Filled-in peeling explorations exhaust every critical Boltzmann planar map
Year: 2026
Type: preprint
Abstract: Let $\mathbf q$ be an admissible critical weight sequence for bipartite Boltzmann planar maps and consider the infinite $\mathbf q$-Boltzmann map with a boundary. Curien proved that a filled-in peeling exploration driven by any algorithm eventually discovers the whole map when $\mathbf q$ is of type $a\in[3/2,5/2]$, and asked whether this holds for every critical weight sequence. We show that it does: for every admissible critical $\mathbf q$, every finite boundary length and every peeling algorithm that never stops, the filled-in exploration exhausts the map almost surely. The type assumption entered Curien's proof through the polynomial tail of the negative jumps of the perimeter process and through its scaling limit. We use instead the exact formula for the negative jumps in terms of the positive ones, which gives the rate at which a fixed boundary edge is swallowed up to constants, and a martingale argument for $\log h^\uparrow$ of the perimeter which shows that these rates have a divergent sum along the exploration.
Author: Shmalo, Yitzchak
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225648669