We consider an irreducible pure jump Markov process with rates Q = (q(x, y)) on Λ ∪ {0} with Λ countable and 0 an absorbing state. A quasi-stationary distribution (qsd) is a probability measure ν on Λ that satisfies: starting with ν, the conditional distribution at time t, given that at time t the process has not been absorbed, is still ν. That is, ν(x) = νP t (x)/( y∈Λ νP t (y)), with P t the transition probabilities for the process with rates Q.A Fleming-Viot (fv) process is a system of N particles moving in Λ. Each particle moves independently with rates Q until it hits the absorbing state 0; but then instantaneously chooses one of the N − 1 particles remaining in Λ and jumps to its position. Between absorptions each particle moves with rates Q independently.Under the condition α := x inf Q(·, x) > sup Q(·, 0) := C we prove existence of qsd for Q; uniqueness has been proven by Jacka and Roberts. When α > 0 the fv process is ergodic for each N. Under α > C the mean normalized densities of the fv unique stationary measure converge to the qsd of Q, as N → ∞; in this limit the variances vanish.

Recent computational studies have shown that speciation can occur under neutral conditions, i.e., when the simulated organisms all have identical fitness. These works bear comparison with mathematical studies of clustering on neutral landscapes in the context of branching and coalescing random walks. Here, we show that sympatric clustering/speciation can occur on a neutral landscape whose dimensions specify only the simulated organisms' phenotypes. We demonstrate that clustering occurs not only in the case of assortative mating, but also in the case of asexual fission; it is not observed in the control case of random mating. We find that the population size and the number of clusters undergo a second-order non-equilibrium phase transition as the maximum mutation size is varied.

No abstract

We introduce a lattice gas model for the merging of two single-lane automobile highways. The merging rules for traffic on the two lanes are deterministic, but the inflow on both lanes is stochastic. Analysing the stationary distribution of this stochastic cellular automaton, we find a discontinuous phase transition from a free-flow phase which depends on the initial state of the road to a jammed phase where all memory of the initial state is lost. Inside the jammed phase we identify dynamical phase transitions in the approach to stationarity. Each dynamical phase is characterized by a fixed number of relaxation cycles which is decreasing as one moves deeper into the jammed phase. In each cycle step, the number of 'desperate' drivers who force their way onto the main road when they reach the end of the on-ramp increases until stationarity.

In this paper we present an algorithm for exact generation of multivariate samples with prespecified marginal distributions and a given correlation matrix, based on a mixture of Fréchet-Hoeffding bounds and marginal products. The algorithm can accommodate any among the theoretically possible correlation coefficients, and explicitly provides a connection between simulation and the minimum correlation attainable for different distribution families. We calculate the minimum correlations in several common distributional examples, including in some that have not been looked at before. As an illustration, we provide the details and results of implementing the algorithm for generating three-dimensional negatively and positively correlated Beta random variables, making it the only non-copula algorithm for correlated Beta simulation in dimensions greater than two. This work has potential for impact in a variety of fields where simulation of multivariate stochastic components is desired.