Percolation is OddStephan Mertens, Otto-von-Guericke University Cristopher Moore, Santa Fe Institute
The Total Number of Spanning Configurations is Always Odd
1 2 3 4 5 6 71 1 1 1 1 1 1 12 3 7 17 41 99 239 5773 7 37 197 1041 5503 29089 1537694 15 175 1985 22193 247759 2764991 308567055 31 781 18621 433809 10056959 232824241 53882741216 63 3367 167337 8057905 384479935 18287614751 8689724109297 127 14197 1461797 144769425 14142942975 1374273318721 133267613878665
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Height
Wid
th
10 20 30 40 50 60
1000.0
107
1011
1015
1019
: # configurations with k occupied sitesAn,m(k)
k
Rn,m(z) =nm
∑k=0
zkAn,m(k)
Rn,m(z) =nm
∑k=0
zkAn,m(k) Pcross(p) =nm
∑k=0
pk(1 − p)nm−kAn,m(k)
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
Rn,m(z) =nm
∑k=0
zkAn,m(k)
Rn,m(z) =nm
∑k=0
zkAn,m(k) Rn,m(−1) = ∑k even
An,m(k) − ∑k odd
An,m(k)
Rn,m(z) =nm
∑k=0
zkAn,m(k) Rn,m(−1) = ∑k even
An,m(k) − ∑k odd
An,m(k)
The paper is organized as follows. We start by proving
(3a) and (3b). We then generalize this result to site per-
colation on the hypercube Zd and, more generally, to carte-
sian graph products. Then we present the most general form
of our result in terms of percolation on graph stacks. Fi-
nally we discuss the computation of Rn,m(�1, 1) for pairs of
matching lattices.
II. The Square Lattice
We compute Rn,m(�1, 1) by constructing a partial matching on
the set of spanning configurations: that is, for most span-
ning configurations � we define a unique partner �0 which
is another spanning configuration, such that �00 = �. More-
over, � and �0 have opposite parity, since they differ at a
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
Table I. Values of Rn,m(�1, 1) for the square lattice with m rows and n
columns, for either open or cylindrical boundary conditions.
4
k odd k even
partial matching
k odd k even
partial matching
The Odd One Out
⌊ m2 ⌋ n + ⌈ m
2 ⌉
The Odd One Out
Rn,m(−1) = ∑k even
An,m(k) − ∑k odd
An,m(k) = (−1)⌊ m2 ⌋n+⌈ m
2 ⌉
The paper is organized as follows. We start by proving
(3a) and (3b). We then generalize this result to site per-
colation on the hypercube Zd and, more generally, to carte-
sian graph products. Then we present the most general form
of our result in terms of percolation on graph stacks. Fi-
nally we discuss the computation of Rn,m(�1, 1) for pairs of
matching lattices.
II. The Square Lattice
We compute Rn,m(�1, 1) by constructing a partial matching on
the set of spanning configurations: that is, for most span-
ning configurations � we define a unique partner �0 which
is another spanning configuration, such that �00 = �. More-
over, � and �0 have opposite parity, since they differ at a
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
Table I. Values of Rn,m(�1, 1) for the square lattice with m rows and n
columns, for either open or cylindrical boundary conditions.
4
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
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; one of the best-written mathematical books I have ever read, period.
Cosma Shalizi, Carnegie Mellonwww.nature-of-computation.org
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; one of the best-written mathematical books I have ever read, period.
Cosma Shalizi, Carnegie Mellonwww.nature-of-computation.org
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