Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotics for the survival probability in a killed branching random walk
431
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $gamma-epsilon$, where $gamma$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $epsilonto 0$, the probability in question decays like $exp{- {beta + o(1)over epsilon^{1/2}}}$, where $beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0<p<{1over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.
rate research
Read More
We first study a model, introduced recently in cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no `obstacle placed there. The obstacles appear at each site with probability $pin [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $mathbb P_p$-measure, as $ntoinfty$, (i) Critical case: P^{omega}(S_n)simfrac{2}{qn}; (ii) Subcritical case: P^{omega}(S_n)= expleft[left( -C_{d,q}cdot frac{n}{(log n)^{2/d}} right)(1+o(1))right], where $C_{d,q}>0$ does not depend on the branching law. Hence, the model exhibits `self-averaging in the critical case but not in the subcritical one. I.e., in (i) the asymptotic tail behavior is the same as in a toy model where space is removed, while in (ii) the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. We utilize a spine decomposition of the branching process as well as some known results on random walks.
We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rateis given by the directional free energy.
We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2times 2$ random matrices.
We work under the A{i}d{e}kon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown that $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+o(log x)$ as $xtoinfty$. Also, we provide necessary and sufficient conditions under which $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+{rm const}+o(1)$ as $xtoinfty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit $Z$. The methodological novelty of the present paper is a three terms representation of a subharmonic function of at most linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.
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).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
