30,466 research outputs found
Two-Dimensional Scaling Limits via Marked Nonsimple Loops
We postulate the existence of a natural Poissonian marking of the double
(touching) points of SLE(6) and hence of the related continuum nonsimple loop
process that describes macroscopic cluster boundaries in 2D critical
percolation. We explain how these marked loops should yield continuum versions
of near-critical percolation, dynamical percolation, minimal spanning trees and
related plane filling curves, and invasion percolation. We show that this
yields for some of the continuum objects a conformal covariance property that
generalizes the conformal invariance of critical systems. It is an open problem
to rigorously construct the continuum objects and to prove that they are indeed
the scaling limits of the corresponding lattice objects.Comment: 25 pages, 5 figure
The Universal Cut Function and Type II Metrics
In analogy with classical electromagnetic theory, where one determines the
total charge and both electric and magnetic multipole moments of a source from
certain surface integrals of the asymptotic (or far) fields, it has been known
for many years - from the work of Hermann Bondi - that energy and momentum of
gravitational sources could be determined by similar integrals of the
asymptotic Weyl tensor. Recently we observed that there were certain overlooked
structures, {defined at future null infinity,} that allowed one to determine
(or define) further properties of both electromagnetic and gravitating sources.
These structures, families of {complex} `slices' or `cuts' of Penrose's null
infinity, are referred to as Universal Cut Functions, (UCF). In particular, one
can define from these structures a (complex) center of mass (and center of
charge) and its equations of motion - with rather surprising consequences. It
appears as if these asymptotic structures contain in their imaginary part, a
well defined total spin-angular momentum of the source. We apply these ideas to
the type II algebraically special metrics, both twisting and twist-free.Comment: 32 page
Study of the de Almeida-Thouless line using power-law diluted one-dimensional Ising spin glasses
We test for the existence of a spin-glass phase transition, the de
Almeida-Thouless line, in an externally-applied (random) magnetic field by
performing Monte Carlo simulations on a power-law diluted one-dimensional Ising
spin glass for very large system sizes. We find that an Almeida-Thouless line
only occurs in the mean field regime, which corresponds, for a short-range spin
glass, to dimension d larger than 6.Comment: 4 pages, 2 figures, 1 tabl
Interfaces (and Regional Congruence?) in Spin Glasses
We present a general theorem restricting properties of interfaces between
thermodynamic states and apply it to the spin glass excitations observed
numerically by Krzakala-Martin and Palassini-Young in spatial dimensions d=3
and 4. We show that such excitations, with interface dimension smaller than d,
cannot yield regionally congruent thermodynamic states. More generally, zero
density interfaces of translation-covariant excitations cannot be pinned (by
the disorder) in any d but rather must deflect to infinity in the thermodynamic
limit. Additional consequences concerning regional congruence in spin glasses
and other systems are discussed.Comment: 4 pages (ReVTeX); 1 figure; submitted to Physical Review Letter
Metastates in mean-field models with random external fields generated by Markov chains
We extend the construction by Kuelske and Iacobelli of metastates in
finite-state mean-field models in independent disorder to situations where the
local disorder terms are are a sample of an external ergodic Markov chain in
equilibrium. We show that for non-degenerate Markov chains, the structure of
the theorems is analogous to the case of i.i.d. variables when the limiting
weights in the metastate are expressed with the aid of a CLT for the occupation
time measure of the chain. As a new phenomenon we also show in a Potts example
that, for a degenerate non-reversible chain this CLT approximation is not
enough and the metastate can have less symmetry than the symmetry of the
interaction and a Gaussian approximation of disorder fluctuations would
suggest.Comment: 20 pages, 2 figure
Simplicity of State and Overlap Structure in Finite-Volume Realistic Spin Glasses
We present a combination of heuristic and rigorous arguments indicating that
both the pure state structure and the overlap structure of realistic spin
glasses should be relatively simple: in a large finite volume with
coupling-independent boundary conditions, such as periodic, at most a pair of
flip-related (or the appropriate number of symmetry-related in the non-Ising
case) states appear, and the Parisi overlap distribution correspondingly
exhibits at most a pair of delta-functions at plus/minus the self-overlap. This
rules out the nonstandard SK picture introduced by us earlier, and when
combined with our previous elimination of more standard versions of the mean
field picture, argues against the possibility of even limited versions of mean
field ordering in realistic spin glasses. If broken spin flip symmetry should
occur, this leaves open two main possibilities for ordering in the spin glass
phase: the droplet/scaling two-state picture, and the chaotic pairs many-state
picture introduced by us earlier. We present scaling arguments which provide a
possible physical basis for the latter picture, and discuss possible reasons
behind numerical observations of more complicated overlap structures in finite
volumes.Comment: 22 pages (LaTeX; needs revtex), 1 figure (PostScript); to appear in
Physical Review
The Brownian Web: Characterization and Convergence
The Brownian Web (BW) is the random network formally consisting of the paths
of coalescing one-dimensional Brownian motions starting from every space-time
point in . We extend the earlier work of Arratia
and of T\'oth and Werner by providing characterization and convergence results
for the BW distribution, including convergence of the system of all coalescing
random walkssktop/brownian web/finale/arXiv submits/bweb.tex to the BW under
diffusive space-time scaling. We also provide characterization and convergence
results for the Double Brownian Web, which combines the BW with its dual
process of coalescing Brownian motions moving backwards in time, with forward
and backward paths ``reflecting'' off each other. For the BW, deterministic
space-time points are almost surely of ``type'' -- {\em zero} paths
into the point from the past and exactly {\em one} path out of the point to the
future; we determine the Hausdorff dimension for all types that actually occur:
dimension 2 for type , 3/2 for and , 1 for , and 0
for and .Comment: 52 pages with 4 figure
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