## Abstract

We consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it observes. We show that up to equivalence the scenery can be reconstructed a.s. from the color record of all particles. To do so, we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to Popov and Pachon [Stochastics 83 (2011) 107-116], where at each time the particles needed to see a window around their current position, and in Löwe and Matzinger [Ann. Appl. Probab. 12 (2002) 1322-1347], where the reconstruction is done for d = 2 with a single particle instead of a branching random walk, but millions of colors are necessary.

Original language | English |
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Pages (from-to) | 651-685 |

Number of pages | 35 |

Journal | Annals of Applied Probability |

Volume | 27 |

Issue number | 2 |

Early online date | 26 May 2017 |

DOIs | |

Publication status | E-pub ahead of print - 26 May 2017 |

Externally published | Yes |

## Keywords

- Branching random walk
- Random walk
- Reconstruction algorithm