6,962 research outputs found
Isomorphism Checking for Symmetry Reduction
In this paper, we show how isomorphism checking can be used as an effective technique for symmetry reduction. Reduced state spaces are equivalent to the original ones under a strong notion of bisimilarity which preserves the multiplicity of outgoing transitions, and therefore also preserves stochastic temporal logics. We have implemented this in a setting where states are arbitrary graphs. Since no efficiently computable canonical representation is known for arbitrary graphs modulo isomorphism, we define an isomorphism-predicting hash function on the basis of an existing partition refinement algorithm. As an example, we report a factorial state space reduction on a model of an ad-hoc network connectivity protocol
Worm Monte Carlo study of the honeycomb-lattice loop model
We present a Markov-chain Monte Carlo algorithm of "worm"type that correctly
simulates the O(n) loop model on any (finite and connected) bipartite cubic
graph, for any real n>0, and any edge weight, including the fully-packed limit
of infinite edge weight. Furthermore, we prove rigorously that the algorithm is
ergodic and has the correct stationary distribution. We emphasize that by using
known exact mappings when n=2, this algorithm can be used to simulate a number
of zero-temperature Potts antiferromagnets for which the Wang-Swendsen-Kotecky
cluster algorithm is non-ergodic, including the 3-state model on the
kagome-lattice and the 4-state model on the triangular-lattice. We then use
this worm algorithm to perform a systematic study of the honeycomb-lattice loop
model as a function of n<2, on the critical line and in the densely-packed and
fully-packed phases. By comparing our numerical results with Coulomb gas
theory, we identify the exact scaling exponents governing some fundamental
geometric and dynamic observables. In particular, we show that for all n<2, the
scaling of a certain return time in the worm dynamics is governed by the
magnetic dimension of the loop model, thus providing a concrete dynamical
interpretation of this exponent. The case n>2 is also considered, and we
confirm the existence of a phase transition in the 3-state Potts universality
class that was recently observed via numerical transfer matrix calculations.Comment: 33 pages, 12 figure
Cyclic Coloring of Plane Graphs with Maximum Face Size 16 and 17
Plummer and Toft conjectured in 1987 that the vertices of every 3-connected
plane graph with maximum face size D can be colored using at most D+2 colors in
such a way that no face is incident with two vertices of the same color. The
conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for
D=16 and D=17
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Two-dimensional O(n) model in a staggered field
Nienhuis' truncated O(n) model gives rise to a model of self-avoiding loops
on the hexagonal lattice, each loop having a fugacity of n. We study such loops
subjected to a particular kind of staggered field w, which for n -> infinity
has the geometrical effect of breaking the three-phase coexistence, linked to
the three-colourability of the lattice faces. We show that at T = 0, for w > 1
the model flows to the ferromagnetic Potts model with q=n^2 states, with an
associated fragmentation of the target space of the Coulomb gas. For T>0, there
is a competition between T and w which gives rise to multicritical versions of
the dense and dilute loop universality classes. Via an exact mapping, and
numerical results, we establish that the latter two critical branches coincide
with those found earlier in the O(n) model on the triangular lattice. Using
transfer matrix studies, we have found the renormalisation group flows in the
full phase diagram in the (T,w) plane, with fixed n. Superposing three
copies of such hexagonal-lattice loop models with staggered fields produces a
variety of one or three-species fully-packed loop models on the triangular
lattice with certain geometrical constraints, possessing integer central
charges 0 <= c <= 6. In particular we show that Benjamini and Schramm's RGB
loops have fractal dimension D_f = 3/2.Comment: 40 pages, 17 figure
Markov chain sampling of the loop models on the infinite plane
It was recently proposed in
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.043322 [Herdeiro &
Doyon Phys.,Rev.,E (2016)] a numerical method showing a precise sampling of the
infinite plane 2d critical Ising model for finite lattice subsections. The
present note extends the method to a larger class of models, namely the
loop gas models for . We argue that even though the Gibbs measure
is non local, it is factorizable on finite subsections when sufficient
information on the loops touching the boundaries is stored. Our results attempt
to show that provided an efficient Markov chain mixing algorithm and an
improved discrete lattice dilation procedure the planar limit of the
models can be numerically studied with efficiency similar to the Ising case.
This confirms that scale invariance is the only requirement for the present
numerical method to work.Comment: v2: added conclusion section, changes in introduction and appendice
Percolation on self-dual polygon configurations
Recently, Scullard and Ziff noticed that a broad class of planar percolation
models are self-dual under a simple condition that, in a parametrized version
of such a model, reduces to a single equation. They state that the solution of
the resulting equation gives the critical point. However, just as in the
classical case of bond percolation on the square lattice, self-duality is
simply the starting point: the mathematical difficulty is precisely showing
that self-duality implies criticality. Here we do so for a generalization of
the models considered by Scullard and Ziff. In these models, the states of the
bonds need not be independent; furthermore, increasing events need not be
positively correlated, so new techniques are needed in the analysis. The main
new ingredients are a generalization of Harris's Lemma to products of partially
ordered sets, and a new proof of a type of Russo-Seymour-Welsh Lemma with
minimal symmetry assumptions.Comment: Expanded; 73 pages, 24 figure
- âŠ