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Circles in Aztec Diamonds. Aztec Diamonds in Groves. Circles in Groves?

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## Circles in Aztec Diamonds. Aztec Diamonds in Groves. Circles in Groves?

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**Circles in Aztec Diamonds. Aztec Diamonds in Groves. Circles**in Groves? Limiting Behavior of Combinatorial Models**Circles in Aztec Diamonds**• 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}.**Circles in Aztec Diamonds**• 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.**Circles in Aztec Diamonds**• 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.**Circles in Aztec Diamonds**• 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.**Circles in Aztec Diamonds**Shuffling:**Circles in Aztec Diamonds**Shuffling: 1. Slide dominoes**Circles in Aztec Diamonds**Shuffling: 1. Slide dominoes 2. Fill in randomly**Circles in Aztec Diamonds**Shuffling: 1. Slide dominoes**Circles in Aztec Diamonds**Shuffling: 1. Slide dominoes 2. Fill in randomly**Circles in Aztec Diamonds**Shuffling: 0. Delete bad blocks**Circles in Aztec Diamonds**Shuffling: 0. Delete bad blocks 1. Slide dominoes**Circles in Aztec Diamonds**Shuffling: 0. Delete bad blocks 1. Slide dominoes 2. Fill in randomly**Circles in Aztec Diamonds**Shuffling: 0. Delete bad blocks**Circles in Aztec Diamonds**Shuffling: 0. Delete bad blocks 1. Slide dominoes**Circles in Aztec Diamonds**Shuffling: 0. Delete bad blocks 1. Slide dominoes 2. Fill in randomly**Circles in Aztec Diamonds**• A domino is called North-going if it migrates north under shuffling, similarly for south, east, and west.**Circles in Aztec Diamonds**• 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.**Circles in Aztec Diamonds**• 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.**Circles in Aztec Diamonds**• We typically color the tiles red, yellow, blue, and green.**Circles in Aztec Diamonds**• A domino is called frozen if it can never be annihilated by further shuffling.**Circles in Aztec Diamonds**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.**Circles in Aztec Diamonds**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).**Circles in Aztec Diamonds**Further Statistics of Aztec diamonds: (Cohn, Elkies, and Propp) – Expectations within the temperate zone**Circles in Aztec Diamonds**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**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. f(1) = 2 f(2) = 8 f(3) = (2f(2)^2)/f(1) = 64 f(4) = (2f(3)^2)/f(2) = 1024**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.**Aztec Diamonds in Groves**> 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) = + + + + + + +**Aztec Diamonds in Groves**• 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?**Aztec Diamonds in Groves**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) = ??**Aztec Diamonds in Groves**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) = Groves**Aztec Diamonds in Groves**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**Aztec Diamonds in Groves**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 trivial initials**Aztec Diamonds in Groves**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**Aztec Diamonds in Groves**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. Kleber initial conditions (4,2,3) Random grove on KI(4,2,3) The grove**Aztec Diamonds in Groves**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. Aztec diamond initial conditions of order 4 Random grove on AD(4) The grove**Aztec Diamonds in Groves**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) Remember that the octahedron recurrence is a special case of the cube recurrence.**Aztec Diamonds in Groves**There is a correspondence between tilings of Aztec diamonds of order n and certain groves on Aztec initial conditions of order n.**Aztec Diamonds in Groves**> 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.**Aztec Diamonds in Groves**The standard initial conditions for a grove look like the compliment of an upside down Q*Bert board. Standard initial conditions of order 8**Aztec Diamonds in Groves**A grove on standard initial conditions**Aztec Diamonds in Groves**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.**Aztec Diamonds in Groves**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).**Aztec Diamonds in Groves**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).**Aztec Diamonds in Groves**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).**Aztec Diamonds in Groves**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).**Aztec Diamonds in Groves**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).**Aztec Diamonds in Groves**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).**Aztec Diamonds in Groves**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).**Aztec Diamonds in Groves**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).