How many T-tessellations on $k$ lines? Existence of associated Gibbs measures on bounded convex domains

TL;DR

This paper establishes bounds on the number of T-tessellations on a fixed set of lines, introduces algorithms for their efficient encoding, and proves the existence of a corresponding random Gibbsian measure.

Contribution

It provides the first combinatorial bounds and algorithms for T-tessellations on fixed lines, and demonstrates the existence of associated Gibbs measures on convex domains.

Findings

Bound on the number of T-tessellations is sharp.

Algorithms efficiently encode and retrieve tessellations.

Existence of a Gibbsian measure for T-tessellations is proven.

Abstract

The paper bounds the number of tessellations with T-shaped vertices on a fixed set of $k$ lines: tessellations are efficiently encoded, and algorithms retrieve them, proving injectivity. This yields existence of a completely random T-tessellation, as defined by Ki\^en Ki\^eu et al., and of its Gibbsian modifications. The combinatorial bound is sharp, but likely pessimistic in typical cases.