27 research outputs found
Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs
For a graph , let be the partition function of the
monomer-dimer system defined by , where is the
number of matchings of size in . We consider graphs of bounded degree
and develop a sublinear-time algorithm for estimating at an
arbitrary value within additive error with high
probability. The query complexity of our algorithm does not depend on the size
of and is polynomial in , and we also provide a lower bound
quadratic in for this problem. This is the first analysis of a
sublinear-time approximation algorithm for a # P-complete problem. Our
approach is based on the correlation decay of the Gibbs distribution associated
with . We show that our algorithm approximates the probability
for a vertex to be covered by a matching, sampled according to this Gibbs
distribution, in a near-optimal sublinear time. We extend our results to
approximate the average size and the entropy of such a matching within an
additive error with high probability, where again the query complexity is
polynomial in and the lower bound is quadratic in .
Our algorithms are simple to implement and of practical use when dealing with
massive datasets. Our results extend to other systems where the correlation
decay is known to hold as for the independent set problem up to the critical
activity
Lee-Yang Problems and The Geometry of Multivariate Polynomials
We describe all linear operators on spaces of multivariate polynomials
preserving the property of being non-vanishing in open circular domains. This
completes the multivariate generalization of the classification program
initiated by P\'olya-Schur for univariate real polynomials and provides a
natural framework for dealing in a uniform way with Lee-Yang type problems in
statistical mechanics, combinatorics, and geometric function theory.
This is an announcement with some of the main results in arXiv:0809.0401 and
arXiv:0809.3087.Comment: To appear in Letters in Mathematical Physics; 8 pages, no figures,
LaTeX2
Rigorous results on superconducting ground states for attractive extended Hubbard models
We show that the exact ground state for a class of extended Hubbard models
including bond-charge, exchange, and pair-hopping terms, is the Yang
"eta-paired" state for any non-vanishing value of the pair-hopping amplitude,
at least when the on-site Coulomb interaction is attractive enough and the
remaining physical parameters satisfy a single constraint. The ground state is
thus rigorously superconducting. Our result holds on a bipartite lattice in any
dimension, at any band filling, and for arbitrary electron hopping.Comment: 12 page
Pocket Monte Carlo algorithm for classical doped dimer models
We study the correlations of classical hardcore dimer models doped with
monomers by Monte Carlo simulation. We introduce an efficient cluster
algorithm, which is applicable in any dimension, for different lattices and
arbitrary doping. We use this algorithm for the dimer model on the square
lattice, where a finite density of monomers destroys the critical confinement
of the two-monomer problem. The monomers form a two-component plasma located in
its high-temperature phase, with the Coulomb interaction screened at finite
densities. On the triangular lattice, a single pair of monomers is not
confined. The monomer correlations are extremely short-ranged and hardly change
with doping.Comment: 6 pages, REVTeX
Kosterlitz Thouless Universality in Dimer Models
Using the monomer-dimer representation of strongly coupled U(N) lattice gauge
theories with staggered fermions, we study finite temperature chiral phase
transitions in (2+1) dimensions. A new cluster algorithm allows us to compute
monomer-monomer and dimer-dimer correlations at zero monomer density (chiral
limit) accurately on large lattices. This makes it possible to show
convincingly, for the first time, that these models undergo a finite
temperature phase transition which belongs to the Kosterlitz-Thouless
universality class. We find that this universality class is unaffected even in
the large N limit. This shows that the mean field analysis often used in this
limit breaks down in the critical region.Comment: 4 pages, 4 figure
Antiferromagnetism in the Exact Ground State of the Half Filled Hubbard Model on the Complete-Bipartite Graph
As a prototype model of antiferromagnetism, we propose a repulsive Hubbard
Hamiltonian defined on a graph \L={\cal A}\cup{\cal B} with and bonds connecting any element of with all the
elements of . Since all the hopping matrix elements associated with
each bond are equal, the model is invariant under an arbitrary permutation of
the -sites and/or of the -sites. This is the Hubbard model
defined on the so called -complete-bipartite graph,
() being the number of elements in (). In this
paper we analytically find the {\it exact} ground state for at
half filling for any ; the repulsion has a maximum at a critical
-dependent value of the on-site Hubbard . The wave function and the
energy of the unique, singlet ground state assume a particularly elegant form
for N \ra \inf. We also calculate the spin-spin correlation function and show
that the ground state exhibits an antiferromagnetic order for any non-zero
even in the thermodynamic limit. We are aware of no previous explicit analytic
example of an antiferromagnetic ground state in a Hubbard-like model of
itinerant electrons. The kinetic term induces non-trivial correlations among
the particles and an antiparallel spin configuration in the two sublattices
comes to be energetically favoured at zero Temperature. On the other hand, if
the thermodynamic limit is taken and then zero Temperature is approached, a
paramagnetic behavior results. The thermodynamic limit does not commute with
the zero-Temperature limit, and this fact can be made explicit by the analytic
solutions.Comment: 19 pages, 5 figures .ep
Super-Hubbard models and applications
We construct XX- and Hubbard- like models based on unitary superalgebras
gl(N|M) generalising Shastry's and Maassarani's approach of the algebraic case.
We introduce the R-matrix of the gl(N|M) XX model and that of the Hubbard model
defined by coupling two independent XX models. In both cases, we show that the
R-matrices satisfy the Yang--Baxter equation, we derive the corresponding local
Hamiltonian in the transfer matrix formalism and we determine the symmetry of
the Hamiltonian. Explicit examples are worked out. In the cases of the gl(1|2)
and gl(2|2) Hubbard models, a perturbative calculation at two loops a la Klein
and Seitz is performed.Comment: 26 page
Optimal designs for rational function regression
We consider optimal non-sequential designs for a large class of (linear and
nonlinear) regression models involving polynomials and rational functions with
heteroscedastic noise also given by a polynomial or rational weight function.
The proposed method treats D-, E-, A-, and -optimal designs in a
unified manner, and generates a polynomial whose zeros are the support points
of the optimal approximate design, generalizing a number of previously known
results of the same flavor. The method is based on a mathematical optimization
model that can incorporate various criteria of optimality and can be solved
efficiently by well established numerical optimization methods. In contrast to
previous optimization-based methods proposed for similar design problems, it
also has theoretical guarantee of its algorithmic efficiency; in fact, the
running times of all numerical examples considered in the paper are negligible.
The stability of the method is demonstrated in an example involving high degree
polynomials. After discussing linear models, applications for finding locally
optimal designs for nonlinear regression models involving rational functions
are presented, then extensions to robust regression designs, and trigonometric
regression are shown. As a corollary, an upper bound on the size of the support
set of the minimally-supported optimal designs is also found. The method is of
considerable practical importance, with the potential for instance to impact
design software development. Further study of the optimality conditions of the
main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory
and additional example
On Ising and dimer models in two and three dimensions
Motivated by recent interest in 2+1 dimensional quantum dimer models, we
revisit Fisher's mapping of two dimensional Ising models to hardcore dimer
models. First, we note that the symmetry breaking transition of the
ferromagetic Ising model maps onto a non-symmetry breaking transition in dimer
language -- instead it becomes a deconfinement transition for test monomers.
Next, we introduce a modification of Fisher's mapping in which a second dimer
model, also equivalent to the Ising model, is defined on a generically
different lattice derived from the dual. In contrast to Fisher's original
mapping, this enables us to reformulate frustrated Ising models as dimer models
with positive weights and we illustrate this by providing a new solution of the
fully frustrated Ising model on the square lattice. Finally, by means of the
modified mapping we show that a large class of three-dimensional Ising models
are precisely equivalent, in the time continuum limit, to particular quantum
dimer models. As Ising models in three dimensions are dual to Ising gauge
theories, this further yields an exact map between the latter and the quantum
dimer models. The paramagnetic phase in Ising language maps onto a deconfined,
topologically ordered phase in the dimer models. Using this set of ideas, we
also construct an exactly soluble quantum eight vertex model.Comment: 10 pages, 9 figures autmatically include
The Lee-Yang and P\'olya-Schur Programs. I. Linear Operators Preserving Stability
In 1952 Lee and Yang proposed the program of analyzing phase transitions in
terms of zeros of partition functions. Linear operators preserving
non-vanishing properties are essential in this program and various contexts in
complex analysis, probability theory, combinatorics, and matrix theory. We
characterize all linear operators on finite or infinite-dimensional spaces of
multivariate polynomials preserving the property of being non-vanishing
whenever the variables are in prescribed open circular domains. In particular,
this solves the higher dimensional counterpart of a long-standing
classification problem originating from classical works of Hermite, Laguerre,
Hurwitz and P\'olya-Schur on univariate polynomials with such properties.Comment: Final version, to appear in Inventiones Mathematicae; 27 pages, no
figures, LaTeX2