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Percolation is Odd Stephan Mertens, Otto-von-Guericke University - - PowerPoint PPT Presentation
Percolation is Odd Stephan Mertens, Otto-von-Guericke University - - PowerPoint PPT Presentation
Percolation is Odd Stephan Mertens, Otto-von-Guericke University Cristopher Moore, Santa Fe Institute <latexit
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The Total Number of Spanning Configurations is Always Odd
1 2 3 4 5 6 7 1 1 1 1 1 1 1 1 2 3 7 17 41 99 239 577 3 7 37 197 1041 5503 29089 153769 4 15 175 1985 22193 247759 2764991 30856705 5 31 781 18621 433809 10056959 232824241 5388274121 6 63 3367 167337 8057905 384479935 18287614751 868972410929 7 127 14197 1461797 144769425 14142942975 1374273318721 133267613878665
<latexit sha1_base64="ra7591rNlC9TFfYc3oiajUfMOE=">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</latexit>Height Width
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10 20 30 40 50 60 1000.0 107 1011 1015 1019
: # configurations with k occupied sites
An,m(k)
k
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Rn,m(z) =
nm
∑
k=0
zkAn,m(k)
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Rn,m(z) =
nm
∑
k=0
zkAn,m(k) Pcross(p) =
nm
∑
k=0
pk(1 − p)nm−kAn,m(k)
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Rn,m(z) =
nm
∑
k=0
zkAn,m(k) Pcross(p) =
nm
∑
k=0
pk(1 − p)nm−kAn,m(k)
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
5 × 5, 11 × 11, 22 × 22
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Rn,m(z) =
nm
∑
k=0
zkAn,m(k)
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Rn,m(z) =
nm
∑
k=0
zkAn,m(k) Rn,m(−1) = ∑
k even
An,m(k) − ∑
k odd
An,m(k)
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Rn,m(z) =
nm
∑
k=0
zkAn,m(k) Rn,m(−1) = ∑
k even
An,m(k) − ∑
k odd
An,m(k)
m n 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1
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k odd k even partial matching
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k odd k even partial matching
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The Odd One Out
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⌊ m 2 ⌋ n + ⌈ m 2 ⌉
The Odd One Out
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Rn,m(−1) = ∑
k even
An,m(k) − ∑
k odd
An,m(k) = (−1)⌊
m 2 ⌋n+⌈ m 2 ⌉
m n 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1
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Other Matching Proofs
A square integer has an odd number of divisors The number of binary trees with leaves is odd A prime has an odd number of representations
2n p = 4n + 1 p = x2 + y2
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Shameless plug
To put it bluntly: this book rocks! It somehow manages to combine the fun of a popular book with the intellectual heft of a textbook. Scott Aaronson, UT Austin This is, simply put, the best-written book on the theory of computation I have ever read;
- ne of the best-written mathematical books I
have ever read, period. Cosma Shalizi, Carnegie Mellon
www.nature-of-computation.org
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Shameless plug
To put it bluntly: this book rocks! It somehow manages to combine the fun of a popular book with the intellectual heft of a textbook. Scott Aaronson, UT Austin This is, simply put, the best-written book on the theory of computation I have ever read;
- ne of the best-written mathematical books I
have ever read, period. Cosma Shalizi, Carnegie Mellon
www.nature-of-computation.org