El círculo artico
Transcript of El círculo artico
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Domino tilingsMathematical tools
A few ideas of the proof
The arctic circle
Vincent Beffara
ISSMYS August 28, 2012
Vincent Beffara The arctic circle
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
What is adomino?
Vincent Beffara The arctic circle
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
What is adomino?
A domino is a rectangle of dimensions 2 1, oriented eithervertically or horizontally.
Vincent Beffara The arctic circle
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
What is a dominotiling?
Vincent Beffara The arctic circle
D i ili D fi i i
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
What is a dominotiling?
Aplanar domainis a bounded region in the plane formed of
adjacent 1 1 squares. Usually it will have no hole.
Vincent Beffara The arctic circle
D i tili D fi iti
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
What is a dominotiling?
Aplanar domainis a bounded region in the plane formed of
adjacent 1 1 squares. Usually it will have no hole.
Adomino tilingof a domain is a way to cover it with dominoswithout overlaps.
Vincent Beffara The arctic circle
Domino tilings Definitions
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
What is arandomdomino tiling?
For a given domain, there are only finitely many ways (possibly
none at all) to cover is with dominos. Assuming that there arecoverings, we can pick one at random, giving the same probabilityto all of them. This is what we will always mean by a randomdomino tiling.
Vincent Beffara The arctic circle
Domino tilings Definitions
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
How does a random domino tiling look like?
Vincent Beffara The arctic circle
Domino tilings Definitions
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
How does a random domino tiling look like?
Vincent Beffara The arctic circle
Domino tilings Definitions
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Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
How to generate a random domino tiling?
There are efficient ways, but they are tricky to describe. Here is asimple one:
Vincent Beffara The arctic circle
Domino tilings Definitions
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gMathematical tools
A few ideas of the proofRandom tilingsOther kinds of tilings
How to generate a random domino tiling?
There are efficient ways, but they are tricky to describe. Here is asimple one:
1 Start from your favorite tiling;
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Domino tilings Definitions
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gMathematical tools
A few ideas of the proofRandom tilingsOther kinds of tilings
How to generate a random domino tiling?
There are efficient ways, but they are tricky to describe. Here is asimple one:
1 Start from your favorite tiling;
2 Look for a pair of adjacent, parallel dominos forming a 2 2square; pick such a pair at random;
Vincent Beffara The arctic circle
Domino tilings Definitions
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gMathematical tools
A few ideas of the proofRandom tilingsOther kinds of tilings
How to generate a random domino tiling?
There are efficient ways, but they are tricky to describe. Here is asimple one:
1 Start from your favorite tiling;
2 Look for a pair of adjacent, parallel dominos forming a 2 2square; pick such a pair at random;
3 Flipthe pair to the other possible configuration within the2 2 square;
Vincent Beffara The arctic circle
Domino tilings Definitions
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Mathematical toolsA few ideas of the proof
Random tilingsOther kinds of tilings
How to generate a random domino tiling?
There are efficient ways, but they are tricky to describe. Here is asimple one:
1 Start from your favorite tiling;
2 Look for a pair of adjacent, parallel dominos forming a 2 2square; pick such a pair at random;
3 Flipthe pair to the other possible configuration within the2 2 square;
4 Go back to step2.
Vincent Beffara The arctic circle
Domino tilingsM h i l l
DefinitionsR d ili
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Mathematical toolsA few ideas of the proof
Random tilingsOther kinds of tilings
How to generate a random domino tiling?
There are efficient ways, but they are tricky to describe. Here is asimple one:
1 Start from your favorite tiling;
2 Look for a pair of adjacent, parallel dominos forming a 2 2square; pick such a pair at random;
3 Flipthe pair to the other possible configuration within the2 2 square;
4 Go back to step2.
5 Except, stop after a long enough time.
Vincent Beffara The arctic circle
Domino tilingsM th ti l t ls
DefinitionsR d tili s
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Mathematical toolsA few ideas of the proof
Random tilingsOther kinds of tilings
Tilings of the Aztec diamond
TheAztec diamondof size n is this domain:
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
DefinitionsRandom tilings
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Mathematical toolsA few ideas of the proof
Random tilingsOther kinds of tilings
Tilings of the Aztec diamond
TheAztec diamondof size n is this domain:
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
DefinitionsRandom tilings
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Mathematical toolsA few ideas of the proof
Random tilingsOther kinds of tilings
video
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
DefinitionsRandom tilings
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Mathematical toolsA few ideas of the proof
Random tilingsOther kinds of tilings
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
DefinitionsRandom tilings
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Mathematical toolsA few ideas of the proof
Random tilingsOther kinds of tilings
Thats a bit strange. What is happening?
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
DefinitionsRandom tilings
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A few ideas of the proofg
Other kinds of tilings
Thats a bit strange. What is happening?
We are selecting a tiling at random, and we see a deterministicshapeappearing.
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
DefinitionsRandom tilings
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A few ideas of the proof Other kinds of tilings
Thats a bit strange. What is happening?
We are selecting a tiling at random, and we see a deterministicshapeappearing. What that means is, most domino tilings of theAztec diamond are frozen outside the disk.
We need to understand a few things now:
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
DefinitionsRandom tilings
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A few ideas of the proof Other kinds of tilings
Thats a bit strange. What is happening?
We are selecting a tiling at random, and we see a deterministicshapeappearing. What that means is, most domino tilings of theAztec diamond are frozen outside the disk.
We need to understand a few things now:Why do the dominos align in the corners?
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A f id f h f
DefinitionsRandom tilingsO h ki d f ili
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A few ideas of the proof Other kinds of tilings
Thats a bit strange. What is happening?
We are selecting a tiling at random, and we see a deterministicshapeappearing. What that means is, most domino tilings of theAztec diamond are frozen outside the disk.
We need to understand a few things now:
Why do the dominos align in the corners?
What is the shape, is it really a circle?
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A f id f th f
DefinitionsRandom tilingsOth ki d f tili
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A few ideas of the proof Other kinds of tilings
Thats a bit strange. What is happening?
We are selecting a tiling at random, and we see a deterministicshapeappearing. What that means is, most domino tilings of theAztec diamond are frozen outside the disk.
We need to understand a few things now:
Why do the dominos align in the corners?
What is the shape, is it really a circle?
How come in the first square there is no frozen region?
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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A few ideas of the proof Other kinds of tilings
Thats a bit strange. What is happening?
We are selecting a tiling at random, and we see a deterministicshapeappearing. What that means is, most domino tilings of theAztec diamond are frozen outside the disk.
We need to understand a few things now:
Why do the dominos align in the corners?
What is the shape, is it really a circle?
How come in the first square there is no frozen region?
How come in the fake Aztec diamond everything is frozen?
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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A few ideas of the proof Other kinds of tilings
Thats a bit strange. What is happening?
We are selecting a tiling at random, and we see a deterministicshapeappearing. What that means is, most domino tilings of theAztec diamond are frozen outside the disk.
We need to understand a few things now:
Why do the dominos align in the corners?
What is the shape, is it really a circle?
How come in the first square there is no frozen region?
How come in the fake Aztec diamond everything is frozen?
The right answer to the last question holds the key to all theothers.
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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A few ideas of the proof Other kinds of tilings
One can tile a hexagon with lozenges
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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A few ideas of the proof Other kinds of tilings
One can tile a hexagon with lozenges
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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p g
One can even tile a more complicated shape
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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p g
One can even tile a more complicated shape
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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To sum up:
In many cases we see a surprising behavior: a deterministic shapeoccurs inside a random object. We would like to understand it:
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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To sum up:
In many cases we see a surprising behavior: a deterministic shapeoccurs inside a random object. We would like to understand it:
Why is there a deterministic shape appearing?
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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To sum up:
In many cases we see a surprising behavior: a deterministic shapeoccurs inside a random object. We would like to understand it:
Why is there a deterministic shape appearing?What is the shape? How to determine it?
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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To sum up:
In many cases we see a surprising behavior: a deterministic shapeoccurs inside a random object. We would like to understand it:
Why is there a deterministic shape appearing?What is the shape? How to determine it?
What kind of mathematical tools can we use?
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
DefinitionsRandom tilingsOther kinds of tilings
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To sum up:
In many cases we see a surprising behavior: a deterministic shapeoccurs inside a random object. We would like to understand it:
Why is there a deterministic shape appearing?What is the shape? How to determine it?
What kind of mathematical tools can we use?
The main idea is that, since we select one out of many tilings, themain step will involvecounting tilingshaving a certain behavior.
Vincent Beffara The arctic circle
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Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
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Easy test: Parity of the number of squares
We will need to count all possible tilings of a given shape if wewant to understand why a circle appears. The first step is to
determine whether a domain has a tiling at all ...
Very easy remark: a domino hastwosquares, so a domain with anoddnumber of squares cannot be covered by dominos.
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
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Easy test: Parity of the number of squares (too easy)
We will need to count all possible tilings of a given shape if wewant to understand why a circle appears. The first step is to
determine whether a domain has a tiling at all ...
Very easy remark: a domino hastwosquares, so a domain with anoddnumber of squares cannot be covered by dominos.
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
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Easy test 2: Bipartition
Here is a nice trick: draw the domain on a (infinite) chess board.It has white and black squares, and every domino has one of each.This gives us a better criterion:
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
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Easy test 2: Bipartition
Here is a nice trick: draw the domain on a (infinite) chess board.It has white and black squares, and every domino has one of each.This gives us a better criterion:
If a domain has a different number of white and black squares, itcannot be covered by dominos.
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
E B
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Easy test 2: Bipartition
Here is a nice trick: draw the domain on a (infinite) chess board.It has white and black squares, and every domino has one of each.This gives us a better criterion:
If a domain has a different number of white and black squares, itcannot be covered by dominos.
We will keep this trick from now on: the domain is assumed to be
drawn on a chess board. This explains the difference in colorsbetween the dominos in the previous picture:
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Th ll 4 f d i
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There are really 4 types of dominos
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
R hl h ili h ?
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Roughly, how many tilings are there anyway?
Take the easy case of a 2n 2n square S. We can give two bounds
on the number of tilings N(S) it admits:
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
R hl h tili th ?
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Roughly, how many tilings are there anyway?
Take the easy case of a 2n 2n square S. We can give two bounds
on the number of tilings N(S) it admits:
Each little square is in a domino, and there are only 4 kinds ofdominos; giving the kind of all these dominos specifies thetiling.
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
R hl h tili th ?
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Roughly, how many tilings are there anyway?
Take the easy case of a 2n 2n square S. We can give two bounds
on the number of tilings N(S) it admits:
Each little square is in a domino, and there are only 4 kinds ofdominos; giving the kind of all these dominos specifies thetiling. (It says nothing about overlaps, so we get a number
which is much bigger than the real one!)
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Roughly how many tilings are there anyway?
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Roughly, how many tilings are there anyway?
Take the easy case of a 2n 2n square S. We can give two bounds
on the number of tilings N(S) it admits:
Each little square is in a domino, and there are only 4 kinds ofdominos; giving the kind of all these dominos specifies thetiling. (It says nothing about overlaps, so we get a number
which is much bigger than the real one!) Hence:
N(S) 44n2.
Vincent Beffara The arctic circle
Domino tilingsMathematical toolsA few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Roughly how many tilings are there anyway?
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Roughly, how many tilings are there anyway?
Take the easy case of a 2n 2n square S. We can give two bounds
on the number of tilings N(S) it admits:
Each little square is in a domino, and there are only 4 kinds ofdominos; giving the kind of all these dominos specifies thetiling. (It says nothing about overlaps, so we get a number
which is much bigger than the real one!) Hence:
N(S) 44n2.
One can look for tilings obtained by gluing 2 2 squares
together, n2
of them. Each small square can be covered intwo ways.
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Roughly how many tilings are there anyway?
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Roughly, how many tilings are there anyway?
Take the easy case of a 2n 2n square S. We can give two bounds
on the number of tilings N(S) it admits:
Each little square is in a domino, and there are only 4 kinds ofdominos; giving the kind of all these dominos specifies thetiling. (It says nothing about overlaps, so we get a number
which is much bigger than the real one!) Hence:
N(S) 44n2.
One can look for tilings obtained by gluing 2 2 squares
together, n2
of them. Each small square can be covered intwo ways. (Of course not every tiling is of that kind, so thenumber we get is much too small!)
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Roughly how many tilings are there anyway?
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Roughly, how many tilings are there anyway?
Take the easy case of a 2n 2n square S. We can give two bounds
on the number of tilings N(S) it admits:
Each little square is in a domino, and there are only 4 kinds ofdominos; giving the kind of all these dominos specifies thetiling. (It says nothing about overlaps, so we get a number
which is much bigger than the real one!) Hence:
N(S) 44n2.
One can look for tilings obtained by gluing 2 2 squares
together, n2
of them. Each small square can be covered intwo ways. (Of course not every tiling is of that kind, so thenumber we get is much too small!) Hence:
N(S) 2n2.
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Roughly how many tilings are there anyway?
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Roughly, how many tilings are there anyway?
2n2 N(S) 44n2 .
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Roughly how many tilings are there anyway?
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Roughly, how many tilings are there anyway?
2n2
N(S) 44n2
.
Hence:
N(S) cn2.
Thats only partly a joke, it isthe right answer.
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
Roughly how many tilings are there anyway?
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Roughly, how many tilings are there anyway?
2n2
N(S) 44n2
.
Hence:
N(S) cn2.
Thats only partly a joke, it isthe right answer. In fact, c is knownexplicitly:
c= exp
4
k=0
(1)k
(2k+ 1)2
5.354 . . .
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
The height function
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The height function
A domino tiling is a complicated mathematical object. We wouldlike to encode it into a simpler one...
The height function is defined by a simple local rule: around ablack square, turning counterclockwise, it increases by 1 alongevery edge,except when crossing a domino(in which case itdecreases by 3).
Of course one needs to check that this is well-defined (that there issuch a function).
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
The height function along the boundary
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The height function along the boundary
One important remark: the height function on the boundary of agiven domaindoes not depend on the tiling. On our 3 favoritedomains, different things occur:
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
The height function along the boundary
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The height function along the boundary
One important remark: the height function on the boundary of agiven domaindoes not depend on the tiling. On our 3 favoritedomains, different things occur:
On the square, the boundary value is almost flat;
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
The height function along the boundary
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g g y
One important remark: the height function on the boundary of agiven domaindoes not depend on the tiling. On our 3 favoritedomains, different things occur:
On the square, the boundary value is almost flat;
On the Aztec diamond, it looks like a jigsaw;
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Counting possible tilingsThe height functionStatement of the arctic circle theorem
The height function along the boundary
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g g y
One important remark: the height function on the boundary of agiven domaindoes not depend on the tiling. On our 3 favoritedomains, different things occur:
On the square, the boundary value is almost flat;
On the Aztec diamond, it looks like a jigsaw;
On the fake Aztec diamond, it looks like a wedge.
Because a height function cannot oscillate much, if the boundarycondition is very wild, then it has little choice inside the domain.
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger;
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger; or equivalently, draw a finer and finer grid on a fixeddomain.
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger; or equivalently, draw a finer and finer grid on a fixeddomain. Then look at the height function generated by a randomtiling of each one.
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger; or equivalently, draw a finer and finer grid on a fixeddomain. Then look at the height function generated by a randomtiling of each one. (And normalize it correctly.)
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger; or equivalently, draw a finer and finer grid on a fixeddomain. Then look at the height function generated by a randomtiling of each one. (And normalize it correctly.)
Theorem
The height function approaches a deterministic, well characterizedshape
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger; or equivalently, draw a finer and finer grid on a fixeddomain. Then look at the height function generated by a randomtiling of each one. (And normalize it correctly.)
Theorem
The height function approaches a deterministic, well characterizedshape (which depends on the domain shape)
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger; or equivalently, draw a finer and finer grid on a fixeddomain. Then look at the height function generated by a randomtiling of each one. (And normalize it correctly.)
Theorem
The height function approaches a deterministic, well characterizedshape (which depends on the domain shape) (and on the details of the boundary).
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Statement of the main theorem
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Take a sequence of domains of the same shape, but larger andlarger; or equivalently, draw a finer and finer grid on a fixeddomain. Then look at the height function generated by a randomtiling of each one. (And normalize it correctly.)
Theorem
The height function approaches a deterministic, well characterizedshape (which depends on the domain shape) (and on the details of the boundary).
Where is the arctic circle?
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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Here is what the theorem says in our favorite domains:
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constant
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constantso the random tiling is uniform;
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constantso the random tiling is uniform;
On the fake arctic circle, the height function is asymptoticallyaffine
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constantso the random tiling is uniform;
On the fake arctic circle, the height function is asymptoticallyaffine (we know that the tiling is completely aligned already);
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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72/104
Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constantso the random tiling is uniform;
On the fake arctic circle, the height function is asymptoticallyaffine (we know that the tiling is completely aligned already);
On the arctic circle, the shape of the height function becomesinteresting:
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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73/104
Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constantso the random tiling is uniform;
On the fake arctic circle, the height function is asymptoticallyaffine (we know that the tiling is completely aligned already);
On the arctic circle, the shape of the height function becomesinteresting:
it is slanted in the corners;
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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74/104
Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constantso the random tiling is uniform;
On the fake arctic circle, the height function is asymptoticallyaffine (we know that the tiling is completely aligned already);
On the arctic circle, the shape of the height function becomesinteresting:
it is slanted in the corners;
and it is flat in the middle
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
Localizing the arctic circle
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Here is what the theorem says in our favorite domains:
On the square, the height function is asymptotically constantso the random tiling is uniform;
On the fake arctic circle, the height function is asymptoticallyaffine (we know that the tiling is completely aligned already);
On the arctic circle, the shape of the height function becomesinteresting:
it is slanted in the corners;
and it is flat in the middleso the random tiling is interesting as well.
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting possible tilings
The height functionStatement of the arctic circle theorem
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Vincent Beffara The arctic circle
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Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings of a given slope
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We saw before the approximate number of tilings in a box of size n
(with a straight boundary, which makes the height functionroughly constant along the boundary).
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings of a given slope
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79/104
We saw before the approximate number of tilings in a box of size n
(with a straight boundary, which makes the height functionroughly constant along the boundary).
Now, tune the boundary to make the height function have a slope:along the boundary,
h(x, y) ax+by.
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings of a given slope
W b f h i b f ili i b f i
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80/104
We saw before the approximate number of tilings in a box of size n
(with a straight boundary, which makes the height functionroughly constant along the boundary).
Now, tune the boundary to make the height function have a slope:along the boundary,
h(x, y) ax+by.
It is still the case that the number of tilings inside is exponential inthe surface of the square, like before.
Vincent Beffara The arctic circle
Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings of a given slope
W b f h i b f ili i b f i
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81/104
We saw before the approximate number of tilings in a box of size n
(with a straight boundary, which makes the height functionroughly constant along the boundary).
Now, tune the boundary to make the height function have a slope:along the boundary,
h(x, y) ax+by.
It is still the case that the number of tilings inside is exponential inthe surface of the square, like before. This can be shown using
sub-additivity.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings of a given slope
W b f th i t b f tili i b f i
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We saw before the approximate number of tilings in a box of size n
(with a straight boundary, which makes the height functionroughly constant along the boundary).
Now, tune the boundary to make the height function have a slope:along the boundary,
h(x, y) ax+by.
It is still the case that the number of tilings inside is exponential inthe surface of the square, like before. This can be shown using
sub-additivity. But the constant is different!
N c(a, b)(2n)2.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The value of c(a, b)
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Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The value of c(a, b)
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Thats where things get complicated...
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The value of c(a, b)
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Thats where things get complicated... but there is an exact
formula for it. Actually this is a very interesting story starting here.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The value of c(a, b)
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Thats where things get complicated... but there is an exact
formula for it. Actually this is a very interesting story starting here.
For completeness:
c(a, b) = expL(p1) +L(p2) +L(p3) +L(p4)
where
L(x) = x
0 log2sin tdt is the Lobachevsky function;
2(p1 p2) =a;
2(p4 p3) =b;p1+p2+p3+p4 = 1;
sin(p1) sin(p2) = sin(p3)sin(p4).
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings given an approximate height function
( )
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Consider a (smooth) shape h for the height function to follow.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings given an approximate height function
C id ( h) h h f h h i h f i f ll
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Consider a (smooth) shape h for the height function to follow.
Locally the shape looks flat, so we know how many possibilitiesthere are for the local tiling: something like
c(a, b)(n)2
where a and bdepend on the local slope.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings given an approximate height function
C id ( h) h h f h h i h f i f ll
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Consider a (smooth) shape h for the height function to follow.
Locally the shape looks flat, so we know how many possibilitiesthere are for the local tiling: something like
c(a, b)(n)2
where a and bdepend on the local slope. Namely, a=xh,b=yh.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
Counting tilings given an approximate height function
C id ( th) h h f th h i ht f ti t f ll
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Consider a (smooth) shape h for the height function to follow.
Locally the shape looks flat, so we know how many possibilitiesthere are for the local tiling: something like
c(a, b)(n)2
where a and bdepend on the local slope. Namely, a=xh,b=yh.
To put things in a formula: The number of tilings giving rise to aheight function close to h is approximately
N(h) exp
log(c(xh, yh)) dxdy
C(h)n
2.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The most likely height function
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N(h) C(h)n2
.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The most likely height function
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N(h) C(h)n2
.
So, the larger N(h) is, the more likely it is to observe a heightfunction that is close to h.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The most likely height function
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N(h) C(h)n2
.
So, the larger N(h) is, the more likely it is to observe a heightfunction that is close to h. But remember, the boundary value ofhis fixed, and h cannot be too wild...
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The most likely height function
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N(h) C(h)n2
.
So, the larger N(h) is, the more likely it is to observe a heightfunction that is close to h. But remember, the boundary value ofhis fixed, and h cannot be too wild...
There is a unique h0 maximizing N(h) under the constraints that hhas to satisfy. h0 is themost likely (asymptotic) height function.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
The most likely height function
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N(h) C(h)n2
.
So, the larger N(h) is, the more likely it is to observe a heightfunction that is close to h. But remember, the boundary value ofhis fixed, and h cannot be too wild...
There is a unique h0 maximizing N(h) under the constraints that hhas to satisfy. h0 is themost likely (asymptotic) height function.
TheoremThe height function of a uniform random domino tiling, in anydomain, is very likely to be close toh0.
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
Large deviations
What is the probability that the height function H of a random
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at s t e p obab ty t at t e e g t u ct o H o a a do
tiling is close to some fixed function h?
Vincent Beffara The arctic circle Domino tilings
Mathematical toolsA few ideas of the proof
Large deviations
What is the probability that the height function Hof a random
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p y g
tiling is close to some fixed function h?
P[H h] {tilings s.t. H h}
{tilings}
C(h)n2
C(h)n2
Vi t B ff Th ti i l
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Domino tilingsMathematical tools
A few ideas of the proof
Large deviations
What is the probability that the height function Hof a random
-
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p y g
tiling is close to some fixed function h?
P[H h] {tilings s.t. H h}
{tilings}
C(h)n2
C(h)n2
But: ifc1
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P[H h] C(h)C(h0)
n
2
.
Vi t B ff Th ti i l Domino tilingsMathematical tools
A few ideas of the proof
Large deviations
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P[H h] C(h)C(h0)
n
2
.
There are two cases now, giving the proof of the theorem:
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Large deviations
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P[H h] C(h)C(h0)
n
2
.
There are two cases now, giving the proof of the theorem:
Ifh=h0, then P[H h] 1;
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Large deviations
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P[H h] C(h)C(h0)
n
2
.
There are two cases now, giving the proof of the theorem:
Ifh=h0, then P[H h] 1;Ifh =h0, then it isexponentially small, like
P[H h] ec(h)n2
where c= log(C(h)/C(h0))> 0.
Vincent Beffara The arctic circle
Domino tilingsMathematical tools
A few ideas of the proof
Large deviations
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P[H h] C(h)C(h0)
n
2
.
There are two cases now, giving the proof of the theorem:
Ifh=h0, then P[H h] 1;Ifh =h0, then it isexponentially small, like
P[H h] ec(h)n2
where c= log(C(h)/C(h0))> 0.
This is known as alarge deviation theorem.
Vincent Beffara The arctic circle