxwell's equations

The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor-valued function on G satisfying certain restrictions, and let {_n} be a sequence of vector-valued functions on G where the L₂-norms of {_n}, {curl _n}, and {div(β _n)} are bounded, and where all _n either satisfy x _n = 0 or (β F_n) = 0 at the boundary ?G of G ( = normal to ?G): then {_n} has a L₂-convergent subsequence. The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ?G if β is interpreted as electric dielectricity ? or as magnetic permeability μ, respectively. These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations.     

A refinement of Vietoris' inequality for sine polynomials

Let where In 1958, Vietoris proved that σ_n (x) is positive for all n ≥ 1 and x ∈ (0, π). We establish the following refinement. The inequalities hold for all natural numbers n and real numbers n ≥ 1 and x ∈ (0, π) if and only if      

Law of large numbers for increasing subsequences of random permutations

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)²) as n → ∞ if and only if . In particular then, the weak law of large numbers holds for Z if ; that is, We also show the following approximation result for the uniform measure U_n on S_n. Define the probability measure μ on S_n by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x₁,x₂,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if . In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Near‐automorphisms of paths

The total relative displacement of a permutation f of vertices of a connected graph G is , where the sum is taken over all unordered pairs of distinct vertices of G. Let π(G) denote the smallest positive value of δ_f(G) among the n! permutations f. Aitken [J Combin Theory Series A 87 (1999), 1–21] proved that π(P_n) = 2n ? 4 for the n‐path P_n, which was conjectured by Chartrand et al. [Proceedings of the 1996 Eighth Quadrennial International Conference on Graph Theory, Combinatorics Algorithms, and Applications I, New Issues Press, Kalamazoo, 1999, pp. 181–192]. This article gives a short proof of the result. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:323‐325, 2011     

A note on the probabilistic analysis of patricia trees

We consider random PATRICIA trees constructed from n i.i.d. sequences of independent equiprobable bits. We study the height H_n (the maximal distance between the root and a leaf), and the minimal fill-up level F_n (the minimum distance between the root and a leaf). We give probabilistic proofs of .     

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)     

Linear deviations of the classes\tilde W_p^\alpha and approximations in spaces of multipliers

We consider the problem of the asymptotically best linear method of approximation in the metric of L_s[?π, π] of the set \(\tilde W_p^\alpha (1)\) of periodic functions with a bounded in L_p[?π, π] fractional derivative, by functions from \(\tilde W_p^\beta (M)\) ,β >α, for sufficiently large M, and the problem about the best approximation in L_s[?π, π] of the operator of differentiation on \(\tilde W_p^\alpha (1)\) by continuous linear operators whose norm (as operators from L_r[?π, π] into L_q[?π, π])does not exceed M. These problems are reduced to the approximation of an individual element in the space of multipliers, and this allows us to obtain estimates that are exact in the sense of the order.     

On an online random k‐SAT model

Given n Boolean variables x₁,…,x_n, a k‐clause is a disjunction of k literals, where a literal is a variable or its negation. Suppose random k‐clauses are generated one at a time and an online algorithm accepts or rejects each clause as it is generated. Our goal is to accept as many randomly generated k‐clauses as possible with the condition that it must be possible to satisfy every clause that is accepted. When cn random k‐clauses on n variables are given, a natural online algorithm known as Online‐Lazy accepts an expected (1 ? )cn + a_kn clauses for some constant a_k. If these clauses are given offline, it is possible to do much better, (1 ? )cn + Ω( )n can be accepted whp . The question of closing the gap between a_k and Ω( ) for the online version remained open. This article shows that for any k ≥ 1, any online algorithm will accept less than (1 ? )cn + (ln 2)n k‐clauses whp , closing the gap between the constant and Ω( ). Furthermore we show that this bound is asymptotically tight as k → ∞. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Expected length of the longest common subsequence for large alphabets

We consider the length L of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k-ary alphabet. Subadditivity arguments yield that E[L]/n converges to a constant γ_k. We prove a conjecture of Sankoff and Mainville from the early 1980s claiming that as k→∞.     

Relative length of longest paths and cycles in 3‐connected graphs

For a graph G, let p(G) denote the order of a longest path in G and c(G) the order of a longest cycle in G, respectively. We show that if G is a 3‐connected graph of order n such that for every independent set {x₁, x₂, x₃, x₄}, then G satisfies c(G) ≥ p(G) ? 1. Using this result, we give several lower bounds to the circumference of a 3‐connected graph. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 137–156, 2001     

Supercritical Branching Brownian Motion and K-P-P Equation In the Critical Speed-Area

If R_t is the position of the rightmost particle at time t in a one dimensional branching brownian motion, whore α is the inverse of the mean life time and m is the mean of the reproduction law. If Z_t denotes the random point measure of particles living at time t, we get in the critical area {c = c₀} The function u(t, x) = P(R_t > x) is studied as a solution of the K-P-P equation for some function f. Conditioned on non-extinction of the spatial tree in the c₀-direction, a limit distribution is obtained and characterized.     

On (ε,k)‐min‐wise independent permutations

A family of permutations of [n] = {1,2,…,n} is (ε,k)‐min‐wise independent if for every nonempty subset X of at most k elements of [n], and for any x ∈ X, the probability that in a random element π of , π(x) is the minimum element of π(X), deviates from 1/∣X∣ by at most ε/∣X∣. This notion can be defined for the uniform case, when the elements of are picked according to a uniform distribution, or for the more general, biased case, in which the elements of are chosen according to a given distribution D. It is known that this notion is a useful tool for indexing replicated documents on the web. We show that even in the more general, biased case, for all admissible k and ε and all large n, the size of must satisfy as well as This improves the best known previous estimates even for the uniform case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

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     

Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity

Let us consider the boundary‐value problem where g: ? → ? is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ?² and ∈ C [0, π ] with ∫^π ₀ (x) sin x dx = 0. If λ ₁ denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ ₁, 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)