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Consider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is
$\Theta(1/n)$
in the minimal
$L_p$
metric for any
$p\in[1,\infty]$
, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions.
It is known that for Kn,n equipped with i.i.d. exp (1) edge costs, the minimum total cost of a perfect matching converges to
$\zeta(2)=\pi^2/6$
in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension
$q \geq 1$
. In this paper we extend those results to all real positive q, confirming the Mézard–Parisi conjecture in the last remaining applicable case.
We introduce a non-increasing tree growth process
$((T_n,{\sigma}_n),\, n\ge 1)$
, where Tn is a rooted labelled tree on n vertices and σn is a permutation of the vertex labels. The construction of (Tn, σn) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ Tn with positive probability. The key feature of the process is that the shape of Tn has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process.
We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n, this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least
$c\ln n$
, c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.
We study the scaling limit of a random forest with prescribed degree sequence in the regime that the largest tree consists of all but a vanishing fraction of nodes. We give a description of the limit of the forest consisting of the small trees, by relating a plane forest to a marked cyclic forest and its corresponding skip-free walk.
We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.
In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.
We show that the Mallows measure on permutations of
$1,\dots ,n$
arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph
$K_{{\mathbb Z},{\mathbb Z}}$
falls into one of two classes: a countable family
$(\sigma _n)_{n\in {\mathbb Z}}$
of tame stable matchings, in which the length of the longest edge crossing k is
$O(\log |k|)$
as
$k\to \pm \infty $
, and an uncountable family of wild stable matchings, in which this length is
$\exp \Omega (k)$
as
$k\to +\infty $
. The tame stable matching
$\sigma _n$
has the law of the Mallows permutation of
${\mathbb Z}$
(as constructed by Gnedin and Olshanski) composed with the shift
$k\mapsto k+n$
. The permutation
$\sigma _{n+1}$
dominates
$\sigma _{n}$
pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.
A uniform recursive tree on n vertices is a random tree where each possible
$(n-1)!$
labelled recursive rooted tree is selected with equal probability. We introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications. We analyse the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.
Let V be an n-set, and let X be a random variable taking values in the power-set of V. Suppose we are given a sequence of random coupons
$X_1, X_2, \ldots $
, where the
$X_i$
are independent random variables with distribution given by X. The covering time T is the smallest integer
$t\geq 0$
such that
$\bigcup_{i=1}^t X_i=V$
. The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focused almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.
In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.
We study random unlabelled k-trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle-pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov–Hausdorff–Prokhorov and Benjamini–Schramm limits that describe their asymptotic geometric shape on a global and local scale as the number of (k + 1)-cliques tends to infinity.
We introduce a model for the spreading of fake news in a community of size n. There are
$j_n = \alpha n - g_n$
active gullible persons who are willing to believe and spread the fake news, the rest do not react to it. We address the question ‘How long does it take for
$r = \rho n - h_n$
persons to become spreaders?’ (The perturbation functions
$g_n$
and
$h_n$
are o(n), and
$0\le \rho \le \alpha\le 1$
.) The setup has a straightforward representation as a convolution of geometric random variables with quadratic probabilities. However, asymptotic distributions require delicate analysis that gives a somewhat surprising outcome. Normalized appropriately, the waiting time has three main phases: (a) away from the depletion of active gullible persons, when
$0< \rho < \alpha$
, the normalized variable converges in distribution to a Gumbel random variable; (b) near depletion, when
$0< \rho = \alpha$
, with
$h_n - g_n \to \infty$
, the normalized variable also converges in distribution to a Gumbel random variable, but the centering function gains weight with increasing perturbations; (c) at almost complete depletion, when
$r = j -c$
, for integer
$c\ge 0$
, the normalized variable converges in distribution to a convolution of two independent generalized Gumbel random variables. The influence of various perturbation functions endows the three main phases with an infinite number of phase transitions at the seam lines.
The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.
For a rumour spreading protocol, the spread time is defined as the first time everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by
$O({n^{1/3}}{\log ^{2/3}}n)$
. This improves the
$O(\sqrt n)$
upper bound of Giakkoupis, Nazari and Woelfel (2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph. We also show that if, for a pair α, β of real numbers, there exist infinitely many graphs for which the two spread times are nα and nβ in expectation, then
$0 \le \alpha \le 1$
and
$\alpha \le \beta \le {1 \over 3} + {2 \over 3} \alpha $
; and we show each such pair α, β is achievable.
We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of weighted combinatorial objects. We describe a general setting characterized by the formation of a giant component. The collection of small fragments is shown to converge in total variation toward a limit object following a Pólya–Boltzmann distribution.
The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with independent and identically distributed signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.
Consider any fixed graph whose edges have been randomly and independently oriented, and write {S ⇝} to indicate that there is an oriented path going from a vertex s ∊ S to vertex i. Narayanan (2016) proved that for any set S and any two vertices i and j, {S ⇝ i} and {S ⇝ j} are positively correlated. His proof relies on the Ahlswede–Daykin inequality, a rather advanced tool of probabilistic combinatorics.
In this short note I give an elementary proof of the following, stronger result: writing V for the vertex set of the graph, for any source set S, the events {S ⇝ i}, i ∊ V, are positively associated, meaning that the expectation of the product of increasing functionals of the family {S ⇝ i} for i ∊ V is greater than the product of their expectations.
Let r ⩾ 2 be a fixed constant and let
$ {\cal H} $
be an r-uniform, D-regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that degrees of pairs of vertices in
$ {\cal H} $
are at most L where L = D/( log N)ω(1). We consider the random greedy algorithm for forming a matching in
$ {\cal H} $
. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of
$ {\cal H} $
that are not saturated by the final matching is at most (L/D)(1/(2(r−1)))+o(1). This point is a natural barrier in the analysis of the random greedy hypergraph matching process.
The accessibility percolation model is investigated on random rooted labeled trees. More precisely, the number of accessible leaves (i.e. increasing paths) Zn and the number of accessible vertices Cn in a random rooted labeled tree of size n are jointly considered in this work. As n → ∞, we prove that (Zn, Cn) converges in distribution to a random vector whose probability generating function is given in an explicit form. In particular, we obtain that the asymptotic distributions of Zn + 1 and Cn are geometric distributions with parameters e/(1 + e) and 1/e, respectively. Much of our analysis is performed in the context of local weak convergence of random rooted labeled trees.
We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are ‘jointly connected’. Bollobás, Riordan, Slivken, and Smith (2017) proved that, when the two graphs are independent binomial random graphs, whether the jigsaw process percolates undergoes a phase transition when the product of the two probabilities is
$\Theta({1}/{(n\ln n)})$
. We show that this threshold is sharp, and that it lies at
${1}/{(4n\ln n)}$
.
In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).