Symmetric Con gurations in the Abelian Sandpile Model

24 Pages·2011·1.98 MB·English

Symmetric Configurations in the Abelian Sandpile
Model
Laura Florescu
David Perkinson
Julia Porcino
Nick Salter
Tianyuan Xu
Reed College
April, 9th,2011
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Self-organized criticality
Bak, Tang and Wiesenfeld invented the idea of “self-organized criticality”,
also observed that builing a sandpile by dropping grains on sand, one
obtains a cone of sand.
connection between the recurrent states of the sandpile model and
the dimer model
connection between the dimer model and viral tiling theory
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Abelian Sandpile Model
Definition
A configuration is recurrent if it is stable and given any configuration a,
there exists a configuration b, s.t. a + b = c.
The recurrent configurations form the sandpile group.
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Symmetric Sandpiles
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Reduced Laplacian
Definition
The reduced Laplacian of a graph encodes the adjacency matrix and its
firings. It is called reduced because we remove the row and column for the
sink vertex.
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Matrix Tree Theorem, KPW
Theorem (Matrix-Tree Theorem)
The number of spanning trees is the determinant of the reduced Laplacian.
Theorem (Kenyon,Propp, Wilson)
Trees on a planar graph are in bijection with perfect matchings on a
corresponding graph.
Furthermore, perfect matchings ⇐⇒ domino tilings.
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Tilings
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Grid graphs without symmetry
Theorem
Let H denote the tilings on a 2m×2n+1 grid graph. Let
2m,2n+1
NG denote the number of recurrents of the m × n grid graph with the
m,n
right corner elements firing 1 to the sink and the right edge elements,
except for the corner elements, firing 0 to the sink.
Then, |H |=NG . In addition, these are counted by the
2m,2n+1 m,n
n(cid:89)/2 (cid:18) (cid:18) (cid:19)(cid:19)
kπ
following Chebyshev polynomial formula: U icos .
m
n+1
k=1
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Chebyshev Polynomials
T (x) = 1 U (x) = 1
0 0
T (x) = x U (x) = 2x
1 1
T (x) = 2xT (x)−T (x). U (x) = 2xU (x)−U (x).
n+1 n n−1 n+1 n n−1
T (x)= det U (x)= det
n n
   
x 1 2x 1
1 2x 1  1 2x 1 
 ... ... .  ... ... .
   
1 2x 1 2x
(2k−1)π Similarly, the roots of U are
The roots of T are x =cos , n
n k 2n x = cos kπ , k = 1,··· ,n.
k = 1,··· ,n. k n+1
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Grid graphs without symmetry
The reduced laplacian of this grid graph with special firings is
 
A −I ··· 0
n n
−In An −In 
 ... ... ... ... 
 
(cid:52) =  −In An −In 
 ... ... ... ... 
 
 −In An −In
0 ··· −I A
n n
where I is the n×n identity matrix and where the A matrices are of
n n
the following form:
 
4 −1 ··· 0
−1 4 −1 
 ... ... ... ... ,
 
 −1 4 −1
0 ··· −1 3
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