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A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q 1,Q 2,…, Q n on a circle. If the counterclockwise way from Q i to Q j on the circle is shorter than the clockwise way, we say Q i dominates Q j . Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2 n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, η i = N i (∞), as a function of N i (0). We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/S = ξ i , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2, the two-leaf star which has three vertices and two edges.

Here we introduce two simple models: simple cubic random packing and random packing by Hamming distance. Consider the packing density γ d of dimension d by cubic random packing. From computer simulations up to dimension 11, γ d +1/γ d seems to approach 1. Also, we give simulation results for random packing by Hamming distance and discuss the behavior of packing density when dimensionality is increased. For the case of Hamming distances of 2 or 3, d–α fits the simulation results of packing density where α is an empirical constant. The variance of packing density is larger when k is even and smaller when k is odd, where k represents Hamming distance.

Let L(t) be the random variable which represents the minimum of length of gaps generated by random packing of unit intervals into [0, tJ. We have P(L(x +1) h)' P(L(y)>h)P(L(x y)?h)dy xo with 0 for 0-x

We investigate a random collision model for competition between types of individuals in a population. There are dominance relations defined for each pair of types such that if two individuals of different types collide then after the collision both are of the dominant type. These dominance relations are represented by an oriented graph, called a tournament. It is shown that tournaments having a particular form are relatively stable, while other tournaments are relatively unstable. A measure of the stability of the stable tournaments is given in the main theorem.

On introduit des integrales invariantes d'un systeme de Lotka-Volterra avec une infinite d'especes. Les valeurs de ces integrales sont donnees par les probabilites de certaines configurations de points aleatoires sur un cercle quand la densite de probabilite sur le cercle satisfait une certaine condition de symetrie