1,125 research outputs found
The bead model and limit behaviors of dimer models
In this paper, we study the bead model: beads are threaded on a set of wires
on the plane represented by parallel straight lines. We add the constraint that
between two consecutive beads on a wire; there must be exactly one bead on each
neighboring wire. We construct a one-parameter family of Gibbs measures on the
bead configurations that are uniform in a certain sense. When endowed with one
of these measures, this model is shown to be a determinantal point process,
whose marginal on each wire is the sine process (given by eigenvalues of large
hermitian random matrices). We prove then that this process appears as a limit
of any dimer model on a planar bipartite graph when some weights degenerate.Comment: Published in at http://dx.doi.org/10.1214/08-AOP398 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices
We study asymptotics of perfect matchings on a large class of graphs called
the contracting square-hexagon lattice, which is constructed row by row from
either a row of a square grid or a row of a hexagonal lattice. We assign the
graph periodic edge weights with period , and consider the
probability measure of perfect matchings in which the probability of each
configuration is proportional to the product of edge weights. We show that the
partition function of perfect matchings on such a graph can be computed
explicitly by a Schur function depending on the edge weights. By analyzing the
asymptotics of the Schur function, we then prove the Law of Large Numbers
(limit shape) and the Central Limit Theorem (convergence to the Gaussian free
field) for the corresponding height functions. We also show that the
distribution of certain type of dimers near the turning corner is the same as
the eigenvalues of Gaussian Unitary Ensemble, and that in the scaling limit
under the boundary condition that each segment of the bottom boundary grows
linearly with respect the dimension of the graph, the frozen boundary is a
cloud curve whose number of tangent points to the bottom boundary of the domain
depends on the size of the period, as well as the number of segments along the
bottom boundary
When rule-based models need to count
Rule-based modelers dislike direct enumeration of cases when more efficient
means of enumeration are available. We present an extension of the Kappa
language which attaches to agents a notion of level. We detail two encodings
that are more concise than the former practice
Le contrat notionnel : efficience et causalité.
The main topic of this paper is about the overall functionnning of the french futures bonds market. Three mains questions are asked: (1) Is the futures market inefficient? (2) Is there any free lunch between the future market and the spot market? (3) Does one of these markets play a leader role? Econometric tests give globally an negative answer to all these three questions. The arbitrage's opportunities which have been detected in the period 86.12-90.06 can be explained by the restrictions to sell-short bonds in the french cash market. The two tests on the rationality of the expectations which have been undertaken (a time series test and a panel data test) yield the same conclusion : one cannot reject the assumption that the market is efficient. Finally, tests of causality between the future market and the cash market on daily data shows strong instantaneous correlation between them. Concerning delayed causality, the future market have played a leader role during the period 90.01-93.06 during which one has observed an overall stability of the futures prices.Futures markets ; Efficiency ; Causality.
The critical Z-invariant Ising model via dimers: the periodic case
We study a large class of critical two-dimensional Ising models namely
critical Z-invariant Ising models on periodic graphs, example of which are the
classical square, triangular and honeycomb lattice at the critical temperature.
Fisher introduced a correspondence between the Ising model and the dimer model
on a decorated graph, thus setting dimer techniques as a powerful tool for
understanding the Ising model. In this paper, we give a full description of the
dimer model corresponding to the critical Z-invariant Ising model. We prove
that the dimer characteristic polynomial is equal (up to a constant) to the
critical Laplacian characteristic polynomial, and defines a Harnack curve of
genus 0. We prove an explicit expression for the free energy, and for the Gibbs
measure obtained as weak limit of Boltzmann measures.Comment: 35 pages, 8 figure
Height representation of XOR-Ising loops via bipartite dimers
The XOR-Ising model on a graph consists of random spin configurations on
vertices of the graph obtained by taking the product at each vertex of the
spins of two independent Ising models. In this paper, we explicitly relate loop
configurations of the XOR-Ising model and those of a dimer model living on a
decorated, bipartite version of the Ising graph. This result is proved for
graphs embedded in compact surfaces of genus g.
Using this fact, we then prove that XOR-Ising loops have the same law as
level lines of the height function of this bipartite dimer model. At
criticality, the height function is known to converge weakly in distribution to
a Gaussian free field.
As a consequence, results of this paper shed a light on the occurrence of the
Gaussian free field in the XOR-Ising model. In particular, they prove a
discrete analogue of Wilson's conjecture, stating that the scaling limit of
XOR-Ising loops are "contour lines" of the Gaussian free field.Comment: 41 pages, 10 figure
Loop statistics in the toroidal honeycomb dimer model
The dimer model on a graph embedded in the torus can be interpreted as a
collection of random self-avoiding loops. In this paper, we consider the
uniform toroidal honeycomb dimer model. We prove that when the mesh of the
graph tends to zero and the aspect of the torus is fixed, the winding number of
the collection of loops converges in law to a two-dimensional discrete Gaussian
distribution. This is known to physicists in more generality from their
analysis of toroidal two-dimensional critical loop models and their mapping to
the massless free field on the torus. This paper contains the first
mathematical proof of this more general physics result in the specific case of
the loop model induced by a toroidal dimer model.Comment: Published in at http://dx.doi.org/10.1214/09-AOP453 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Statistical mechanics on isoradial graphs
Isoradial graphs are a natural generalization of regular graphs which give,
for many models of statistical mechanics, the right framework for studying
models at criticality. In this survey paper, we first explain how isoradial
graphs naturally arise in two approaches used by physicists: transfer matrices
and conformal field theory. This leads us to the fact that isoradial graphs
provide a natural setting for discrete complex analysis, to which we dedicate
one section. Then, we give an overview of explicit results obtained for
different models of statistical mechanics defined on such graphs: the critical
dimer model when the underlying graph is bipartite, the 2-dimensional critical
Ising model, random walk and spanning trees and the q-state Potts model.Comment: 22 page
New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized
The definitional equality of an intensional type theory is its test of type
compatibility. Today's systems rely on ordinary evaluation semantics to compare
expressions in types, frustrating users with type errors arising when
evaluation fails to identify two `obviously' equal terms. If only the machine
could decide a richer theory! We propose a way to decide theories which
supplement evaluation with `-rules', rearranging the neutral parts of
normal forms, and report a successful initial experiment.
We study a simple -calculus with primitive fold, map and append operations on
lists and develop in Agda a sound and complete decision procedure for an
equational theory enriched with monoid, functor and fusion laws
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