Convergence in Law for the Branching Random Walk Seen from Its Tip

Abstract

Abstract. Considering a critical branching random walk on the real line. In a recent paper, Aïdékon [2] developed a powerful method to obtain the convergence in law of its minimum after a log-factor translation. By an adaptation of this method, we show that the point process formed by the branching random walk seen from the minimum converges in law to a decorated Poisson point process. This result, confirming a conjecture of Brunet and Derrida [10], can be viewed as a discrete analog of the corresponding results for the branching brownian motion, previously established by Arguin et al. [5] [6] and Aïdékon et al. [3].

Freezing and Decorated Poisson Point Processes

“…It also gives the independence of the derivative martingale and the limiting shifted process. Combined with Remark 3.2 there, Theorem 2.3 of [41] implies that for those functions L ξ [ f | · ] ≈ Gum. Essentially, using an approximation argument the equivalence is extended to f ∈ C + c (R) in order to show that this process is exponentially-stable (to be accurate, the approximation is done in terms of the characteristic function and not the Laplace transform).…”

“…For the purpose of comparison, in the case of BBM we can relate our results to studying the limiting behavior as t → ∞ and then y → ∞, and the approach of Arguin et al can be seen as taking the limits simultaneously by defining (1) and letting t → ∞. Madaule [41] proved that the extremal process of the BRW is an SDPPP of exponential density. Theorem 2.3 of [41], which as the author notes is the key step to the main result, expresses the Laplace functional of the extremal process shifted by the derivative martingale, the definition of which is similar to that in the case of BBM, for functions of the form f (x) = i≤k θ i exp(β i x) with β i larger then a critical value.…”

“…In fact, as we discuss in Sect. 3, a study of the Laplace functional is the main step in the approach taken for all the models for which the limiting extremal process is known to satisfy (SDP): the BBM [6], the two-speed BBM [10], the variable speed BBM [11], and the branching random walk (BRW) [41]. In particular, Proposition 3.2 of [6] gives exactly Condition (SUS) above.…”

“…Madaule [41] proved that the extremal process of the BRW is an SDPPP of exponential density. Theorem 2.3 of [41], which as the author notes is the key step to the main result, expresses the Laplace functional of the extremal process shifted by the derivative martingale, the definition of which is similar to that in the case of BBM, for functions of the form f (x) = i≤k θ i exp(β i x) with β i larger then a critical value. It also gives the independence of the derivative martingale and the limiting shifted process.…”

Oscillations in the height of the Yule tree and application to the binary search tree

“…Furthermore, the asymptotic behaviour of E(F t ) is especially interesting because it is related to a more general question, namely the convergence of the extremal point process. In the case of the branching Brownian motion, it has been shown independently by Aïdékon, Berestycki, Brunet and Shi [3] and Arguin, Bovier and Kistler [5] that the extremal point process of branching Brownian motion converges and Madaule [29] proved the analogous result for branching random walks with non-lattice support. The lattice case has not been dealt with and the behaviour of E(F t ) could shed a first light on this case.…”