Large Deviations in randomly coloured random graphs

Models of random graphs are considered where the presence or absence of an edge depends on the random types (colours) of its vertices, so that whether or not edges are present can be dependent. The principal objective is to study large deviations in the number of edges. These graphs provide a natural example with two different non-degenerate large deviation regimes, one arising from large deviations in the colourings followed by typical edge placement and the other from large deviation in edge placement. A secondary objective is to illustrate the use of a general result on large deviations for mixtures

Survival, extinction and approximation of discrete-time branching random walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.Comment: 32 pages, a few misprints have been correcte

Sperm precedence in the domestic fowl

The aim of this study was to examine last-male sperm precedence in the domestic fowl. We used sperm from two different genotypes to assign paternity: and in seven experiments females were artificially inseminated with either equal or unequal numbers of sperm at intervals of 4 or 24 h. We were unable to replicate the results of a previous study by Compton et al. (1978) in which a strong last-male precedence effect had been recorded when two equal sized inseminations were made 4 h apart. We observed no marked last-male sperm precedence and our results did not differ significantly from that predicted by the passive sperm loss model, in which a last-male effect is determined by the rate at which sperm are lost from the female tract and the interval between successive inseminations. The most likely explanation for the disparity between our result and Compton et al.'s is a difference in the timing of inseminations. The implications of this for studies of sperm competition in birds is discussed

Accurately quantifying the shape of birds’ eggs

Describing the range of avian egg shapes quantitatively has long been recognized as difficult. A variety of approaches has been adopted, some of which aim to capture the shape accurately and some to provide intelligible indices of shape. The objectives here are to show that a (four‐parameter) method proposed by Preston (1953, The Auk, 70, 160) is the best option for quantifying egg shape, to provide and document an R program for applying this method to suitable photographs of eggs, to illustrate that intelligible shape indices can be derived from the summary this method provides, to review shape indices that have been proposed, and to report on the errors introduced using photographs of eggs at rest rather than horizontal

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

Exact asymptotics of the freezing transition of a logarithmically correlated random energy model

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation - thus translating Bramson's work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point

Uniform Convergence Of Martingales In The One-Dimensional Branching Random Walk

In the supercritical branching random walk an initial person has children whose positions are given by a point process Z (1) . Each of these then has children in the same way, with the positions of children in each family, relative to their parent's, being given by independent copies of Z (1) , and so on. For any value of its argument, , the Laplace transform of the point process of n th generation people, normalized by its expected value, is a martingale, the usual branching process martingale being a special case. Here it is shown that under certain conditions these martingales converge uniformly in , almost surely and in mean. A consequence of this result is that the limit is, in an appropriate region, analytic in . 1 Introduction. This paper considers the one dimensional supercritical branching random walk. The process starts with a single initial ancestor at the origin. She has children, forming the ørst generation, with their positions on the real line, R, being given by ..

The growth of iterates of multivariate generating functions

The vector-valued function of a -vector has components . For each , is the (multivariate) Laplace transform of a discrete measure concentrated on with only a finite number of atoms. The main objective is to give conditions for the functional iterates of to grow like for a suitable . The initial stimulus was provided by results of Miller and O'Sullivan (1992) on enumeration issues in `context free languages', results which can be improved using the theory developed here. The theory also allows certain results in Jones (2004) on multitype branching to be proved under significantly weaker conditions