Large components of bipartite random mappings

Annales De L Institut Henri Poincare-probabilites Et Statistiques, 2008

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, a k ∼ Ck p−1 , k → ∞, p > 0, where C is a positive constant. The measures considered are associated with the generalized Maxwell-Boltzmann models in statistical mechanics, reversible coagulation-fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation.

Journal of Applied Probability, 1989

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

Discussiones Mathematicae Graph Theory, 1996

The asymptotic distributions of the number of vertices of a given degree in random graphs, where the probabilities of edges may not be the same, are given. Using the method of Poisson convergence, distributions in a general and particular cases (complete, almost regular and bipartite graphs) are obtained.

Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.

International Journal of Statistics and Probability, 2014

In this article we correct [DM10, Remark 3] and improve upon the proof of [DM10, Theorem 2.5]; the large deviation principle (LDP) for empirical neigbourhood distribution of symbolled random graphs conditioned to a given empirical symbol distribution and empirical pair distribution. We show that the LDP for the empirical degree measure of the classical Erdős-Rényi graph is a special case of [DM10, Theorem 2.5]. From the LDP for the empirical degree measure, we derive the LDP for the the proportion of isolated vertices in the classical Erdős-Rényi graph.

Electronic Journal of Probability, 2013

We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to deal with the normal approximation of the geometric U -statistics introduced by Reitzner and Schulte (2011). In particular, we shall provide a new analytic characterization of geometric random graphs whose edge-counting statistics exhibit asymptotic Gaussian fluctuations, and describe a new form of Poisson convergence for stationary random graphs with sparse connections. In a companion paper, the above analysis is extended to general sequences of U -statistics with rescaled kernels.

Random Structures and Algorithms, 2005

We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1 − p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold p c = p c (G, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV 1/3 , where λ is fixed and positive. We show that for any such model, there is a phase transition at p c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p c . In particular, we show that the largest cluster inside a scaling window of size |p − p c | = Θ(Ω −1 V −1/3 ) is of size Θ(V 2/3 ), while below this scaling window, it is much smaller, of order O( −2 log(V 3 )), with = Ω(p c − p). We also obtain an upper bound O(Ω(p − p c )V ) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p − p c )). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n-cube and certain Hamming cubes, as well as the spread-out n-dimensional torus for n > 6.

2005

In this paper we consider a random mapping,T n , of the finite set {1, 2, ..., n} into itself for which the digraph representationĜ n is constructed by: (1) selecting a random number,L n , of cyclic vertices, (2) constructing a uniform random forest of size n with the selected cyclic vertices as roots, and forming 'cycles' of trees by applying a random permutation to the selected cyclic vertices. We investigatê k n , the size of a 'typical' component ofĜ n , and, under the assumption that the random permutation on the cyclical vertices is uniform, we obtain the asymptotic distribution ofk n conditioned onL n = m(n). As an application of our results, we show in Section 3 that provided L n is of order much larger than √ n, then the joint distribution of the normalized order statistics of the component sizes ofĜ n converges to the Poisson-Dirichlet(1) distribution as n → ∞. Other applications and generalizations are also discussed in Section 3.

Combinatorics, …, 2000

Combinatorics, Probability & Computing, 1997

Assemblies are labelled combinatorial objects that can be decomposed into components. Examples of assemblies include set partitions, permutations and random mappings. In addition, a distribution from population genetics called the Ewens sampling formula may be treated as an assembly. Each assembly has a size n, and the sum of the sizes of the components sums to n. When the uniform distribution is put on all assemblies of size n, the process of component counts is equal in distribution to a process of independent Poisson variables Z i conditioned on the event that a weighted sum of the independent variables is equal to n. Logarithmic assemblies are assemblies characterized by some θ > 0 for which i EZ i → θ. Permutations and random mappings are logarithmic assemblies; set partitions are not a logarithmic assembly. Suppose b = b(n) is a sequence of positive integers for which b/n → β ∈ [0, 1]. For logarithmic assemblies, the total variation distance d b (n) between the laws of the first b coordinates of the component counting process and of the first b coordinates of the independent processes converges to a constant H(β). An explicit formula for H(β) is given for β ∈ (0, 1] in terms of a limit process which depends only on the parameter θ. Also, it is shown that d b (n) → 0 if and only if b/n → 0, generalizing results of Arratia, Barbour and Tavaré for the Ewens sampling formula. Local limit theorems for weighted sums of the Z i are used to prove these results.