72 research outputs found
Conformal Invariance in Percolation, Self-Avoiding Walks and Related Problems
Over the years, problems like percolation and self-avoiding walks have
provided important testing grounds for our understanding of the nature of the
critical state. I describe some very recent ideas, as well as some older ones,
which cast light both on these problems themselves and on the quantum field
theories to which they correspond. These ideas come from conformal field
theory, Coulomb gas mappings, and stochastic Loewner evolution.Comment: Plenary talk given at TH-2002, Paris. 21 pages, 9 figure
Center clusters in the Yang-Mills vacuum
Properties of local Polyakov loops for SU(2) and SU(3) lattice gauge theory
at finite temperature are analyzed. We show that spatial clusters can be
identified where the local Polyakov loops have values close to the same center
element. For a suitable definition of these clusters the deconfinement
transition can be characterized by the onset of percolation in one of the
center sectors. The analysis is repeated for different resolution scales of the
lattice and we argue that the center clusters have a continuum limit.Comment: Table added. Final version to appear in JHE
Fracturing ranked surfaces
Discretized landscapes can be mapped onto ranked surfaces, where every
element (site or bond) has a unique rank associated with its corresponding
relative height. By sequentially allocating these elements according to their
ranks and systematically preventing the occupation of bridges, namely elements
that, if occupied, would provide global connectivity, we disclose that bridges
hide a new tricritical point at an occupation fraction , where
is the percolation threshold of random percolation. For any value of in the
interval , our results show that the set of bridges has a
fractal dimension in two dimensions. In the limit , a self-similar fracture is revealed as a singly connected line
that divides the system in two domains. We then unveil how several seemingly
unrelated physical models tumble into the same universality class and also
present results for higher dimensions
Scale invariance and universality of force networks in static granular matter
Force networks form the skeleton of static granular matter. They are the key
ingredient to mechanical properties, such as stability, elasticity and sound
transmission, which are of utmost importance for civil engineering and
industrial processing. Previous studies have focused on the global structure of
external forces (the boundary condition), and on the probability distribution
of individual contact forces. The disordered spatial structure of the force
network, however, has remained elusive so far. Here we report evidence for
scale invariance of clusters of particles that interact via relatively strong
forces. We analyzed granular packings generated by molecular dynamics
simulations mimicking real granular matter; despite the visual variation, force
networks for various values of the confining pressure and other parameters have
identical scaling exponents and scaling function, and thus determine a
universality class. Remarkably, the flat ensemble of force configurations--a
simple generalization of equilibrium statistical mechanics--belongs to the same
universality class, while some widely studied simplified models do not.Comment: 15 pages, 4 figures; to appear in Natur
One-sided versus two-sided stochastic descriptions
It is well-known that discrete-time finite-state Markov Chains, which are
described by one-sided conditional probabilities which describe a dependence on
the past as only dependent on the present, can also be described as
one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for
finite-spin models, which are described by two-sided conditional probabilities.
In such Markov Fields the time interpretation of past and future is being
replaced by the space interpretation of an interior volume, surrounded by an
exterior to the left and to the right.
If we relax the Markov requirement to weak dependence, that is, continuous
dependence, either on the past (generalising the Markov-Chain description) or
on the external configuration (generalising the Markov-Field description), it
turns out this equivalence breaks down, and neither class contains the other.
In one direction this result has been known for a few years, in the opposite
direction a counterexample was found recently. Our counterexample is based on
the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.Comment: 13 pages, Contribution for "Statistical Mechanics of Classical and
Disordered Systems
Tutte polynomial of pseudofractal scale-free web
The Tutte polynomial of a graph is a 2-variable polynomial which is quite
important in both combinatorics and statistical physics. It contains various
numerical invariants and polynomial invariants, such as the number of spanning
trees, the number of spanning forests, the number of acyclic orientations, the
reliability polynomial, chromatic polynomial and flow polynomial. In this
paper, we study and gain recursive formulas for the Tutte polynomial of
pseudofractal scale-free web (PSW) which implies logarithmic complexity
algorithm is obtained to calculate the Tutte polynomial of PSW although it is
NP-hard for general graph. We also obtain the rigorous solution for the the
number of spanning trees of PSW by solving the recurrence relations derived
from Tutte polynomial, which give an alternative approach for explicitly
determining the number of spanning trees of PSW. Further more, we analysis the
all-terminal reliability of PSW and compare the results with that of Sierpinski
gasket which has the same number of nodes and edges with PSW. In contrast with
the well-known conclusion that scale-free networks are more robust against
removal of nodes than homogeneous networks (e.g., exponential networks and
regular networks). Our results show that Sierpinski gasket (which is a regular
network) are more robust against random edge failures than PSW (which is a
scale-free network). Whether it is true for any regular networks and scale-free
networks, is still a unresolved problem.Comment: 19pages,7figures. arXiv admin note: text overlap with arXiv:1006.533
Probabilistic Interaction Network of Evidence Algorithm and its Application to Complete Labeling of Peak Lists from Protein NMR Spectroscopy
The process of assigning a finite set of tags or labels to a collection of observations, subject to side conditions, is notable for its computational complexity. This labeling paradigm is of theoretical and practical relevance to a wide range of biological applications, including the analysis of data from DNA microarrays, metabolomics experiments, and biomolecular nuclear magnetic resonance (NMR) spectroscopy. We present a novel algorithm, called Probabilistic Interaction Network of Evidence (PINE), that achieves robust, unsupervised probabilistic labeling of data. The computational core of PINE uses estimates of evidence derived from empirical distributions of previously observed data, along with consistency measures, to drive a fictitious system M with Hamiltonian H to a quasi-stationary state that produces probabilistic label assignments for relevant subsets of the data. We demonstrate the successful application of PINE to a key task in protein NMR spectroscopy: that of converting peak lists extracted from various NMR experiments into assignments associated with probabilities for their correctness. This application, called PINE-NMR, is available from a freely accessible computer server (http://pine.nmrfam.wisc.edu). The PINE-NMR server accepts as input the sequence of the protein plus user-specified combinations of data corresponding to an extensive list of NMR experiments; it provides as output a probabilistic assignment of NMR signals (chemical shifts) to sequence-specific backbone and aliphatic side chain atoms plus a probabilistic determination of the protein secondary structure. PINE-NMR can accommodate prior information about assignments or stable isotope labeling schemes. As part of the analysis, PINE-NMR identifies, verifies, and rectifies problems related to chemical shift referencing or erroneous input data. PINE-NMR achieves robust and consistent results that have been shown to be effective in subsequent steps of NMR structure determination
Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman
In the introduction to this volume, we discuss some of the highlights of the
research career of Chuck Newman. This introduction is divided into two main
sections, the first covering Chuck's work in statistical mechanics and the
second his work in percolation theory, continuum scaling limits, and related
topics.Comment: 38 pages (including many references), introduction to Festschrift in
honor of C.M. Newma
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