THE PHASE TRANSITION FOR DYADIC TILINGS
Artikel i vetenskaplig tidskrift, 2014

A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independent of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n -> infinity, as conjectured by Joel Spencer in 1999. In particular, we prove that if p = 7/8, such a tiling exists with probability at least 1 - (3/4)(n). The proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling.

UNIT SQUARE

generating

function

ZERO-ONE LAW

phase transition

tiling

percolation

Dyadic rectangle

Författare

O. Angel

A. E. Holroyd

G. Kozma

Johan Wästlund

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

P. Winkler

Transactions of the American Mathematical Society

0002-9947 (ISSN) 1088-6850 (eISSN)

Vol. 366 2 1029-1046

Ämneskategorier

Matematik

DOI

10.1090/S0002-9947-2013-05923-5