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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