6,903 research outputs found
Pfaffian Correlation Functions of Planar Dimer Covers
The Pfaffian structure of the boundary monomer correlation functions in the
dimer-covering planar graph models is rederived through a combinatorial /
topological argument. These functions are then extended into a larger family of
order-disorder correlation functions which are shown to exhibit Pfaffian
structure throughout the bulk. Key tools involve combinatorial switching
symmetries which are identified through the loop-gas representation of the
double dimer model, and topological implications of planarity.Comment: Revised figures; corrected misprint
Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function
A streamlined derivation of the Kac-Ward formula for the planar Ising model's
partition function is presented and applied in relating the kernel of the
Kac-Ward matrices' inverse with the correlation functions of the Ising model's
order-disorder correlation functions. A shortcut for both is facilitated by the
Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also
extended here to produce a family of non planar interactions on
for which the partition function and the order-disorder correlators are
solvable at special values of the coupling parameters/temperature.Comment: An extension of the Kac-Ward determinantal formula beyond planarity
was added (Section 5). To appear in Journal of Statistical Physic
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Combinatorial RNA Design: Designability and Structure-Approximating Algorithm
In this work, we consider the Combinatorial RNA Design problem, a minimal
instance of the RNA design problem which aims at finding a sequence that admits
a given target as its unique base pair maximizing structure. We provide
complete characterizations for the structures that can be designed using
restricted alphabets. Under a classic four-letter alphabet, we provide a
complete characterization of designable structures without unpaired bases. When
unpaired bases are allowed, we provide partial characterizations for classes of
designable/undesignable structures, and show that the class of designable
structures is closed under the stutter operation. Membership of a given
structure to any of the classes can be tested in linear time and, for positive
instances, a solution can be found in linear time. Finally, we consider a
structure-approximating version of the problem that allows to extend bands
(helices) and, assuming that the input structure avoids two motifs, we provide
a linear-time algorithm that produces a designable structure with at most twice
more base pairs than the input structure.Comment: CPM - 26th Annual Symposium on Combinatorial Pattern Matching, Jun
2015, Ischia Island, Italy. LNCS, 201
Revisiting the combinatorics of the 2D Ising model
We provide a concise exposition with original proofs of combinatorial
formulas for the 2D Ising model partition function, multi-point fermionic
observables, spin and energy density correlations, for general graphs and
interaction constants, using the language of Kac-Ward matrices. We also give a
brief account of the relations between various alternative formalisms which
have been used in the combinatorial study of the planar Ising model: dimers and
Grassmann variables, spin and disorder operators, and, more recently,
s-holomorphic observables. In addition, we point out that these formulas can be
extended to the double-Ising model, defined as a pointwise product of two Ising
spin configurations on the same discrete domain, coupled along the boundary.Comment: Minor change in the notation (definition of eta). 55 pages, 4 figure
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