10,882 research outputs found
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
The Bernardi process and torsor structures on spanning trees
Let G be a ribbon graph, i.e., a connected finite graph G together with a
cyclic ordering of the edges around each vertex. By adapting a construction due
to O. Bernardi, we associate to any pair (v,e) consisting of a vertex v and an
edge e adjacent to v a bijection between spanning trees of G and elements of
the set Pic^g(G) of degree g divisor classes on G, where g is the genus of G.
Using the natural action of the Picard group Pic^0(G) on Pic^g(G), we show that
the Bernardi bijection gives rise to a simply transitive action \beta_v of
Pic^0(G) on the set of spanning trees which does not depend on the choice of e.
A plane graph has a natural ribbon structure (coming from the
counterclockwise orientation of the plane), and in this case we show that
\beta_v is independent of v as well. Thus for plane graphs, the set of spanning
trees is naturally a torsor for the Picard group. Conversely, we show that if
\beta_v is independent of v then G together with its ribbon structure is
planar. We also show that the natural action of Pic^0(G) on spanning trees of a
plane graph is compatible with planar duality.
These findings are formally quite similar to results of Holroyd et al. and
Chan-Church-Grochow, who used rotor-routing to construct an action r_v of
Pic^0(G) on the spanning trees of a ribbon graph G, which they show is
independent of v if and only if G is planar. It is therefore natural to ask how
the two constructions are related. We prove that \beta_v = r_v for all vertices
v of G when G is a planar ribbon graph, i.e. the two torsor structures
(Bernardi and rotor-routing) on the set of spanning trees coincide. In
particular, it follows that the rotor-routing torsor is compatible with planar
duality. We conjecture that for every non-planar ribbon graph G, there exists a
vertex v with \beta_v \neq r_v.Comment: 25 pages. v2: numerous revisions based on referee comments. v3:
substantial additional revisions; final version to appear in IMR
Sandpiles, spanning trees, and plane duality
Let G be a connected, loopless multigraph. The sandpile group of G is a
finite abelian group associated to G whose order is equal to the number of
spanning trees in G. Holroyd et al. used a dynamical process on graphs called
rotor-routing to define a simply transitive action of the sandpile group of G
on its set of spanning trees. Their definition depends on two pieces of
auxiliary data: a choice of a ribbon graph structure on G, and a choice of a
root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon
graph, it has a canonical rotor-routing action associated to it, i.e., the
rotor-routing action is actually independent of the choice of root vertex.
It is well-known that the spanning trees of a planar graph G are in canonical
bijection with those of its planar dual G*, and furthermore that the sandpile
groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing
actions, of the sandpile group of G on its spanning trees, and of the sandpile
group of G* on its spanning trees, compatible under plane duality? In this
paper, we give an affirmative answer to this question, which had been
conjectured by Baker.Comment: 13 pages, 9 figure
Critical Ising model and spanning trees partition functions
We prove that the squared partition function of the two-dimensional critical
Ising model defined on a finite, isoradial graph , is equal to
times the partition function of spanning trees of the graph
, where is the graph extended along the boundary; edges
of are assigned Kenyon's [Ken02] critical weights, and boundary edges of
have specific weights. The proof is an explicit construction,
providing a new relation on the level of configurations between two classical,
critical models of statistical mechanics.Comment: 38 pages, 26 figure
The looping rate and sandpile density of planar graphs
We give a simple formula for the looping rate of loop-erased random walk on a
finite planar graph. The looping rate is closely related to the expected amount
of sand in a recurrent sandpile on the graph. The looping rate formula is
well-suited to taking limits where the graph tends to an infinite lattice, and
we use it to give an elementary derivation of the (previously computed) looping
rate and sandpile densities of the square, triangular, and honeycomb lattices,
and compute (for the first time) the looping rate and sandpile densities of
many other lattices, such as the kagome lattice, the dice lattice, and the
truncated hexagonal lattice (for which the values are all rational), and the
square-octagon lattice (for which it is transcendental)
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