The beauty of mathematics · The beauty of mathematics Pope John Paul II College of Education...

The beauty of mathematics

Pope John Paul II College of Education

Pondicherry, 23 Feb 2012 Xavier ViennotCNRS, Bordeaux, France

part V: The arctic circle theorem

algorithm for a random tiling

probabilities

§ 12 the arctic circle theorem

4 colors for the dimers

the arctic circle theorem

algorithm for a random tilingwith bijections

http://images.math.cnrs.fr/Pavages-aleatoires-par-touillage.html

from "Images des mathématiques"

Elise Janvresse and Thierry de la Rue, 17 Feb 2009

w = w1 w1 = 0w1 = 1

21

w = w1w2 wi = 0wi = 1

2(1+2)

23 = 8

w = w1w2w3 wi = 0wi = 1

2(1+2+3)

26 = 64

w = w1w2w3w4 wi = 0wi = 1

2(1+2+3+4)

210 = 1024

w = w1w2w3w4w5wi = 0wi = 1

2(1+2+3+4+5)

215 = 32.768

w = w1w2w3w4w5w6

wi = 0wi = 1

2(1+2+3+4+5+6)

221 = 2.097.152

random tiling on a rectangle

Effets de bordsÀ gauche : un pavage typique du diamant aztèque d’ordre 100, sur lequel on observe le phénomène du cercle arctique. À droite : un

pavage typique du carré de côté 60.

Thank you !