730 research outputs found
Holographic Algorithm with Matchgates Is Universal for Planar CSP Over Boolean Domain
We prove a complexity classification theorem that classifies all counting
constraint satisfaction problems (CSP) over Boolean variables into exactly
three categories: (1) Polynomial-time tractable; (2) P-hard for general
instances, but solvable in polynomial-time over planar graphs; and (3)
P-hard over planar graphs. The classification applies to all sets of local,
not necessarily symmetric, constraint functions on Boolean variables that take
complex values. It is shown that Valiant's holographic algorithm with
matchgates is a universal strategy for all problems in category (2).Comment: 94 page
Susceptibility amplitude ratios in the two-dimensional Potts model and percolation
The high-temperature susceptibility of the -state Potts model behaves as
as , while for one may define
both longitudinal and transverse susceptibilities, with the same power law but
different amplitudes and . We extend a previous analytic
calculation of the universal ratio in two dimensions to the
low-temperature ratio , and test both predictions with Monte
Carlo simulations for and 4. The data for are inconclusive owing to
large corrections to scaling, while for they appear consistent with the
prediction for , but not with that for . A
simple extrapolation of our analytic results to indicates a similar
discrepancy with the corresponding measured quantities in percolation. We point
out that stronger assumptions were made in the derivation of the ratio
, and our work suggests that these may be unjustified.Comment: 17 pages, late
Interacting dimers on the honeycomb lattice: An exact solution of the five-vertex model
The problem of close-packed dimers on the honeycomb lattice was solved by
Kasteleyn in 1963. Here we extend the solution to include interactions between
neighboring dimers in two spatial lattice directions. The solution is obtained
by using the method of Bethe ansatz and by converting the dimer problem into a
five-vertex problem. The complete phase diagram is obtained and it is found
that a new frozen phase, in which the attracting dimers prevail, arises when
the interaction is attractive. For repulsive dimer interactions a new
first-order line separating two frozen phases occurs. The transitions are
continuous and the critical behavior in the disorder regime is found to be the
same as in the case of noninteracting dimers characterized by a specific heat
exponent \a=1/2.Comment: latex, 29 pages + 7 figure
Counting dimer coverings on self-similar Schreier graphs
We study partition functions for the dimer model on families of finite graphs
converging to infinite self-similar graphs and forming approximation sequences
to certain well-known fractals. The graphs that we consider are provided by
actions of finitely generated groups by automorphisms on rooted trees, and thus
their edges are naturally labeled by the generators of the group. It is thus
natural to consider weight functions on these graphs taking different values
according to the labeling. We study in detail the well-known example of the
Hanoi Towers group , closely related to the Sierpi\'nski gasket.Comment: 29 pages. Final version, to appear in European Journal of
Combinatoric
Dimer statistics on the M\"obius strip and the Klein bottle
Closed-form expressions are obtained for the generating function of
close-packed dimers on a simple quartic lattice embedded on a
M\"obius strip and a Klein bottle. Finite-size corrections are also analyzed
and compared with those under cylindrical and free boundary conditions.
Particularly, it is found that, for large lattices of the same size and with a
square symmetry, the number of dimer configurations on a M\"obius strip is
70.2% of that on a cylinder. We also establish two identities relating dimer
generating functions for M\"obius strips and cylinders.Comment: 12 pages, 2 figs included, accepted by Phys. Lett.
Equilibriumlike invaded cluster algorithm: critical exponents and dynamical properties
We present a detailed study of the Equilibriumlike invaded cluster algorithm
(EIC), recently proposed as an extension of the invaded cluster (IC) algorithm,
designed to drive the system to criticality while still preserving the
equilibrium ensemble. We perform extensive simulations on two special cases of
the Potts model and examine the precision of critical exponents by including
the leading corrections. We show that both thermal and magnetic critical
exponents can be obtained with high accuracy compared to the best available
results. The choice of the auxiliary parameters of the algorithm is discussed
in context of dynamical properties. We also discuss the relation to the
Li-Sokal bound for the dynamical exponent .Comment: 11 pages, 13 figures, accepted for publication in Phys. Rev.
Single-cluster dynamics for the random-cluster model
We formulate a single-cluster Monte Carlo algorithm for the simulation of the
random-cluster model. This algorithm is a generalization of the Wolff
single-cluster method for the -state Potts model to non-integer values
. Its results for static quantities are in a satisfactory agreement with
those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which
involves a full cluster decomposition of random-cluster configurations. We
explore the critical dynamics of this algorithm for several two-dimensional
Potts and random-cluster models. For integer , the single-cluster algorithm
can be reduced to the Wolff algorithm, for which case we find that the
autocorrelation functions decay almost purely exponentially, with dynamic
exponents , and for , and
4 respectively. For non-integer , the dynamical behavior of the
single-cluster algorithm appears to be very dissimilar to that of the SWCM
algorithm. For large critical systems, the autocorrelation function displays a
range of power-law behavior as a function of time. The dynamic exponents are
relatively large. We provide an explanation for this peculiar dynamic behavior.Comment: 7 figures, 4 table
Asymptotic energy of graphs
The energy of a simple graph arising in chemical physics, denoted by
, is defined as the sum of the absolute values of eigenvalues of
. We consider the asymptotic energy per vertex (say asymptotic energy) for
lattice systems. In general for a type of lattice in statistical physics, to
compute the asymptotic energy with toroidal, cylindrical, Mobius-band,
Klein-bottle, and free boundary conditions are different tasks with different
hardness. In this paper, we show that if is a sequence of finite
simple graphs with bounded average degree and a sequence of spanning
subgraphs of such that almost all vertices of and have
the same degrees, then and have the same asymptotic energy. Thus,
for each type of lattices with toroidal, cylindrical, Mobius-band,
Klein-bottle, and free boundary conditions, we have the same asymptotic energy.
As applications, we obtain the asymptotic formulae of energies per vertex of
the triangular, , and hexagonal lattices with toroidal, cylindrical,
Mobius-band, Klein-bottle, and free boundary conditions simultaneously.Comment: 15 pages, 3 figure
Quadri-tilings of the plane
We introduce {\em quadri-tilings} and show that they are in bijection with
dimer models on a {\em family} of graphs arising from rhombus
tilings. Using two height functions, we interpret a sub-family of all
quadri-tilings, called {\em triangular quadri-tilings}, as an interface model
in dimension 2+2. Assigning "critical" weights to edges of , we prove an
explicit expression, only depending on the local geometry of the graph ,
for the minimal free energy per fundamental domain Gibbs measure; this solves a
conjecture of \cite{Kenyon1}. We also show that when edges of are
asymptotically far apart, the probability of their occurrence only depends on
this set of edges. Finally, we give an expression for a Gibbs measure on the
set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs
measures, and conjecture it to be that of minimal free energy per fundamental
domain.Comment: Revised version, minor changes. 30 pages, 13 figure
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