Search

We introduce the following model for the evolution of a population. At every discrete time $j\geq 0$ exactly one individual is introduced in the population and is assigned a death probability $c_j$ sampled from $C$, a fixed probability distribution. We think of $c_j$ as a genetic marker of this individual. At every time $n\geq 1$ every individual in the population dies or not independently of each other with its corresponding death probability $c_j$. We show that the population size goes to infinity if and only if $E(1/C)=\infty$. This is in sharp contrast with the model with constant $c$ and with the model in random environment (same random $c_n$ for all individuals at time $n$). Both of these models are always positive recurrent. Thus, genetic diversity does seem to help survival! We also study the point process associated with our model. We show that the limit point process has an accumulation point near 0 for the $c'$s. For certain $C$ distributions, including the uniform, the limit process properly rescaled is also shown to converge to a non-homogeneous Poisson process.

We consider a discrete time population model for which each individual alive at time n survives independently of everybody else at time n + 1 with probability β n . The sequence ( β n ) is i.i.d. and constitutes our random environment. Moreover, at every time n we add Z n individuals to the population. The sequence ( Z n ) is also i.i.d. We find sufficient conditions for null recurrence and transience (positive recurrence has been addressed by Neuts 1994 J. Appl. Probab. 31 48–58). We apply our results to a particular ( Z n ) distribution and deterministic β. This particular case shows a rather unusual phase transition in β in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.

This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the sum over all loops of the integral of each loop against a given one-form on the graph. An extension of this result to the noncommutative case of loop holonomies is also discussed. As an application of the first result, we derive a central limit theorem for windings of loops around the faces of a planar graph. More precisely, we show that the winding field generated by a random walk loop soup, when appropriately normalized, has a Gaussian limit as the loop soup intensity tends to ∞, and we give an explicit formula for the covariance kernel of the limiting field. We also derive a Spitzer-type law for windings of the Brownian loop soup, i.e., we show that the total winding around a point of all loops of diameter larger than δ, when multiplied by \(1/\log \delta \), converges in distribution to a Cauchy random variable as δ → 0. The random variables analyzed in this work have various interpretations, which we highlight throughout the paper.

We propose a variation of the GMS model of evolution of species. In this version, as in the GMS model, at each birth, the new species in the system is labeled with a random fitness mark, but in our variation, to each extinction event is associated a random threshold mark and all species with fitness lower than the threshold are removed from the system. We present necessary and suficient criteria for the recurrence and transience of the empty configuration of species; we show the existence of a long time limit distribution of species in the system, and present necessary and suficient criteria for the finiteness of the number of species in that distribution. There is a remarkable symmetry between both sets of criteria., 11 pages

We propose a metric space of coalescing pairs of paths on which we are able to prove (more or less) directly convergence of objects such as the persistence probability in the (one dimensional, nearest neighbor, symmetric) voter model or the diffusively rescaled weight distribution in a silo model (as well as the equivalent output distribution in a river basin model), interpreted in terms of (dual) diffusively rescaled coalescing random walks, to corresponding objects defined in terms of the Brownian web., 22 pages

We take up the issue of deriving the limit as $n\to\infty$ of the GREM-like K process on a tree with $n$ levels. Under specific conditions on the parameters of the process, implying the martingality of a modification of the underlying clock process sequences, we obtain infinite level clock processes as nontrivial limits of the finite level clocks, and use them to construct a process on a suitable product space which is then shown to be the limit of the $n$ level K processes as $n\to\infty$. Some properties of the limiting, infinite level K process are established, like an expression for the asymptotic empirical measure of cylinders, giving information on the prospective equilibrium measure of the infinite level dynamics., 42 pages, 2 figures

We discuss anisotropic scaling limits of long-range dependent linear random fields X on \(\mathbb {Z}^2\) with arbitrary dependence axis (direction in the plane along which the moving-average coefficients decay at a smallest rate). The scaling limits \(V^X_{\gamma }\) are random fields on \(\mathbb {R}^2_+\) defined as the limits (in the sense of finite-dimensional distributions) of partial sums of X taken over rectangles with sides increasing along horizontal and vertical directions at rates λ and λγ respectively as λ →∞ for arbitrary fixed γ > 0. The scaling limits generally depend on γ and constitute a one-dimensional family \(\{V^X_{\gamma }, \gamma >0\}\) of random fields. The scaling transition occurs at some \(\gamma ^X_0 >0\) if \(V^X_\gamma \) are different and do not depend on γ for \( \gamma > \gamma ^X_0 \) and \(\gamma < \gamma ^X_0\). We prove that the fact of ‘oblique’ dependence axis (or incongruous scaling) dramatically changes the scaling transition in the above model so that \(\gamma _0^X = 1\) independently of other parameters, contrasting the results in Pilipauskaitė and Surgailis (2017) on the scaling transition under congruous scaling.