15,548 research outputs found
High-Precision Entropy Values for Spanning Trees in Lattices
Shrock and Wu have given numerical values for the exponential growth rate of
the number of spanning trees in Euclidean lattices. We give a new technique for
numerical evaluation that gives much more precise values, together with
rigorous bounds on the accuracy. In particular, the new values resolve one of
their questions.Comment: 7 pages. Revision mentions alternative approach. Title changed
slightly. 2nd revision corrects first displayed equatio
Microwave properties of DyBa_2Cu_3O_(7-x) monodomains and related compounds in magnetic fields
We present a microwave characterization of a DyBaCuO
single domain, grown by the top-seeded melt-textured technique. We report the
(a,b) plane field-induced surface resistance, , at 48.3 GHz,
measured by means of a cylindrical metal cavity in the end-wall-replacement
configuration. Changes in the cavity quality factor Q against the applied
magnetic field yield at fixed temperatures. The temperature
range [70 K ; T_c] was explored. The magnetic field 0.8 T was
applied along the c axis. The field dependence of does not
exhibit the steep, step-like increase at low fields typical of weak-links. This
result indicates the single-domain character of the sample under investigation.
exhibits a nearly square-root dependence on H, as expected for
fluxon motion. From the analysis of the data in terms of motion of Abrikosov
vortices we estimate the temperature dependences of the London penetration
depth and the vortex viscosity , and their zero-temperature
values 165 nm and 3 10 Nsm, which are
found in excellent agreement with reported data in YBaCuO
single crystals. Comparison of microwave properties with those of related
samples indicate the need for reporting data as a function of T/T_c in order to
obtain universal laws.Comment: 6 pages, 4 figures, LaTeX, submitted to Journal of Applied Physic
Phase diagram of self-assembled rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations
Monte Carlo simulations and finite-size scaling analysis have been carried
out to study the critical behavior in a two-dimensional system of particles
with two bonding sites that, by decreasing temperature or increasing density,
polymerize reversibly into chains with discrete orientational degrees of
freedom and, at the same time, undergo a continuous isotropic-nematic (IN)
transition. A complete phase diagram was obtained as a function of temperature
and density. The numerical results were compared with mean field (MF) and real
space renormalization group (RSRG) analytical predictions about the IN
transformation. While the RSRG approach supports the continuous nature of the
transition, the MF solution predicts a first-order transition line and a
tricritical point, at variance with the simulation results.Comment: 12 pages, 10 figures, supplementary informatio
Slow movement of a random walk on the range of a random walk in the presence of an external field
In this article, a localisation result is proved for the biased random walk
on the range of a simple random walk in high dimensions (d \geq 5). This
demonstrates that, unlike in the supercritical percolation setting, a slowdown
effect occurs as soon a non-trivial bias is introduced. The proof applies a
decomposition of the underlying simple random walk path at its cut-times to
relate the associated biased random walk to a one-dimensional random walk in a
random environment in Sinai's regime
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
The present paper is devoted to the study of sample paths of G-Brownian
motion and stochastic differential equations (SDEs) driven by G-Brownian motion
from the view of rough path theory. As the starting point, we show that
quasi-surely, sample paths of G-Brownian motion can be enhanced to the second
level in a canonical way so that they become geometric rough paths of roughness
2 < p < 3. This result enables us to introduce the notion of rough differential
equations (RDEs) driven by G-Brownian motion in the pathwise sense under the
general framework of rough paths. Next we establish the fundamental relation
between SDEs and RDEs driven by G-Brownian motion. As an application, we
introduce the notion of SDEs on a differentiable manifold driven by GBrownian
motion and construct solutions from the RDE point of view by using pathwise
localization technique. This is the starting point of introducing G-Brownian
motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin.
The last part of this paper is devoted to such construction for a wide and
interesting class of G-functions whose invariant group is the orthogonal group.
We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian
motion of independent interest
Phase Transitions on Nonamenable Graphs
We survey known results about phase transitions in various models of
statistical physics when the underlying space is a nonamenable graph. Most
attention is devoted to transitive graphs and trees
Dynamical percolation on general trees
H\"aggstr\"om, Peres, and Steif (1997) have introduced a dynamical version of
percolation on a graph . When is a tree they derived a necessary and
sufficient condition for percolation to exist at some time . In the case
that is a spherically symmetric tree, H\"aggstr\"om, Peres, and Steif
(1997) derived a necessary and sufficient condition for percolation to exist at
some time in a given target set . The main result of the present paper
is a necessary and sufficient condition for the existence of percolation, at
some time , in the case that the underlying tree is not necessary
spherically symmetric. This answers a question of Yuval Peres (personal
communication). We present also a formula for the Hausdorff dimension of the
set of exceptional times of percolation.Comment: 24 pages; to appear in Probability Theory and Related Field
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