463 research outputs found
Spectral Curve of Periodic Fisher Graphs
We study the spectral curves of dimer models on periodic Fisher graphs,
obtained from a ferromagnetic Ising model on . The spectral curve
is defined by the zero locus of the determinant of a modified weighted
adjacency matrix. We prove that either they are disjoint from the unit torus
() or they intersect at a
single real point.Comment: 27 page
Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices
We study asymptotics of perfect matchings on a large class of graphs called
the contracting square-hexagon lattice, which is constructed row by row from
either a row of a square grid or a row of a hexagonal lattice. We assign the
graph periodic edge weights with period , and consider the
probability measure of perfect matchings in which the probability of each
configuration is proportional to the product of edge weights. We show that the
partition function of perfect matchings on such a graph can be computed
explicitly by a Schur function depending on the edge weights. By analyzing the
asymptotics of the Schur function, we then prove the Law of Large Numbers
(limit shape) and the Central Limit Theorem (convergence to the Gaussian free
field) for the corresponding height functions. We also show that the
distribution of certain type of dimers near the turning corner is the same as
the eigenvalues of Gaussian Unitary Ensemble, and that in the scaling limit
under the boundary condition that each segment of the bottom boundary grows
linearly with respect the dimension of the graph, the frozen boundary is a
cloud curve whose number of tangent points to the bottom boundary of the domain
depends on the size of the period, as well as the number of segments along the
bottom boundary
Self-avoiding walks and amenability
The connective constant of an infinite transitive graph is the
exponential growth rate of the number of self-avoiding walks from a given
origin. The relationship between connective constants and amenability is
explored in the current work.
Various properties of connective constants depend on the existence of
so-called 'graph height functions', namely: (i) whether is a local
function on certain graphs derived from , (ii) the equality of and
the asymptotic growth rate of bridges, and (iii) whether there exists a
terminating algorithm for approximating to a given degree of accuracy.
In the context of amenable groups, it is proved that the Cayley graphs of
infinite, finitely generated, elementary amenable groups support graph height
functions, which are in addition harmonic. In contrast, the Cayley graph of the
Grigorchuk group, which is amenable but not elementary amenable, does not have
a graph height function.
In the context of non-amenable, transitive graphs, a lower bound is presented
for the connective constant in terms of the spectral bottom of the graph. This
is a strengthening of an earlier result of the same authors. Secondly, using a
percolation inequality of Benjamini, Nachmias, and Peres, it is explained that
the connective constant of a non-amenable, transitive graph with large girth is
close to that of a regular tree. Examples are given of non-amenable groups
without graph height functions, of which one is the Higman group.Comment: v2 differs from v1 in the inclusion of further material concerning
non-amenable graphs, notably an improved lower bound for the connective
constan
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