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Markov random fields and the Pivot property
Nishant Chandgotia 1 Tom Meyerovitch2
1University of British Columbia
2Ben-Gurion University
July 2013
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Outline
Nearest neighbour shifts of finite type
Topological Markov fields
Markov random fields and Gibbs measures with nearestneighbour interactions
Pivot property
Examples: 3-coloured chessboard and the Square Island shift.
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Consider a torus R2/Z2 with the map [ 1 11 0 ].
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[ 1 11 0 ] has two eigenvalues (∼ 1.618 and −.618).
Vector with eigen value ~ 1.618
Vector with eigen value ~ -0.618
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We can divide the torus into 3 parts by extending theeigendirections. These are called Markov partitions.
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Trajectories of points of the torus under the map [ 1 11 0 ] can be
coded via the partition it visits.
This gives us an ‘isomorphism’between the torus with the map [ 1 1
1 0 ] and the set of bi-infinitesequences in red, green and blue where consecutive colours cannotbe the same and red cannot be followed by green.
This phenomenon is much more general: Any automorphism of thetorus(with no eigenvalues of modulus 1) can be coded in a similarway.
![Page 7: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/7.jpg)
Trajectories of points of the torus under the map [ 1 11 0 ] can be
coded via the partition it visits. This gives us an ‘isomorphism’between the torus with the map [ 1 1
1 0 ] and the set of bi-infinitesequences in red, green and blue where consecutive colours cannotbe the same and red cannot be followed by green.
This phenomenon is much more general: Any automorphism of thetorus(with no eigenvalues of modulus 1) can be coded in a similarway.
![Page 8: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/8.jpg)
Trajectories of points of the torus under the map [ 1 11 0 ] can be
coded via the partition it visits. This gives us an ‘isomorphism’between the torus with the map [ 1 1
1 0 ] and the set of bi-infinitesequences in red, green and blue where consecutive colours cannotbe the same and red cannot be followed by green.
This phenomenon is much more general: Any automorphism of thetorus(with no eigenvalues of modulus 1) can be coded in a similarway.
![Page 9: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/9.jpg)
Nearest Neighbour Shifts of Finite Type
Let A be a finite alphabet set.
A shape F is any finite subset ofZd . A pattern is a function f : F −→ A. Given F , a list ofpatterns, a shift space XF is given by
XF = {x ∈ AZd | patterns from F do not occur in x}.
A nearest neighbour shift of finite type is a shift space such that Fcan be given by patterns on ‘edges’.
![Page 10: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/10.jpg)
Nearest Neighbour Shifts of Finite Type
Let A be a finite alphabet set. A shape F is any finite subset ofZd .
A pattern is a function f : F −→ A. Given F , a list ofpatterns, a shift space XF is given by
XF = {x ∈ AZd | patterns from F do not occur in x}.
A nearest neighbour shift of finite type is a shift space such that Fcan be given by patterns on ‘edges’.
![Page 11: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/11.jpg)
Nearest Neighbour Shifts of Finite Type
Let A be a finite alphabet set. A shape F is any finite subset ofZd . A pattern is a function f : F −→ A.
Given F , a list ofpatterns, a shift space XF is given by
XF = {x ∈ AZd | patterns from F do not occur in x}.
A nearest neighbour shift of finite type is a shift space such that Fcan be given by patterns on ‘edges’.
![Page 12: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/12.jpg)
Nearest Neighbour Shifts of Finite Type
Let A be a finite alphabet set. A shape F is any finite subset ofZd . A pattern is a function f : F −→ A. Given F , a list ofpatterns,
a shift space XF is given by
XF = {x ∈ AZd | patterns from F do not occur in x}.
A nearest neighbour shift of finite type is a shift space such that Fcan be given by patterns on ‘edges’.
![Page 13: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/13.jpg)
Nearest Neighbour Shifts of Finite Type
Let A be a finite alphabet set. A shape F is any finite subset ofZd . A pattern is a function f : F −→ A. Given F , a list ofpatterns, a shift space XF is given by
XF = {x ∈ AZd | patterns from F do not occur in x}.
A nearest neighbour shift of finite type is a shift space such that Fcan be given by patterns on ‘edges’.
![Page 14: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/14.jpg)
Nearest Neighbour Shifts of Finite Type
Let A be a finite alphabet set. A shape F is any finite subset ofZd . A pattern is a function f : F −→ A. Given F , a list ofpatterns, a shift space XF is given by
XF = {x ∈ AZd | patterns from F do not occur in x}.
A nearest neighbour shift of finite type is a shift space such that Fcan be given by patterns on ‘edges’.
![Page 15: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/15.jpg)
Examples:
The full shift: F is empty. XF = AZd.
The hard square model: A = {0, 1} and F = {11}di=1.The n-coloured chessboard: A = {0, 1, 2, . . . , n− 1} andF = {00, 11, 22, . . . , }di=1.
0 0 0 0
0000
0 0 0 0
0000
1
1
1
11
1
1
1
1111
1 1 1 1
1 1 1 1
1111
1 1 1 1
1 1 1 1
2 2
2 2
22
2 2
0 0
00
0 0
00
Figure : The 3-coloured chessboard in 2 dimensions
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Examples:
The full shift: F is empty. XF = AZd.
The hard square model: A = {0, 1} and F = {11}di=1.
The n-coloured chessboard: A = {0, 1, 2, . . . , n− 1} andF = {00, 11, 22, . . . , }di=1.
0 0 0 0
0000
0 0 0 0
0000
1
1
1
11
1
1
1
1111
1 1 1 1
1 1 1 1
1111
1 1 1 1
1 1 1 1
2 2
2 2
22
2 2
0 0
00
0 0
00
Figure : The 3-coloured chessboard in 2 dimensions
![Page 17: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/17.jpg)
Examples:
The full shift: F is empty. XF = AZd.
The hard square model: A = {0, 1} and F = {11}di=1.The n-coloured chessboard: A = {0, 1, 2, . . . , n− 1} andF = {00, 11, 22, . . . , }di=1.
0 0 0 0
0000
0 0 0 0
0000
1
1
1
11
1
1
1
1111
1 1 1 1
1 1 1 1
1111
1 1 1 1
1 1 1 1
2 2
2 2
22
2 2
0 0
00
0 0
00
Figure : The 3-coloured chessboard in 2 dimensions
![Page 18: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/18.jpg)
Examples:
The full shift: F is empty. XF = AZd.
The hard square model: A = {0, 1} and F = {11}di=1.The n-coloured chessboard: A = {0, 1, 2, . . . , n− 1} andF = {00, 11, 22, . . . , }di=1.
0 0 0 0
0000
0 0 0 0
0000
1
1
1
11
1
1
1
1111
1 1 1 1
1 1 1 1
1111
1 1 1 1
1 1 1 1
2 2
2 2
22
2 2
0 0
00
0 0
00
Figure : The 3-coloured chessboard in 2 dimensions
![Page 19: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/19.jpg)
1 Dimension vs Higher Dimensions
A lot is known about 1 dimensional nearest neighbour shifts offinite type
because they can be recoded as walks on graphs (vertexshifts).For instance walks on
10
give us the hard square shift space. (A = {0, 1} and F = {11}).However in higher dimensions given A and F there is no algorithmto decide whether the nearest neighbour shift of finite type isnon-empty!!! This is not an issue if the space has a ‘safe symbol’.
![Page 20: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/20.jpg)
1 Dimension vs Higher Dimensions
A lot is known about 1 dimensional nearest neighbour shifts offinite type because they can be recoded as walks on graphs (vertexshifts).
For instance walks on
10
give us the hard square shift space. (A = {0, 1} and F = {11}).However in higher dimensions given A and F there is no algorithmto decide whether the nearest neighbour shift of finite type isnon-empty!!! This is not an issue if the space has a ‘safe symbol’.
![Page 21: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/21.jpg)
1 Dimension vs Higher Dimensions
A lot is known about 1 dimensional nearest neighbour shifts offinite type because they can be recoded as walks on graphs (vertexshifts).For instance walks on
10
give us the hard square shift space. (A = {0, 1} and F = {11}).However in higher dimensions given A and F there is no algorithmto decide whether the nearest neighbour shift of finite type isnon-empty!!! This is not an issue if the space has a ‘safe symbol’.
![Page 22: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/22.jpg)
1 Dimension vs Higher Dimensions
A lot is known about 1 dimensional nearest neighbour shifts offinite type because they can be recoded as walks on graphs (vertexshifts).For instance walks on
10
give us the hard square shift space. (A = {0, 1} and F = {11}).
However in higher dimensions given A and F there is no algorithmto decide whether the nearest neighbour shift of finite type isnon-empty!!! This is not an issue if the space has a ‘safe symbol’.
![Page 23: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/23.jpg)
1 Dimension vs Higher Dimensions
A lot is known about 1 dimensional nearest neighbour shifts offinite type because they can be recoded as walks on graphs (vertexshifts).For instance walks on
10
give us the hard square shift space. (A = {0, 1} and F = {11}).However in higher dimensions given A and F there is no algorithmto decide whether the nearest neighbour shift of finite type isnon-empty!!!
This is not an issue if the space has a ‘safe symbol’.
![Page 24: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/24.jpg)
1 Dimension vs Higher Dimensions
A lot is known about 1 dimensional nearest neighbour shifts offinite type because they can be recoded as walks on graphs (vertexshifts).For instance walks on
10
give us the hard square shift space. (A = {0, 1} and F = {11}).However in higher dimensions given A and F there is no algorithmto decide whether the nearest neighbour shift of finite type isnon-empty!!! This is not an issue if the space has a ‘safe symbol’.
![Page 25: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/25.jpg)
Suppose X is a nearest neighbour shift of finite type on alphabetA. A symbol ? ∈ A is called a safe symbol if it can sit adjacent toany other symbol in A.
For instance, the hard square model (A = {0, 1}, F = {11}) hasa safe symbol viz. 0 but the 3-coloured chessboard (A = {0, 1, 2}and F = {00, 11, 22}) does not have any safe symbol.
![Page 26: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/26.jpg)
Suppose X is a nearest neighbour shift of finite type on alphabetA. A symbol ? ∈ A is called a safe symbol if it can sit adjacent toany other symbol in A.
For instance, the hard square model (A = {0, 1}, F = {11}) hasa safe symbol viz. 0
but the 3-coloured chessboard (A = {0, 1, 2}and F = {00, 11, 22}) does not have any safe symbol.
![Page 27: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/27.jpg)
Suppose X is a nearest neighbour shift of finite type on alphabetA. A symbol ? ∈ A is called a safe symbol if it can sit adjacent toany other symbol in A.
For instance, the hard square model (A = {0, 1}, F = {11}) hasa safe symbol viz. 0 but the 3-coloured chessboard (A = {0, 1, 2}and F = {00, 11, 22}) does not have any safe symbol.
![Page 28: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/28.jpg)
Topological Markov Fields
A topological Markov field is a shift space X ⊂ AZdwith the
‘conditional independence’ property: for all finite subsets F ⊂ Zd ,x , y ∈ X satisfying x = y on ∂F , z ∈ AZd
given by
z =
{x on F
y on F c
is also an element of X .
x y z
F F F
Every nearest neighbour shift of finite type is a topological Markovfield.
![Page 29: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/29.jpg)
Topological Markov Fields
A topological Markov field is a shift space X ⊂ AZdwith the
‘conditional independence’ property: for all finite subsets F ⊂ Zd ,x , y ∈ X satisfying x = y on ∂F , z ∈ AZd
given by
z =
{x on F
y on F c
is also an element of X .
x y z
F F F
Every nearest neighbour shift of finite type is a topological Markovfield.
![Page 30: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/30.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield.
However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where the horizontal constraintscome from X and the vertical direction is constant. If x and yagree on ∂F , they must agree on F . Therefore Y is a topologicalMarkov field. There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 31: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/31.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield. However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where the horizontal constraintscome from X and the vertical direction is constant. If x and yagree on ∂F , they must agree on F . Therefore Y is a topologicalMarkov field. There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 32: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/32.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield. However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where
the horizontal constraintscome from X and the vertical direction is constant. If x and yagree on ∂F , they must agree on F . Therefore Y is a topologicalMarkov field. There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 33: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/33.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield. However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where the horizontal constraintscome from X
and the vertical direction is constant. If x and yagree on ∂F , they must agree on F . Therefore Y is a topologicalMarkov field. There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 34: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/34.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield. However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where the horizontal constraintscome from X and the vertical direction is constant.
If x and yagree on ∂F , they must agree on F . Therefore Y is a topologicalMarkov field. There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 35: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/35.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield. However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where the horizontal constraintscome from X and the vertical direction is constant. If x and yagree on ∂F , they must agree on F .
Therefore Y is a topologicalMarkov field. There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 36: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/36.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield. However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where the horizontal constraintscome from X and the vertical direction is constant. If x and yagree on ∂F , they must agree on F . Therefore Y is a topologicalMarkov field.
There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 37: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/37.jpg)
Topological Markov Fields
Every nearest neighbour shift of finite type is a topological Markovfield. However not every topological Markov field is a nearestneighbour shift of finite type.
Consider any one-dimensional shift space X . Make atwo-dimensional shift space Y where the horizontal constraintscome from X and the vertical direction is constant. If x and yagree on ∂F , they must agree on F . Therefore Y is a topologicalMarkov field. There are uncountably many such shift spaces butthere are only countably many nearest neighbour shift of finitetype!!
![Page 38: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/38.jpg)
Markov Random Fields
The measure-theoretic version of this ‘conditional independence’ iscalled a Markov random field.
A Markov random field is a shift-invariant Borel probability measureµ on AZd
with the property that for all finite A,B ⊂ Zd such that∂A ⊂ B ⊂ Ac and a ∈ AA, b ∈ AB satisfying µ([b]B) > 0
µ([a]A
∣∣∣ [b]B) = µ([a]A
∣∣∣ [b]∂A).The support of every Markov random field is a topological Markovfield.
![Page 39: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/39.jpg)
Markov Random Fields
The measure-theoretic version of this ‘conditional independence’ iscalled a Markov random field.
A Markov random field is a shift-invariant Borel probability measureµ on AZd
with the property that for all finite A,B ⊂ Zd such that∂A ⊂ B ⊂ Ac and a ∈ AA, b ∈ AB satisfying µ([b]B) > 0
µ([a]A
∣∣∣ [b]B) = µ([a]A
∣∣∣ [b]∂A).
The support of every Markov random field is a topological Markovfield.
![Page 40: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/40.jpg)
Markov Random Fields
The measure-theoretic version of this ‘conditional independence’ iscalled a Markov random field.
A Markov random field is a shift-invariant Borel probability measureµ on AZd
with the property that for all finite A,B ⊂ Zd such that∂A ⊂ B ⊂ Ac and a ∈ AA, b ∈ AB satisfying µ([b]B) > 0
µ([a]A
∣∣∣ [b]B) = µ([a]A
∣∣∣ [b]∂A).The support of every Markov random field is a topological Markovfield.
![Page 41: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/41.jpg)
Markov Random Fields
The measure-theoretic version of this ‘conditional independence’ iscalled a Markov random field.
A Markov random field is a shift-invariant Borel probability measureµ on AZd
with the property that for all finite A,B ⊂ Zd such that∂A ⊂ B ⊂ Ac and a ∈ AA, b ∈ AB satisfying µ([b]B) > 0
µ([a]A
∣∣∣ [b]B) = µ([a]A
∣∣∣ [b]∂A).The set of conditional measures µ([·]A
∣∣∣ [b]∂A) for all A ⊂ Zd
finite and b ∈ A∂A is called specification for the measure µ.
Thespecification might contain a huge lot of data!!!!
![Page 42: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/42.jpg)
Markov Random Fields
The measure-theoretic version of this ‘conditional independence’ iscalled a Markov random field.
A Markov random field is a shift-invariant Borel probability measureµ on AZd
with the property that for all finite A,B ⊂ Zd such that∂A ⊂ B ⊂ Ac and a ∈ AA, b ∈ AB satisfying µ([b]B) > 0
µ([a]A
∣∣∣ [b]B) = µ([a]A
∣∣∣ [b]∂A).The set of conditional measures µ([·]A
∣∣∣ [b]∂A) for all A ⊂ Zd
finite and b ∈ A∂A is called specification for the measure µ.Thespecification might contain a huge lot of data!!!!
![Page 43: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/43.jpg)
Given a shift space X define a nearest neighbour interaction on Xas a shift-invariant function V : B(X ) −→ R supported onconfigurations on edges and vertices.
A Gibbs state with a nearest neighbor interaction V is a Markovrandom field µ such that for all x ∈ supp(µ) and A,B ⊂ Zd finitesatisfying ∂A ⊂ B ⊂ Ac
µ([x ]A
∣∣∣ [x ]B) =
∏C⊂A∪∂A
eV ([x ]C )
ZA,x |∂A
where ZA,x |∂A is the uniquely determined normalising factor so thatµ(X ) = 1, dependent upon A and x |∂A.
The specification of a Gibbs measure with a nearest neighbourinteraction has a finite description: all we need is the interaction V .
![Page 44: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/44.jpg)
Given a shift space X define a nearest neighbour interaction on Xas a shift-invariant function V : B(X ) −→ R supported onconfigurations on edges and vertices.
A Gibbs state with a nearest neighbor interaction V is a Markovrandom field µ such that for all x ∈ supp(µ) and A,B ⊂ Zd finitesatisfying ∂A ⊂ B ⊂ Ac
µ([x ]A
∣∣∣ [x ]B) =
∏C⊂A∪∂A
eV ([x ]C )
ZA,x |∂A
where ZA,x |∂A is the uniquely determined normalising factor so thatµ(X ) = 1, dependent upon A and x |∂A.
The specification of a Gibbs measure with a nearest neighbourinteraction has a finite description: all we need is the interaction V .
![Page 45: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/45.jpg)
Given a shift space X define a nearest neighbour interaction on Xas a shift-invariant function V : B(X ) −→ R supported onconfigurations on edges and vertices.
A Gibbs state with a nearest neighbor interaction V is a Markovrandom field µ such that for all x ∈ supp(µ) and A,B ⊂ Zd finitesatisfying ∂A ⊂ B ⊂ Ac
µ([x ]A
∣∣∣ [x ]B) =
∏C⊂A∪∂A
eV ([x ]C )
ZA,x |∂A
where ZA,x |∂A is the uniquely determined normalising factor so thatµ(X ) = 1, dependent upon A and x |∂A.
The specification of a Gibbs measure with a nearest neighbourinteraction has a finite description: all we need is
the interaction V .
![Page 46: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/46.jpg)
Given a shift space X define a nearest neighbour interaction on Xas a shift-invariant function V : B(X ) −→ R supported onconfigurations on edges and vertices.
A Gibbs state with a nearest neighbor interaction V is a Markovrandom field µ such that for all x ∈ supp(µ) and A,B ⊂ Zd finitesatisfying ∂A ⊂ B ⊂ Ac
µ([x ]A
∣∣∣ [x ]B) =
∏C⊂A∪∂A
eV ([x ]C )
ZA,x |∂A
where ZA,x |∂A is the uniquely determined normalising factor so thatµ(X ) = 1, dependent upon A and x |∂A.
The specification of a Gibbs measure with a nearest neighbourinteraction has a finite description: all we need is the interaction V .
![Page 47: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/47.jpg)
Question: When is a Markov random field Gibbs with somenearest neighbour interaction?
(Hammersley-Clifford theorem) Every Markov random field whosesupport has a safe symbol is Gibbs with some nearest neighbourinteraction.
This is the property of the specification rather than the actualmeasure!
Question: How can we weaken the hypothesis?
![Page 48: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/48.jpg)
Question: When is a Markov random field Gibbs with somenearest neighbour interaction?
(Hammersley-Clifford theorem) Every Markov random field whosesupport has a safe symbol is Gibbs with some nearest neighbourinteraction.
This is the property of the specification rather than the actualmeasure!
Question: How can we weaken the hypothesis?
![Page 49: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/49.jpg)
Question: When is a Markov random field Gibbs with somenearest neighbour interaction?
(Hammersley-Clifford theorem) Every Markov random field whosesupport has a safe symbol is Gibbs with some nearest neighbourinteraction.
This is the property of the specification rather than the actualmeasure!
Question: How can we weaken the hypothesis?
![Page 50: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/50.jpg)
Question: When is a Markov random field Gibbs with somenearest neighbour interaction?
(Hammersley-Clifford theorem) Every Markov random field whosesupport has a safe symbol is Gibbs with some nearest neighbourinteraction.
This is the property of the specification rather than the actualmeasure!
Question: How can we weaken the hypothesis?
![Page 51: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/51.jpg)
Pivot Property
A shift space X is said to satisfy the pivot property if for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onat most a single site.
A shift space X is said to satisfy thegeneralised pivot property if there exists K > 0 such that for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onlyon a region of diameter at most K .
Examples:
Any shift space with a safe symbol.
r-coloured checkerboard for r 6= 4, 5.
Domino tilings.
![Page 52: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/52.jpg)
Pivot Property
A shift space X is said to satisfy the pivot property if for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onat most a single site. A shift space X is said to satisfy thegeneralised pivot property if there exists K > 0 such that for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onlyon a region of diameter at most K .
Examples:
Any shift space with a safe symbol.
r-coloured checkerboard for r 6= 4, 5.
Domino tilings.
![Page 53: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/53.jpg)
Pivot Property
A shift space X is said to satisfy the pivot property if for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onat most a single site. A shift space X is said to satisfy thegeneralised pivot property if there exists K > 0 such that for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onlyon a region of diameter at most K .
Examples:
Any shift space with a safe symbol.
r-coloured checkerboard for r 6= 4, 5.
Domino tilings.
![Page 54: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/54.jpg)
Pivot Property
A shift space X is said to satisfy the pivot property if for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onat most a single site. A shift space X is said to satisfy thegeneralised pivot property if there exists K > 0 such that for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onlyon a region of diameter at most K .
Examples:
Any shift space with a safe symbol.
r-coloured checkerboard for r 6= 4, 5.
Domino tilings.
![Page 55: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/55.jpg)
Pivot Property
A shift space X is said to satisfy the pivot property if for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onat most a single site. A shift space X is said to satisfy thegeneralised pivot property if there exists K > 0 such that for allx , y ∈ X which differ only on finitely many sites there exists achain x = x1, x2, x3, . . . , xn = y ∈ X such that x i , x i+1 differ onlyon a region of diameter at most K .
Examples:
Any shift space with a safe symbol.
r-coloured checkerboard for r 6= 4, 5.
Domino tilings.
![Page 56: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/56.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
0 1
10
2 0
20
0
1
0
2
1
0 1
0 1
0
2
0
1
1
0
0
1
2
2
1
1
0
1
0
21
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
![Page 57: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/57.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
0 1
10
2 0
20
0
1
0
2
1
0 1
0 1
0
2
0
1
1
0
0
1
2
2
1
1
0
1
0
21
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
![Page 58: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/58.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
0 1
10
2 0
20
0
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
0
1
0
21
1
![Page 59: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/59.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
0 1
10
2 0
20
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
0
1
0
21
10
![Page 60: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/60.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
0 1
10
2 0
20
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
0
1
0
21
12
![Page 61: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/61.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
0 1
10
2 0
20
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
0
1
0
21
12
![Page 62: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/62.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
0
1
0
21
12
2
![Page 63: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/63.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
0
1
0
21
12
2
![Page 64: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/64.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
1
0
21
12
2
2
![Page 65: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/65.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
0
1
1
0
0
1
2
2
1
1
1
0
21
12
2
2
![Page 66: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/66.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
12
2
2
2
![Page 67: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/67.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
12
2
2
2
![Page 68: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/68.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
![Page 69: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/69.jpg)
The 3-coloured Chessboard
The 3-coloured chessboard has the pivot property.
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
1 0
0
2 0
120
1
10
2 0
20
1
0
2
1
0 1
0 1
0
1
1
0
0
1
2
2
1
1
1
0
21
2
2
2
2
0
![Page 70: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/70.jpg)
Suppose µ is a Markov random field whose support has the pivotproperty.
Then given x , y ∈ supp(µ) that differ exactly on F thereexists a chain x = x1, x2, . . . , xn = y where x i , x i+1 differ exactlyat a site mi ∈ Z2 and consequently
µ([x ]F | [x ]∂F )µ([y ]F | [x ]∂F )
=n−1∏i=1
µ([x i ]F | [x i ]∂F )µ([x i+1]F | [x i ]∂F )
=n−1∏i=1
µ([x i ]mi | [x i ]∂mi)
µ([x i+1]mi | [x i ]∂mi)
.
Therefore the entire specification is determined by finitely many
parameters viz.µ([x ]0∪∂0)µ([y ]0∪∂0)
for configurations x , y which differ only at
0, the origin.
Theorem
Given a shift space with the pivot property the space ofspecifications on that shift space can be parametrised by finitelymany parameters.
![Page 71: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/71.jpg)
Suppose µ is a Markov random field whose support has the pivotproperty. Then given x , y ∈ supp(µ) that differ exactly on F thereexists a chain x = x1, x2, . . . , xn = y where x i , x i+1 differ exactlyat a site mi ∈ Z2
and consequently
µ([x ]F | [x ]∂F )µ([y ]F | [x ]∂F )
=n−1∏i=1
µ([x i ]F | [x i ]∂F )µ([x i+1]F | [x i ]∂F )
=n−1∏i=1
µ([x i ]mi | [x i ]∂mi)
µ([x i+1]mi | [x i ]∂mi)
.
Therefore the entire specification is determined by finitely many
parameters viz.µ([x ]0∪∂0)µ([y ]0∪∂0)
for configurations x , y which differ only at
0, the origin.
Theorem
Given a shift space with the pivot property the space ofspecifications on that shift space can be parametrised by finitelymany parameters.
![Page 72: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/72.jpg)
Suppose µ is a Markov random field whose support has the pivotproperty. Then given x , y ∈ supp(µ) that differ exactly on F thereexists a chain x = x1, x2, . . . , xn = y where x i , x i+1 differ exactlyat a site mi ∈ Z2 and consequently
µ([x ]F | [x ]∂F )µ([y ]F | [x ]∂F )
=n−1∏i=1
µ([x i ]F | [x i ]∂F )µ([x i+1]F | [x i ]∂F )
=n−1∏i=1
µ([x i ]mi | [x i ]∂mi)
µ([x i+1]mi | [x i ]∂mi)
.
Therefore the entire specification is determined by finitely many
parameters viz.µ([x ]0∪∂0)µ([y ]0∪∂0)
for configurations x , y which differ only at
0, the origin.
Theorem
Given a shift space with the pivot property the space ofspecifications on that shift space can be parametrised by finitelymany parameters.
![Page 73: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/73.jpg)
Suppose µ is a Markov random field whose support has the pivotproperty. Then given x , y ∈ supp(µ) that differ exactly on F thereexists a chain x = x1, x2, . . . , xn = y where x i , x i+1 differ exactlyat a site mi ∈ Z2 and consequently
µ([x ]F | [x ]∂F )µ([y ]F | [x ]∂F )
=n−1∏i=1
µ([x i ]F | [x i ]∂F )µ([x i+1]F | [x i ]∂F )
=n−1∏i=1
µ([x i ]mi | [x i ]∂mi)
µ([x i+1]mi | [x i ]∂mi)
.
Therefore the entire specification is determined by finitely many
parameters viz.
µ([x ]0∪∂0)µ([y ]0∪∂0)
for configurations x , y which differ only at
0, the origin.
Theorem
Given a shift space with the pivot property the space ofspecifications on that shift space can be parametrised by finitelymany parameters.
![Page 74: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/74.jpg)
Suppose µ is a Markov random field whose support has the pivotproperty. Then given x , y ∈ supp(µ) that differ exactly on F thereexists a chain x = x1, x2, . . . , xn = y where x i , x i+1 differ exactlyat a site mi ∈ Z2 and consequently
µ([x ]F | [x ]∂F )µ([y ]F | [x ]∂F )
=n−1∏i=1
µ([x i ]F | [x i ]∂F )µ([x i+1]F | [x i ]∂F )
=n−1∏i=1
µ([x i ]mi | [x i ]∂mi)
µ([x i+1]mi | [x i ]∂mi)
.
Therefore the entire specification is determined by finitely many
parameters viz.µ([x ]0∪∂0)µ([y ]0∪∂0)
for configurations x , y which differ only at
0, the origin.
Theorem
Given a shift space with the pivot property the space ofspecifications on that shift space can be parametrised by finitelymany parameters.
![Page 75: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/75.jpg)
Suppose µ is a Markov random field whose support has the pivotproperty. Then given x , y ∈ supp(µ) that differ exactly on F thereexists a chain x = x1, x2, . . . , xn = y where x i , x i+1 differ exactlyat a site mi ∈ Z2 and consequently
µ([x ]F | [x ]∂F )µ([y ]F | [x ]∂F )
=n−1∏i=1
µ([x i ]F | [x i ]∂F )µ([x i+1]F | [x i ]∂F )
=n−1∏i=1
µ([x i ]mi | [x i ]∂mi)
µ([x i+1]mi | [x i ]∂mi)
.
Therefore the entire specification is determined by finitely many
parameters viz.µ([x ]0∪∂0)µ([y ]0∪∂0)
for configurations x , y which differ only at
0, the origin.
Theorem
Given a shift space with the pivot property the space ofspecifications on that shift space can be parametrised by finitelymany parameters.
![Page 76: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/76.jpg)
Question: Suppose we are given a nearest neighbour shift offinite type with the pivot property. Is there an algorithm todetermine the number of parameters which describes thespecification?
![Page 77: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/77.jpg)
Thus a specification supported on the 3-coloured chessboard is
determined the quantities v1 =µ(
[1
1 0 11
])
µ(
[1
1 2 11
]),
v2 =µ(
[2
2 1 22
])
µ(
[2
2 0 22
])
and
v3 =µ(
[0
0 2 00
])
µ(
[0
0 1 00
]). If µ is a Gibbs measure with nearest neighbour
interaction V then
v1 = exp(V (01) + V (10) + V ( 01 ) + V ( 01 )
−V (21)− V (12)− V ( 21 )− V ( 12 )),
v2 = exp(V (12) + V (21) + V ( 21 ) + V ( 12 )
−V (02)− V (20)− V ( 02 )− V ( 20 )),
v3 = exp(V (02) + V (20) + V ( 20 ) + V ( 02 )
−V (01)− V (10)− V ( 01 )− V ( 10 )).
Thus µ is Gibbs if and only if v1v2v3 = 1.
![Page 78: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/78.jpg)
Thus a specification supported on the 3-coloured chessboard is
determined the quantities v1 =µ(
[1
1 0 11
])
µ(
[1
1 2 11
]), v2 =
µ(
[2
2 1 22
])
µ(
[2
2 0 22
])
and
v3 =µ(
[0
0 2 00
])
µ(
[0
0 1 00
]). If µ is a Gibbs measure with nearest neighbour
interaction V then
v1 = exp(V (01) + V (10) + V ( 01 ) + V ( 01 )
−V (21)− V (12)− V ( 21 )− V ( 12 )),
v2 = exp(V (12) + V (21) + V ( 21 ) + V ( 12 )
−V (02)− V (20)− V ( 02 )− V ( 20 )),
v3 = exp(V (02) + V (20) + V ( 20 ) + V ( 02 )
−V (01)− V (10)− V ( 01 )− V ( 10 )).
Thus µ is Gibbs if and only if v1v2v3 = 1.
![Page 79: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/79.jpg)
Thus a specification supported on the 3-coloured chessboard is
determined the quantities v1 =µ(
[1
1 0 11
])
µ(
[1
1 2 11
]), v2 =
µ(
[2
2 1 22
])
µ(
[2
2 0 22
])
and
v3 =µ(
[0
0 2 00
])
µ(
[0
0 1 00
]).
If µ is a Gibbs measure with nearest neighbour
interaction V then
v1 = exp(V (01) + V (10) + V ( 01 ) + V ( 01 )
−V (21)− V (12)− V ( 21 )− V ( 12 )),
v2 = exp(V (12) + V (21) + V ( 21 ) + V ( 12 )
−V (02)− V (20)− V ( 02 )− V ( 20 )),
v3 = exp(V (02) + V (20) + V ( 20 ) + V ( 02 )
−V (01)− V (10)− V ( 01 )− V ( 10 )).
Thus µ is Gibbs if and only if v1v2v3 = 1.
![Page 80: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/80.jpg)
Thus a specification supported on the 3-coloured chessboard is
determined the quantities v1 =µ(
[1
1 0 11
])
µ(
[1
1 2 11
]), v2 =
µ(
[2
2 1 22
])
µ(
[2
2 0 22
])
and
v3 =µ(
[0
0 2 00
])
µ(
[0
0 1 00
]). If µ is a Gibbs measure with nearest neighbour
interaction V then
v1 = exp(V (01) + V (10) + V ( 01 ) + V ( 01 )
−V (21)− V (12)− V ( 21 )− V ( 12 )),
v2 = exp(V (12) + V (21) + V ( 21 ) + V ( 12 )
−V (02)− V (20)− V ( 02 )− V ( 20 )),
v3 = exp(V (02) + V (20) + V ( 20 ) + V ( 02 )
−V (01)− V (10)− V ( 01 )− V ( 10 )).
Thus µ is Gibbs if and only if v1v2v3 = 1.
![Page 81: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/81.jpg)
Thus a specification supported on the 3-coloured chessboard is
determined the quantities v1 =µ(
[1
1 0 11
])
µ(
[1
1 2 11
]), v2 =
µ(
[2
2 1 22
])
µ(
[2
2 0 22
])
and
v3 =µ(
[0
0 2 00
])
µ(
[0
0 1 00
]). If µ is a Gibbs measure with nearest neighbour
interaction V then
v1 = exp(V (01) + V (10) + V ( 01 ) + V ( 01 )
−V (21)− V (12)− V ( 21 )− V ( 12 )),
v2 = exp(V (12) + V (21) + V ( 21 ) + V ( 12 )
−V (02)− V (20)− V ( 02 )− V ( 20 )),
v3 = exp(V (02) + V (20) + V ( 20 ) + V ( 02 )
−V (01)− V (10)− V ( 01 )− V ( 10 )).
Thus µ is Gibbs if and only if v1v2v3 = 1.
![Page 82: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/82.jpg)
Therefore the Hammersley-Clifford type conclusion fails forspecifications of the 3-coloured chessboard
but every fullysupported Markov random field corresponds to the parameterssatisfying v1v2v3 = 1.
Thus the Hammersley-Clifford type conclusion holds for fullysupported measures.
What if the pivot property does not hold? Every 1 dimensionalnearest neighbour shift of finite type has the generalised pivotproperty.
![Page 83: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/83.jpg)
Therefore the Hammersley-Clifford type conclusion fails forspecifications of the 3-coloured chessboard but
every fullysupported Markov random field corresponds to the parameterssatisfying v1v2v3 = 1.
Thus the Hammersley-Clifford type conclusion holds for fullysupported measures.
What if the pivot property does not hold? Every 1 dimensionalnearest neighbour shift of finite type has the generalised pivotproperty.
![Page 84: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/84.jpg)
Therefore the Hammersley-Clifford type conclusion fails forspecifications of the 3-coloured chessboard but every fullysupported Markov random field corresponds to the parameterssatisfying v1v2v3 = 1.
Thus the Hammersley-Clifford type conclusion holds for fullysupported measures.
What if the pivot property does not hold? Every 1 dimensionalnearest neighbour shift of finite type has the generalised pivotproperty.
![Page 85: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/85.jpg)
Therefore the Hammersley-Clifford type conclusion fails forspecifications of the 3-coloured chessboard but every fullysupported Markov random field corresponds to the parameterssatisfying v1v2v3 = 1.
Thus the Hammersley-Clifford type conclusion holds for fullysupported measures.
What if the pivot property does not hold? Every 1 dimensionalnearest neighbour shift of finite type has the generalised pivotproperty.
![Page 86: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/86.jpg)
Therefore the Hammersley-Clifford type conclusion fails forspecifications of the 3-coloured chessboard but every fullysupported Markov random field corresponds to the parameterssatisfying v1v2v3 = 1.
Thus the Hammersley-Clifford type conclusion holds for fullysupported measures.
What if the pivot property does not hold?
Every 1 dimensionalnearest neighbour shift of finite type has the generalised pivotproperty.
![Page 87: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/87.jpg)
Therefore the Hammersley-Clifford type conclusion fails forspecifications of the 3-coloured chessboard but every fullysupported Markov random field corresponds to the parameterssatisfying v1v2v3 = 1.
Thus the Hammersley-Clifford type conclusion holds for fullysupported measures.
What if the pivot property does not hold? Every 1 dimensionalnearest neighbour shift of finite type has the generalised pivotproperty.
![Page 88: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/88.jpg)
Square Island Shift
Inspiration from checkerboard island shift by Quas and Sahin.
Theallowed nearest neighbour configurations are all the nearestneighbour configurations in
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![Page 89: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/89.jpg)
Square Island Shift
Inspiration from checkerboard island shift by Quas and Sahin. Theallowed nearest neighbour configurations are all the nearestneighbour configurations in
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![Page 90: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/90.jpg)
Square Island Shift
Inspiration from checkerboard island shift by Quas and Sahin. Theallowed nearest neighbour configurations are all the nearestneighbour configurations in
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There are two kinds of squares: ones with red dots and oneswithout red dots which float in a sea of blanks.
![Page 91: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/91.jpg)
The Square Island shift does not have the generalised pivotproperty.
There is no way to switch from a big square with red dots to a bigsquare without red dots making single site changes( or even biggerregional changes).
There exists a Markov random field supported on the shift spacewhich is not Gibbs for any finite-range interaction.
Question: Can more uniform mixing conditions help?
![Page 92: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/92.jpg)
The Square Island shift does not have the generalised pivotproperty.
There is no way to switch from a big square with red dots to a bigsquare without red dots making single site changes( or even biggerregional changes).
There exists a Markov random field supported on the shift spacewhich is not Gibbs for any finite-range interaction.
Question: Can more uniform mixing conditions help?
![Page 93: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/93.jpg)
The Square Island shift does not have the generalised pivotproperty.
There is no way to switch from a big square with red dots to a bigsquare without red dots making single site changes( or even biggerregional changes).
There exists a Markov random field supported on the shift spacewhich is not Gibbs for any finite-range interaction.
Question: Can more uniform mixing conditions help?
![Page 94: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/94.jpg)
The Square Island shift does not have the generalised pivotproperty.
There is no way to switch from a big square with red dots to a bigsquare without red dots making single site changes( or even biggerregional changes).
There exists a Markov random field supported on the shift spacewhich is not Gibbs for any finite-range interaction.
Question: Can more uniform mixing conditions help?
![Page 95: Markov random elds and the Pivot propertymath.huji.ac.il/~nishant/Research_files/Presentation... · 2018-10-07 · Examples: 3-coloured chessboard and the Square Island shift. Consider](https://reader035.vdocuments.site/reader035/viewer/2022070616/5d17697f88c99309378daf72/html5/thumbnails/95.jpg)
Thank You!
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