37 research outputs found
Fermions and Loops on Graphs. II. Monomer-Dimer Model as Series of Determinants
We continue the discussion of the fermion models on graphs that started in
the first paper of the series. Here we introduce a Graphical Gauge Model (GGM)
and show that : (a) it can be stated as an average/sum of a determinant defined
on the graph over (binary) gauge field; (b) it is equivalent
to the Monomer-Dimer (MD) model on the graph; (c) the partition function of the
model allows an explicit expression in terms of a series over disjoint directed
cycles, where each term is a product of local contributions along the cycle and
the determinant of a matrix defined on the remainder of the graph (excluding
the cycle). We also establish a relation between the MD model on the graph and
the determinant series, discussed in the first paper, however, considered using
simple non-Belief-Propagation choice of the gauge. We conclude with a
discussion of possible analytic and algorithmic consequences of these results,
as well as related questions and challenges.Comment: 11 pages, 2 figures; misprints correcte
Boundary monomers in the dimer model
The correlation functions of an arbitrary number of boundary monomers in the
system of close-packed dimers on the square lattice are computed exactly in the
scaling limit. The equivalence of the 2n-point correlation functions with those
of a complex free fermion is proved, thereby reinforcing the description of the
monomer-dimer model by a conformal free field theory with central charge c=1.Comment: 15 pages, 2 figure
Fermions and Loops on Graphs. I. Loop Calculus for Determinant
This paper is the first in the series devoted to evaluation of the partition
function in statistical models on graphs with loops in terms of the
Berezin/fermion integrals. The paper focuses on a representation of the
determinant of a square matrix in terms of a finite series, where each term
corresponds to a loop on the graph. The representation is based on a fermion
version of the Loop Calculus, previously introduced by the authors for
graphical models with finite alphabets. Our construction contains two levels.
First, we represent the determinant in terms of an integral over anti-commuting
Grassman variables, with some reparametrization/gauge freedom hidden in the
formulation. Second, we show that a special choice of the gauge, called BP
(Bethe-Peierls or Belief Propagation) gauge, yields the desired loop
representation. The set of gauge-fixing BP conditions is equivalent to the
Gaussian BP equations, discussed in the past as efficient (linear scaling)
heuristics for estimating the covariance of a sparse positive matrix.Comment: 11 pages, 1 figure; misprints correcte
A Pfaffian formula for monomer-dimer partition functions
We consider the monomer-dimer partition function on arbitrary finite planar
graphs and arbitrary monomer and dimer weights, with the restriction that the
only non-zero monomer weights are those on the boundary. We prove a Pfaffian
formula for the corresponding partition function. As a consequence of this
result, multipoint boundary monomer correlation functions at close packing are
shown to satisfy fermionic statistics. Our proof is based on the celebrated
Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the
authors, and a careful labeling and directing procedure of the vertices and
edges of the graph.Comment: Added referenc
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Mini-Workshop: Dimers, Ising and Spanning Trees beyond the Critical Isoradial Case (online meeting)
The goal of this mini-workshop is to gather specialists of the dimer, Ising and spanning tree models around recent and ongoing progress in two directions. One is understanding the connection to the spectral curve of these models in the cases when the curve has positive genus. The other is the introduction of universal embeddings associated to these models. We aim to use these new tools to progress in the study of scaling limits
Exact solution of the dimer model: Corner free energy, correlation functions and combinatorics
In this work, some classical results of the pfaffian theory of the dimer
model based on the work of Kasteleyn, Fisher and Temperley are introduced in a
fermionic framework. Then we shall detail the bosonic formulation of the model
{\it via} the so-called height mapping and the nature of boundary conditions is
unravelled. The complete and detailed fermionic solution of the dimer model on
the square lattice with an arbitrary number of monomers is presented, and
finite size effect analysis is performed to study surface and corner effects,
leading to the extrapolation of the central charge of the model. The solution
allows for exact calculations of monomer and dimer correlation functions in the
discrete level and the scaling behavior can be inferred in order to find the
set of scaling dimensions and compare to the bosonic theory which predict
particular features concerning corner behaviors. Finally, some combinatorial
and numerical properties of partition functions with boundary monomers are
discussed, proved and checked with enumeration algorithms.Comment: Final version to be published in Nuclear Physics B (53 pages and a
lot of figures
Dulmage-Mendelsohn percolation: Geometry of maximally-packed dimer models and topologically-protected zero modes on diluted bipartite lattices
The classic combinatorial construct of {\em maximum matchings} probes the
random geometry of regions with local sublattice imbalance in a site-diluted
bipartite lattice. We demonstrate that these regions, which host the monomers
of any maximum matching of the lattice, control the localization properties of
a zero-energy quantum particle hopping on this lattice. The structure theory of
Dulmage and Mendelsohn provides us a way of identifying a complete and
non-overlapping set of such regions. This motivates our large-scale
computational study of the Dulmage-Mendelsohn decomposition of site-diluted
bipartite lattices in two and three dimensions. Our computations uncover an
interesting universality class of percolation associated with the end-to-end
connectivity of such monomer-carrying regions with local sublattice imbalance,
which we dub {\em Dulmage-Mendelsohn percolation}. Our results imply the
existence of a monomer percolation transition in the classical statistical
mechanics of the associated maximally-packed dimer model and the existence of a
phase with area-law entanglement entropy of arbitrary many-body eigenstates of
the corresponding quantum dimer model. They also have striking implications for
the nature of collective zero-energy Majorana fermion excitations of bipartite
networks of Majorana modes localized on sites of diluted lattices, for the
character of topologically-protected zero-energy wavefunctions of the bipartite
random hopping problem on such lattices, and thence for the corresponding
quantum percolation problem, and for the nature of low-energy magnetic
excitations in bipartite quantum antiferromagnets diluted by a small density of
nonmagnetic impurities.Comment: minor typos and errors fixed; further clarifications added. no
substantive changes in result