Reconstruction of a multidimensional scenery with a branching random walk

TL;DR

This paper proves that a multidimensional scenery can be almost surely reconstructed from the color records of particles in a branching random walk, requiring at least 2d+1 colors with i.i.d. uniform distribution.

Contribution

It improves previous methods by enabling scenery reconstruction from all particles' records without window constraints, using fewer colors in higher dimensions.

Findings

Reconstruction is almost sure with at least 2d+1 colors.

No window constraints needed for particles' observations.

Applicable to multidimensional settings with branching random walks.

Abstract

In this paper 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 is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this 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 [22] where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of a branching random walk, but millions of colors are necessary.