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

By Thomas Madaule. 2011

116 citations recorded by the source corpus.

Consider a critical branching random walk on the real line. In a recent paper, Aïdékon (2011) 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 (J Stat Phys 143:420–446, 2011), can be viewed as a discrete analog of the corresponding results for the branching Brownian motion, previously established by Arguin et al. (2010, 2011) and Aïdékon et al. (2011).

Subfields: Finance, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/2b61ac8c158a0fc147abc141af986798c85222d1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. P. Maillard
2. Thomas Madaule
3. Zhan Shi
4. Rémi Rhodes
5. Vincent Vargas
6. Elie Aidékon
7. J. Berestycki
8. E. Brunet

In Mathematical Physics

1. Thomas Madaule
2. Zhan Shi
3. Rémi Rhodes
4. P. Maillard
5. Vincent Vargas
6. Elie Aidékon
7. J. Berestycki
8. E. Brunet

In Molecular Biology

1. P. Maillard
2. Zhan Shi
3. Thomas Madaule
4. Elie Aidékon
5. J. Berestycki
6. E. Brunet
7. Rémi Rhodes
8. Vincent Vargas

Citations in the corpus

• Branching random walks (2015, background): “By comparison to analoguous results for branching random walks, many deep results have been recently established for Gaussian free fields and more general logarithmically correlated Gaussian fields (Bolthausen, Deuschel and Giacomin [61], Madaule [183], Biskup and Louidor [60], Ding, Roy and Zeitoun”
• A note on stable point processes occurring in branching Brownian motion (2011, methodology): “Once the convergence of the point process is established, one now readily sees that the limiting process is exp-1-stable [8, 18]: Take two BBMs and denote their derivative martingale limits by W and W ′, respectively.”
• Branching Brownian motion seen from its tip (2011, methodology): “Finally we mention that Madaule [28] has proved the analogue of our Theorem 2.1 for non-lattice branching random walks by using the recent result in [1] on the maximum of branching random walks.”
• Maximum of a log-correlated Gaussian field (2013, background): “We expect to establish the convergence of the random measure formed by the particles near to the maximum, just like in the BRW case (see [1], [23], [11] and [19] for an explicit connection between branching random walk and this model).”
• Gaussian multiplicative chaos and applications: A review (2013, background): “onjectured to be involved in the glassy phase and freezing phenomena observed in log-correlated random potentials. The reader may consult [33,67,68] for an account of physics motivations and results, [19,23,109,138] for rigorous results in the case of discrete models and [17,57] for precise conjectu”