4 research outputs found
Approximability of the Eight-Vertex Model
We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics.
We introduce a quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on 4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold
Counting Perfect Matchings and the Eight-Vertex Model
We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in [Jin-Yi Cai et al., 2018].
In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In another region of the parameter space which is larger than the region that is previously known to admit Fully Polynomial Randomized Approximation Scheme (FPRAS), we show that computing the partition function can be reduced to counting perfect matchings (which is valid for both exact and approximate counting). Moreover, we give a complete characterization of nonnegatively weighted (not necessarily planar) 4-ary matchgates, which has been open for several years. The key ingredient of our proof is a geometric lemma.
We also identify a region of the parameter space where approximating the partition function on planar 4-regular graphs is feasible but on general 4-regular graphs is equivalent to approximately counting perfect matchings. To our best knowledge, these are the first problems that exhibit this dichotomic behavior between the planar and the nonplanar settings in approximate counting
From logarithmic delocalization of the six-vertex height function under sloped boundary conditions to weakened crossing probability estimates for the Ashkin-Teller, generalized random-cluster, and -cubic models
To obtain Russo-Seymour-Welsh estimates for the height function of the
six-vertex model under sloped boundary conditions, which can be leveraged to
demonstrate that the height function logarithmically delocalizes under a
broader class of boundary conditions, we formulate crossing probability
estimates in strips of the square lattice and the cylinder, for parameters
satisfying , , and ,
in which each of the first two conditions respectively relate to invariance
under vertical and diagonal reflections enforced through the symmetry for domains in strips of the square lattice, and satisfaction of
FKG, for the height function and for its absolute value. To determine whether
arguments for estimating crossing probabilities of the height function for flat
boundary conditions from a recent work due to Duminil-Copin, Karila, Manolescu,
and Oulamara remain applicable for sloped boundary conditions, from the set of
possible slopes given by the rational points from , we
analyze sloped Gibbs states, which do not have infinitely many disjointly
oriented circuits. In comparison to Russo-Seymour-Welsh arguments for flat
boundary conditions, arguments for sloped boundary conditions present
additional complications for both planar and cylindrical settings, in which
crossing events that are considered in the strip, and then extended to the
annulus and cylinder, must be maintained across rectangles of large aspect
ratio, in spite of the fact that some proportion of faces within the strip
freeze with positive probability. Once considerations have been raised for
estimating crossing probabilities in the strip, and in the cylinder, which
relate to the fraction of points about which cylindrical domains are situated
that are deemed to satisfy the \textit{good} property, we investigate other
models of interest.Comment: 171 pages, 51 figure