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Register a new userThe sustainability probability for the critical Derrida-Retaux model
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We are interested in the recursive model $(Y_n, , nge 0)$ studied by Collet, Eckmann, Glaser and Martin [9] and by Derrida and Retaux [12]. We prove that at criticality, the probability ${bf P}(Y_n>0)$ behaves like $n^{-2 + o(1)}$ as $n$ goes to infinity; this gives a weaker confirmation of predictions made in [9], [12] and [6]. Our method relies on studying the number of pivotal vertices and open paths, combined with a delicate coupling argument.
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We introduce particle systems in one or more dimensions in which particles perform branching Brownian motion and the population size is kept constant equal to $N > 1$, through the following selection mechanism: at all times only the $N$ fittest particles survive, while all the other particles are removed. Fitness is measured with respect to some given score function $s:R^d to R$. For some choices of the function $s$, it is proved that the cloud of particles travels at positive speed in some possibly random direction. In the case where $s$ is linear, we show under some assumptions on the initial configuration that the shape of the cloud scales like $log N$ in the direction parallel to motion but at least $c(log N)^{3/2}$ in the orthogonal direction for some $c > 0$. We conjecture that the exponent 3/2 is sharp. This result is equivalent to the following result of independent interest: in one-dimensional systems, the genealogical time is greater than $c(log N)^3$, thereby contributing a step towards the original predictions of Brunet and Derrida. We discuss several open problems and also explain how our results can be viewed as a rigorous justification of Weismanns arguments for the role of recombination in population genetics.
The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds $I_0$) the extinction time in the critical model is of order $sqrt{N}$. Here we improve that result by proving the convergence of $i_N(t)=I(sqrt{N}t)/sqrt{N}$ to a limiting diffusion. We do this by showing that within a short time, this four dimensional process collapses to two dimensions: the number of $SI$ and $II$ partnerships are constant multiples of the the number of infected singles. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude $sqrt{N}$ on the original time scale and averages out of the limit theorem for $i_N(t)$. As a by-product of our proof we show that if $tau_N$ is the extinction time of $i_N(t)$ (on the $sqrt{N}$ time scale) then $tau_N$ has a limit.
We consider a class of branching-selection particle systems on $R$ similar to the one considered by E. Brunet and B. Derrida in their 1997 paper Shift in the velocity of a front due to a cutoff. Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size $N$ of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate $(log N)^{-2}$. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of $N$ independent branching random walks killed below a linear space-time barrier.
We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $-sqrt{2}$. Kesten (1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $t$. These bounds improve upon results of Kesten (1978), and partially confirm nonrigorous predictions of Derrida and Simon (2007).
The frog model on the rooted d-ary tree changes from transient to recurrent as the number of frogs per site is increased. We prove that the location of this transition is on the same order as the degree of the tree.
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