76 research outputs found
Conformal loop ensembles and the stress-energy tensor
We give a construction of the stress-energy tensor of conformal field theory
(CFT) as a local "object" in conformal loop ensembles CLE_\kappa, for all
values of \kappa in the dilute regime 8/3 < \kappa <= 4 (corresponding to the
central charges 0 < c <= 1, and including all CFT minimal models). We provide a
quick introduction to CLE, a mathematical theory for random loops in simply
connected domains with properties of conformal invariance, developed by
Sheffield and Werner (2006). We consider its extension to more general regions
of definition, and make various hypotheses that are needed for our construction
and expected to hold for CLE in the dilute regime. Using this, we identify the
stress-energy tensor in the context of CLE. This is done by deriving its
associated conformal Ward identities for single insertions in CLE probability
functions, along with the appropriate boundary conditions on simply connected
domains; its properties under conformal maps, involving the Schwarzian
derivative; and its one-point average in terms of the "relative partition
function." Part of the construction is in the same spirit as, but widely
generalizes, that found in the context of SLE_{8/3} by the author, Riva and
Cardy (2006), which only dealt with the case of zero central charge in simply
connected hyperbolic regions. We do not use the explicit construction of the
CLE probability measure, but only its defining and expected general properties.Comment: 49 pages, 3 figures. This is a concatenated, reduced and simplified
version of arXiv:0903.0372 and (especially) arXiv:0908.151
Trivial, Critical and Near-critical Scaling Limits of Two-dimensional Percolation
It is natural to expect that there are only three possible types of scaling
limits for the collection of all percolation interfaces in the plane: (1) a
trivial one, consisting of no curves at all, (2) a critical one, in which all
points of the plane are surrounded by arbitrarily large loops and every
deterministic point is almost surely surrounded by a countably infinite family
of nested loops with radii going to zero, and (3) an intermediate one, in which
every deterministic point of the plane is almost surely surrounded by a largest
loop and by a countably infinite family of nested loops with radii going to
zero. We show how one can prove this using elementary arguments, with the help
of known scaling relations for percolation.
The trivial limit corresponds to subcritical and supercritical percolation,
as well as to the case when the density p approaches the critical probability,
p_c, sufficiently slowly as the lattice spacing is sent to zero. The second
type corresponds to critical percolation and to a faster approach of p to p_c.
The third, or near-critical, type of limit corresponds to an intermediate speed
of approach of p to p_c. The fact that in the near-critical case a
deterministic point is a.s. surrounded by a largest loop demonstrates the
persistence of a macroscopic correlation length in the scaling limit and the
absence of scale invariance.Comment: 15 pages, 3 figure
Free and Open Source Software underpinning the European Forest Data Centre
Worldwide, governments are growingly focusing on free and open source software (FOSS) as a move toward transparency and the freedom to run, copy, study, change and improve the software. The European Commission (EC) is also supporting the development of FOSS [...]. In addition to the financial savings, FOSS contributes to scientific knowledge freedom in computational science (CS) and is increasingly rewarded in the science-policy interface within the emerging paradigm of open science. Since complex computational science applications may be affected by software uncertainty, FOSS may help to mitigate part of the impact of software errors by CS community- driven open review, correction and evolution of scientific code. The continental scale of EC science-based policy support implies wide networks of scientific collaboration. Thematic information systems also may benefit from this approach within reproducible integrated modelling. This is supported by the EC strategy on FOSS: "for the development of new information systems, where deployment is foreseen by parties outside of the EC infrastructure, [F]OSS will be the preferred choice and in any case used whenever possible". The aim of this contribution is to highlight how a continental scale information system may exploit and integrate FOSS technologies within the transdisciplinary research underpinning such a complex system. A European example is discussed where FOSS innervates both the structure of the information system itself and the inherent transdisciplinary research for modelling the data and information which constitute the system content. [...
Dimension (in)equalities and H\"older continuous curves in fractal percolation
We relate various concepts of fractal dimension of the limiting set C in
fractal percolation to the dimensions of the set consisting of connected
components larger than one point and its complement in C (the "dust"). In two
dimensions, we also show that the set consisting of connected components larger
than one point is a.s. the union of non-trivial H\"older continuous curves, all
with the same exponent. Finally, we give a short proof of the fact that in two
dimensions, any curve in the limiting set must have Hausdorff dimension
strictly larger than 1.Comment: 22 pages, 3 figures, accepted for publication in Journal of
Theoretical Probabilit
Metastability in zero-temperature dynamics: Statistics of attractors
The zero-temperature dynamics of simple models such as Ising ferromagnets
provides, as an alternative to the mean-field situation, interesting examples
of dynamical systems with many attractors (absorbing configurations, blocked
configurations, zero-temperature metastable states). After a brief review of
metastability in the mean-field ferromagnet and of the droplet picture, we
focus our attention onto zero-temperature single-spin-flip dynamics of
ferromagnetic Ising models. The situations leading to metastability are
characterized. The statistics and the spatial structure of the attractors thus
obtained are investigated, and put in perspective with uniform a priori
ensembles. We review the vast amount of exact results available in one
dimension, and present original results on the square and honeycomb lattices.Comment: 21 pages, 6 figures. To appear in special issue of JPCM on Granular
Matter edited by M. Nicodem
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
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