1,004 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
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