1,309 research outputs found
Diffusion Limited Aggregation on a Cylinder
We consider the DLA process on a cylinder G x N. It is shown that this
process "grows arms", provided that the base graph G has small enough mixing
time. Specifically, if the mixing time of G is at most (log|G|)^(2-\eps), the
time it takes the cluster to reach the m-th layer of the cylinder is at most of
order m |G|/loglog|G|. In particular we get examples of infinite Cayley graphs
of degree 5, for which the DLA cluster on these graphs has arbitrarily small
density.
In addition, we provide an upper bound on the rate at which the "arms" grow.
This bound is valid for a large class of base graphs G, including discrete tori
of dimension at least 3.
It is also shown that for any base graph G, the density of the DLA process on
a G-cylinder is related to the rate at which the arms of the cluster grow. This
implies, that for any vertex transitive G, the density of DLA on a G-cylinder
is bounded by 2/3.Comment: 1 figur
Error- and Loss-Tolerances of Surface Codes with General Lattice Structures
We propose a family of surface codes with general lattice structures, where
the error-tolerances against bit and phase errors can be controlled
asymmetrically by changing the underlying lattice geometries. The surface codes
on various lattices are found to be efficient in the sense that their threshold
values universally approach the quantum Gilbert-Varshamov bound. We find that
the error-tolerance of surface codes depends on the connectivity of underlying
lattices; the error chains on a lattice of lower connectivity are easier to
correct. On the other hand, the loss-tolerance of surface codes exhibits an
opposite behavior; the logical information on a lattice of higher connectivity
has more robustness against qubit loss. As a result, we come upon a fundamental
trade-off between error- and loss-tolerances in the family of the surface codes
with different lattice geometries.Comment: 5pages, 3 figure
Cardy's Formula for Certain Models of the Bond-Triangular Type
We introduce and study a family of 2D percolation systems which are based on
the bond percolation model of the triangular lattice. The system under study
has local correlations, however, bonds separated by a few lattice spacings act
independently of one another. By avoiding explicit use of microscopic paths, it
is first established that the model possesses the typical attributes which are
indicative of critical behavior in 2D percolation problems. Subsequently, the
so called Cardy-Carleson functions are demonstrated to satisfy, in the
continuum limit, Cardy's formula for crossing probabilities. This extends the
results of S. Smirnov to a non-trivial class of critical 2D percolation
systems.Comment: 49 pages, 7 figure
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
On C*-algebras related to constrained representations of a free group
We consider representations of the free group on two generators such
that the norm of the sum of the generators and their inverses is bounded by
. These -constrained representations determine a C*-algebra
for each . We prove that these C*-algebras form a
continuous bundle of C*-algebras over and calculate their K-groups.Comment: 9 page
Stability of Influence Maximization
The present article serves as an erratum to our paper of the same title,
which was presented and published in the KDD 2014 conference. In that article,
we claimed falsely that the objective function defined in Section 1.4 is
non-monotone submodular. We are deeply indebted to Debmalya Mandal, Jean
Pouget-Abadie and Yaron Singer for bringing to our attention a counter-example
to that claim.
Subsequent to becoming aware of the counter-example, we have shown that the
objective function is in fact NP-hard to approximate to within a factor of
for any .
In an attempt to fix the record, the present article combines the problem
motivation, models, and experimental results sections from the original
incorrect article with the new hardness result. We would like readers to only
cite and use this version (which will remain an unpublished note) instead of
the incorrect conference version.Comment: Erratum of Paper "Stability of Influence Maximization" which was
presented and published in the KDD1
A derivative formula for the free energy function
We consider bond percolation on the lattice. Let be the
number of open clusters in . It is well known that converges to the free energy function at the zero field.
In this paper, we show that converges to
.Comment: 8 pages 1 figur
Auto-tail dependence coefficients for stationary solutions of linear stochastic recurrence equations and for GARCH(1,1)
We examine the auto-dependence structure of strictly stationary solutions of linear stochastic recurrence equations and of strictly stationary GARCH(1, 1) processes from the point of view of ordinary and generalized tail dependence coefficients. Since such processes can easily be of infinite variance, a substitute for the usual auto-correlation function is needed
Bridge Decomposition of Restriction Measures
Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding
walks in the upper half plane, we show that the conjectured scaling limit of
the half-plane SAW, the SLE(8/3) process, also has an appropriately defined
bridge decomposition. This continuum decomposition turns out to entirely be a
consequence of the restriction property of SLE(8/3), and as a result can be
generalized to the wider class of restriction measures. Specifically we show
that the restriction hulls with index less than one can be decomposed into a
Poisson Point Process of irreducible bridges in a way that is similar to Ito's
excursion decomposition of a Brownian motion according to its zeros.Comment: 24 pages, 2 figures. Final version incorporates minor revisions
suggested by the referee, to appear in Jour. Stat. Phy
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