Aztec diamond

Circles in Aztec Diamonds. Aztec

Diamonds in Groves. Circles in

Groves?

Limiting Behavior of Combinatorial

Models

Circles in Aztec Diamonds

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An Aztec diamond of order n is defined as the union of those lattice squares whose interiors lie inside the region {(x,y) : |x+y|<= n+1}.


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An Aztec diamond of order n is defined as the union of those lattice squares whose interiors lie inside the region {(x,y) : |x+y|<= n+1}.

• A domino tiling of an Aztec diamond is a way to cover the region with 2 by 1 rectangles (dominoes) so that none of the dominoes overlap, and none of the dominoes extend outside the boundary of the region .


•

An Aztec diamond of order n is defined as the union of those lattice squares whose interiors lie inside the region {(x,y) :|x+y|<= n+1}.

• A domino tiling of an Aztec diamond is a way to cover the region with 2 by 1 rectangles (dominoes) so that none of the dominoes overlap, and none of the dominoes extend outside the boundary of the region .


•

The number of domino tilings of an Aztec diamond is 2^(n(n+1)/2). Any of these tilings can be generated uniformly at random by a procedure called domino shuffling described in a paper of Elkies,

Kuperberg, Larsen, and Propp.


Shuffling:


Shuffling:

1. Slide dominoes


Shuffling:


2. Fill in randomly


Shuffling:



Shuffling:




Shuffling:

0. Delete bad blocks


Shuffling:




Shuffling:





Shuffling:



Shuffling:




Shuffling:





•

A domino is called North-going if it migrates north under shuffling, similarly for south, east, and west.


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Equivalently, they may be defined by the checkerboard coloring of the plane. A north going domino has a black square on the left and a white square on the right. A south going domino has a white square on the left and a black square on the right. Similarly for east and west.


•

Equivalently, they may be defined by the checkerboard coloring of the plane. A north going domino has a black square on the left and a white square on the right. A south going domino has a white square on the left and a black square on the right. Similarly for east and west.


•

We typically color the tiles red, yellow, blue, and green.


•

A domino is called frozen if it can never be annihilated by further shuffling.


The Arctic Circle Theorem (Jockusch, Propp, Shor): As n (the order of the Aztec diamond) goes to infinity, the expected shape of the boundary between the frozen region and temperate zone is a circle.


The Arctic Circle Theorem (Jockusch, Propp, Shor): Examine the growth model on Young diagrams where each growth position has independent probability ½ of adding a box.

This has limiting shape of a quarter-circle (suitably scaled).


Further Statistics of Aztec diamonds:

(Cohn, Elkies, and Propp) – Expectations within the temperate zone


Further Statistics of Aztec diamonds:

(Johansson) – Fluctuations about the circle. The method of non-intersecting paths, or Brownian motion model yields a link to random matrices and Tracy-Widom distribution. Johansson ultimately equated this model to the random growth model for the Young diagram.

Aztec Diamonds in Groves f(1) = 2 f(2) = 8 f(3) = (2f(2)^2)/f(1) = 64 f(4) = (2f(3)^2)/f(2) = 1024

Aztec diamonds can be enumerated by the octahedron recurrence. Let f(n) = the number of Aztec diamonds of order n. Then f(n)f(n-2) = 2f(n-1)^2.

Aztec Diamonds in Groves

Polynomial version of octahedron recurrence: f(i,j,k)f(i,j,k-2) = f(i-1,j,k-1)f(i+1,j,k-1)+f(i,j-1,k-1)f(i,j+1,k-

1) where f(i,j,k) = x(i,j,k) if k=0,-1. Otherwise f(i,j,n) encodes all the tilings of an Aztec diamond of order n. The rational functions that are generated are not just rational in the x(i,j,k), they are Laurent polynomials.

=


> f(0,0,2); x(0, 0, 0) x(2, 0, 0) x(-2, 0, 0) x(2, 0, 0) x(-1, 1, 0) x(-1, -1, 0)

--------------------------------+ ----------------------------------x(1, 0, -1) x(-1, 0, -1) x(1, 0, -1) x(-1, 0, -1) x(1, 1, 0) x(1, -1, 0) x(-2, 0, 0) x(1, 1, 0) x(1, -1, 0) x(-1, 1, 0) x(-1, -1, 0)

+ ---------------------------------+ ----------------------------------------------x(1, 0, -1) x(-1, 0, -1) x(0, 0, 0) x(1, 0, -1) x(-1, 0, -1) x(1, 1, 0) x(-1, 1, 0) x(1, -1, 0) x(-1, -1, 0) x(1, 1, 0) x(-1, 1, 0) x(0, -2, 0)

+ ----------------------------------------------+ ---------------------------------x(0, 0, 0) x(0, 1, -1) x(0, -1, -1) x(0, 1, -1) x(0, -1, -1) x(0, 2, 0) x(1, -1, 0) x(-1, -1, 0) x(0, 0, 0) x(0, 2, 0) x(0, -2, 0)

+ ----------------------------------+ --------------------------------x(0, 1, -1) x(0, -1, -1) x(0, 1, -1) x(0, -1, -1)

+ + +

+ + + +

Octahedron Recurrence : f(i,j,k)f(i,j,k-2) = f(i-1,j,k-1)f(i+1,j,k-1)+f(i,j-1,k-1)f(i,j+1,k-1)


•

Cube Recurrence : f(i,j,k)f(i-1,j-1,k-1) = f(i-1,j,k)f(i,j-1,k-1)+f(i,j-1,k)f(i-1,j,k-1)+f(i-1,j-1,k)f(i,j,k-1)

•

The cube recurrence is a generalization of the octahedron recurrence. As shown by Fomin and

Zelevinsky using cluster algebra methods, it also produces Laurent polynomials. But what do the polynomials encode?


= ??



= Groves



Trivial initial conditions

A grove is a new combinatorial object, due to Carroll and Speyer, given by the cube recurrence. Groves can be viewed as forests that live on a very special planar region or more intuitively, on a three dimensional surface with lattice point corners (- a big pile of cubes). What the surface looks like is specified by some initial conditions.


Trivial initial conditions 

Unique grove on trivial initials

A grove is a new combinatorial object given by the cube recurrence. Groves can be viewed as forests that live on a very special planar region, given by some specified initial conditions. However, note that the forests have severely restricted behavior.


Trivial initial conditions 

Unique grove on trvial initials



The grove



Kleber initial conditions (4,2,3)

Random grove on KI(4,2,3)

The grove



Aztec diamond initial conditions of order 4

Random grove on AD(4)

The grove



Octahedron Recurrence : f(i,j,k)f(i,j,k-2) = f(i-1,j,k-1)f(i+1,j,k-1)+f(i,j-1,k-1)f(i,j+1,k-1)


Remember that the octahedron recurrence is a special case of the cube recurrence.


There is a correspondence between tilings of Aztec diamonds of order n and certain groves on Aztec initial conditions of order n .


> f(0,0,2); x(0,0,0) x(2,0,0) x(-2,0,0) x(2,0,0) x(-1,1,0) x(-1,-1,0)

--------------------------+ ----------------------------x(1,0,-1) x(-1,0,-1) x(1,0,-1) x(-1,0,-1) x(1,1,0) x(1,-1,0) x(-2,0,0) x(1,1,0) x(1,-1,0) x(-1,1,0) x(-1,-1,0)

+ ---------------------------- + --------------------------------------x(1,0,-1) x(-1,0,-1) x(0,0,0) x(1,0,-1) x(-1,0,-1) x(1,1,0) x(-1,1,0) x(1,-1,0) x(-1,-1,0) x(1,1,0) x(-1,1,0) x(0,-2,0)

+ --------------------------------------- + ---------------------------x(0,0,0) x(0,1,-1) x(0,-1,-1) x(0,1,-1) x(0,-1,-1) x(0,2,0) x(1,-1,0) x(-1,-1,0) x(0,0,0) x(0,2,0) x(0,-2,0)

+ -----------------------------+ ---------------------------x(0,1,-1) x(0,-1,-1) x(0,1,-1) x(0,-1,-1)

Because the octahedron recurrence is a special case of the cube recurrence, there is actually an injection from the set of tilings of Aztec diamonds into the set of groves on Aztec initial conditions.


Standard initial conditions of order 8

The standard initial conditions for a grove look like the compliment of an upside down Q*Bert board.


A grove on standard initial conditions


Groves on standard initial conditions are better represented in a triangular lattice. Notice that we may ignore the short edges. This representation is called a simplified grove .


Like domino shuffling, there is a way to generate groves on standard initial conditions uniformly at random, called grove shuffling (or cube-popping in general).














Like domino shuffling, there is a way to generate groves on standard initial conditions uniformly at random, called grove shuffling (or cube-popping in general ).





Circles in Groves?

Four representations of a randomly generated grove of order 20.

With grove shuffling we can generate large random groves fairly quickly.

Circles in Groves?

Representation of an order 200 grove.

With grove shuffling we can generate large random groves fairly quickly.

Circles in Groves?

The most promising method of attacking the grove problem seems to be by projecting down one of the colors in a corner,

Circles in Groves?

The most promising method of attacking the grove problem seems to be by projecting down one of the colors in a corner, isolating the frozen region, and making the situation look like a Young diagram model with growth probability equal ½ .

Circles in Groves?

Projection of frozen region of a random grove of order 20 above, Young diagram growth model after 20 growth stages below (p= ½ ).

The most promising method of attacking the grove problem seems to be by projecting the frozen region down to a Young diagram model with growth probability equal ½ .

Circles in Groves?

Projection of frozen region of a random grove of order 100 above, Young diagram growth model after 100 growth stages below (p= ½ ).

The most promising method of attacking the grove problem seems to be by projecting the frozen region down to a Young diagram model with growth probability equal ½ .

Circles in Groves?

Looking at a given grove whose projection is above, the observed probabilities of growth are sometimes zero and sometimes 2/3, but never ½! However, I think that if we can take the weighted probabilities over all groves with this projection, then we will find the total probability is equal to the infinite sum of (1/3)^k, k=1 to infinity. What is this sum? ½.

Circles in Groves?

Other peculiarities…

Non-intersecting paths for groves…

Circles in Groves?

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