arXiv:2403.17147v2 Announce Type: replace Abstract: Swarm robotics relies on robots with minimal sensing and computation that achieve self-organization at the level of the collective to address a variety of tasks. Yet swarm robotics still struggles with global tasks that require understanding the broader spatial context in which robots operate, such as identifying the shape of the arena in which they are embedded. Here, we introduce spectral swarm robotics, a framework in which robots recover global spectral descriptors of the swarm's geometric structure through strictly local interactions. To do so, robots emulate diffusion to locally estimate Laplacian spectral invariants, quantities normally obtained only through centralized computation. These spectral values serve as compact descriptors of the environment. In the present proof of concept, robots estimated the second Laplacian eigenvalue and used it to classify arenas from a predefined set of shapes with distinct spectral signatures. We show that estimating a single eigenvalue is sufficient for one-shot classification among selected confined environments with distinct spectral signatures. We uncovered a scaling law linking swarm size (a global parameter) and interaction radius (a local parameter) that ensures robust estimation. We validated the approach experimentally on two distinct hardware platforms, Kilobots and Pogobots, achieving reliable classification of arena shapes under realistic noise and sparse-connectivity conditions. Last, we discuss how locally recovering Laplacian spectral invariants opens avenues for swarm robotics in settings where producing efficient collective behaviors strongly depends on the global structure of the swarm.