domino tilings of the chessboard an introduction to sampling and counting dana randall schools of...
TRANSCRIPT
![Page 1: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/1.jpg)
Domino Tilings of the Chessboard
An Introduction to Sampling and Counting
Dana RandallSchools of
Computer Scienceand Mathematics
Georgia Tech
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Building short walls
How many ways are there to build a 2 x n wall with 1 x 2 bricks?
2
n
21
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Building short walls
21How many ways are there to
build a 2 x n wall with 1 x 2 bricks?
2
n
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n=1 :
n=2 :
n=3 :
n=0 :
n=4 :
Building short walls
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Building short wallsn=0 :
n=1 :
n=2 :
n=3 :
n=4 :
The number of walls equal:
fn = 1, 1, 2, 3, 5
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Building short wallsn=0 :
n=1 :
n=2 :
n=3 :
n=4 :
The number of walls equal:
fn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
?
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Building short wallsn=0 : n=1 : n=2 : n=3 :
n=4 :
The number of walls equals:
fn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
?
n
2
2
fn ={
![Page 8: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/8.jpg)
Building short wallsn=0 : n=1 : n=2 : n=3 :
n=4 :
The number of walls equals:
fn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
?
n
2
2
fn ={
![Page 9: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/9.jpg)
Building short wallsn=0 : n=1 : n=2 : n=3 :
n=4 :
The number of walls equals:
fn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
?
n
2
2
fn-1
fn-2
fn ={
![Page 10: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/10.jpg)
The Fibonacci NumbersThe number of walls equals: fn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
fn-1
fn-2 fn ={
fn = fn-1 + fn-2 , f0 = f1 = 1
fn = (φn + (1-φ)n) /√5 ,
where: (“golden ratio”)
φ = 1+√5 2
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Domino Tilings Given a region R on the infinite chessboard,
cover with non-overlapping 2 x 1 dominos.
Where is a tiling? Do any even exist?
How many tilings are there?
What does a typical tiling look like?
When do we stop our algorithms?
Why do we care?
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Where is a tiling? Do any exist?
n
n ★ Only if n is even!
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Where is a tiling? Do any exist?• The Area of R must be even
n
n
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Where is a tiling? Do any exist?
n
n ★ There must be an equal number of black and white squares.
• The Area of R must be even
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Where is a tiling? Do any exist?
Is this enough?
• The Area of R must be even• With an equal number of white and black squares
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Where is a tiling? Do any exist?
Is this enough?
?
. . .
There is an efficient algorithm to decide if R is tileable and to find one if it is. [Thurston]
• The Area of R must be even• With an equal number of white and black squares
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Domino Tilings
Where is a tiling? Do any even exist?
How many tilings are there?
What does a typical tiling look like?
When do we stop our algorithms?
Why do we care?
![Page 18: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/18.jpg)
How many tilings are there?
≈ φnShort 2 x n
walls
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How many tilings are there?
≈ φnShort 2 x n
walls
Aztec Diamonds
n
# = 1 2 8 64
n=0 n=1 n=2 n=3
= 2n(n+1)/2
[Elkies, Kuperberg, Larson, Propp]
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How many tilings are there?
≈ φnShort 2 x n
walls
Aztec Diamonds
n= 2n(n+1)/2
![Page 21: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/21.jpg)
How many tilings are there?
≈ φnShort 2 x n
walls
Aztec Diamonds
Square n x n walls
n
?
= 2n(n+1)/2
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How many tilings are there? Square n x n walls
2 (Area/4) < #<
# < 4 (Area)<
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How many tilings are there?
#≈ φnShort 2 x n
walls
Aztec Diamonds
Square n x n walls 2Area/4 < # < 4Area
#=2n(n+1)/2
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How many: An Algorithm
• Mark alternating vertical edges;
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How many: An Algorithm
• Mark alternating vertical edges;• Use marked tiles;• Markings must line up!
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How many: An Algorithm
t1
t2
t3
s1
s2
s3
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How many: An Algorithm
t1
t2
t3
s1
s2
s3
s1
s2
s3
t1
t2
t3
We want to count non-intersecting sets ofpaths from si to ti . [R.]
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How many: An Algorithm
We want to count non-intersecting sets ofpaths from si to ti .
Let aij be the number of paths from si to tj .
s1
s2
s3
t1
t2
t3
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How many: An Algorithm
s1
s2
s3
t1
t2
t3
Let aij be the number of paths from si to tj .
We want to count non-intersecting sets ofpaths from si to ti .
1 4 16
1 3 12 52 1 5 24
1 7 40
1 9 1 10
52 40 10
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How many: An Algorithm
s1
s2
s3
t1
t2
t3
We want to count non-intersecting sets ofpaths from si to ti .
1 8
1 7 40
1 5 25
0 3 13 62 1 5 24
1 6 30
52 40 1040 62 30
Let aij be the number of paths from si to tj .
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How many: An Algorithm
s1
s2
s3
t1
t2
t3
We want to count non-intersecting sets ofpaths from si to ti .
1 1 10
1 8
1 6 30
1 4 16
0 2 6 22
52 40 1053 62 3010 30 22
Let aij be the number of paths from si to tj .
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How many: An Algorithm
s1
s2
s3
t1
t2
t3
We want to count non-intersecting sets ofpaths from si to ti .
52 40 1053 62 3010 30 22
= 1728.
[Gessel, Viennot]
Let aij be the number of paths from si to tj .
Det
This is the number domino tilings!!
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Proof sketch for two paths:
a11 x a22 counts what we want + extra stuff.
a12 x a21 also counts the extra stuff.
Therefore (a11 x a22) - (a12 x a21) counts real tilings.
= #
(This is the 2 x 2 determinant!)
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Domino Tilings
Where is a tiling? Do any even exist?
How many tilings are there?
What does a typical tiling look like?
When do we stop our algorithms?
Why do we care?
![Page 35: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/35.jpg)
Why Mathematicians Care
Arctic Circle Theorem [Jockush, Propp, Shor]
The Aztec Diamond
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On the chessboard On the hexagonal lat. “Domino tilings” “Lozenge tilings” (little “cubes”)
What about tilings on lattices?
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Why Mathematicians Care
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Why Mathematicians Care
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Why do we care?
• Mathematics: Discover patterns
• Chemistry, Biology: Estimate probabilities
• Physics: Count and calculate other functions
to study a physical system
• Nanotechnology: Model growth processes
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Why Physicists Care
“Dimer models”: diatomic molecules adhering to the surface of a crystal.
The count (“partition function”) determines: specific heat, entropy, free energy, …
What does “nature” compute?
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Domino Tilings
Where is a tiling? Do any even exist?
How many tilings are there?
What does a typical tiling look like?
When do we stop our algorithms?
Why do we care?
![Page 42: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/42.jpg)
What does a typical tiling look like?
![Page 43: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/43.jpg)
What does a typical tiling look like?
“Mix” them up!
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Markov chain for Lozenge Tilings
Repeat:
Pick v in the lattice region;
Add / remove the ``cube’’ .
at v w.p. ½, if possible.
v v
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Markov chain for Lozenge Tilings
v
1. The state space is connected.
2. If we do this long enough, each tiling will be
equally likely.3. How long is “long enough” ?
v
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Domino Tilings
Where is a tiling? Do any even exist?
How many tilings are there?
What does a typical tiling look like?
When do we stop our algorithms?
Why do we care?
![Page 47: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/47.jpg)
When do we stop our algorithms?
v v
Thm: The lozenge Markov chain is “rapidly mixing.” [Luby, R., Sinclair]
2n, 3n n2, 10√n, …
(exponential)
n2, n log n,n10, …
(polynomial)
3. How long is “long enough” ?
![Page 48: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/48.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
![Page 49: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/49.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a 2 x 2 square;
![Page 50: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/50.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a 2 x 2 square;• Rotate, if possible;
![Page 51: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/51.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a 2 x 2 square;• Rotate, if possible;• Otherwise do nothing.
![Page 52: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/52.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a vtx and a color;
![Page 53: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/53.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a vtx and a color;• Recolor, if possible;
![Page 54: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/54.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a vtx and a color;• Recolor, if possible;• Otherwise do nothing.
![Page 55: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/55.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a vtx v and a bit b;
![Page 56: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/56.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a vtx v and a bit b;• If b=1, try to add v
![Page 57: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/57.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a vtx v and a bit b;• If b=1, try to add v;• If b=0, try to remove v;
![Page 58: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/58.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
• Pick a vtx v and a bit b;• If b=1, try to add v;• If b=0, try to remove v;• O.w. do nothing.
![Page 59: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/59.jpg)
What about other models?
Potts model Hardcore model Dimer model
Domino tilings k-colorings Independent sets
Thm: All of these chains are rapidly mixing.
![Page 60: Domino Tilings of the Chessboard An Introduction to Sampling and Counting Dana Randall Schools of Computer Science and Mathematics Georgia Tech](https://reader038.vdocument.in/reader038/viewer/2022110116/551aa07255034643688b63d8/html5/thumbnails/60.jpg)
HOWEVER . . . Three-colorings:The local chain is fast for 3-colorings in 2-d [LRS]
but slow for 3-colorings in sufficiently high dimension. [Galvin, Kahn, R, Sorkin], [Galvin, R]
Independent Sets
The local chain is fast for sparse Ind Sets in 2-d [Luby, Vigoda],…, [Weitz]
but slow for dense Ind Sets. [R.]
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Dense
Weighted Independent Sets
Sparse
Fast Phase Transition Slow
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Why?
n2/2 l
n2/2 l
(n2/2-n/2) l
“Even” “Odd”
#R/#B
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Summary Where
How
What
When
Why
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Summary Where
How
What
When
Why
Algebra
Algorithms
Combinatorics
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Summary Where
How
What
When
Why
Algebra
Algorithms
Combinatorics
Geometry
Probability
Physics
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Summary Where
How
What
When
Why
Algebra
Algorithms
Combinatorics
Geometry
Probability
Physics
Chemistry
Nanotechnology
Biology
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THANK YOU !
Math
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THANK YOU !
Math + CS
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THANK YOU !
Math + CS
+ Biology,
Chemistry,
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THANK YOU !
+ Physics,
Nanotechnology,
. . .
Math + CS
+ Biology,
Chemistry,
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THANK YOU !