31,837 research outputs found
Statistical hyperbolicity in groups
In this paper, we introduce a geometric statistic called the "sprawl" of a
group with respect to a generating set, based on the average distance in the
word metric between pairs of words of equal length. The sprawl quantifies a
certain obstruction to hyperbolicity. Group presentations with maximum sprawl
(i.e., without this obstruction) are called statistically hyperbolic. We first
relate sprawl to curvature and show that nonelementary hyperbolic groups are
statistically hyperbolic, then give some results for products, for
Diestel-Leader graphs and lamplighter groups. In free abelian groups, the word
metrics asymptotically approach norms induced by convex polytopes, causing the
study of sprawl to reduce to a problem in convex geometry. We present an
algorithm that computes sprawl exactly for any generating set, thus quantifying
the failure of various presentations of Z^d to be hyperbolic. This leads to a
conjecture about the extreme values, with a connection to the classic Mahler
conjecture.Comment: 14 pages, 5 figures. This is split off from the paper "The geometry
of spheres in free abelian groups.
A New Lower Bound for Semigroup Orthogonal Range Searching
We report the first improvement in the space-time trade-off of lower bounds
for the orthogonal range searching problem in the semigroup model, since
Chazelle's result from 1990. This is one of the very fundamental problems in
range searching with a long history. Previously, Andrew Yao's influential
result had shown that the problem is already non-trivial in one
dimension~\cite{Yao-1Dlb}: using units of space, the query time must
be where is the
inverse Ackermann's function, a very slowly growing function.
In dimensions, Bernard Chazelle~\cite{Chazelle.LB.II} proved that the
query time must be where .
Chazelle's lower bound is known to be tight for when space consumption is
`high' i.e., . We have two main results.
The first is a lower bound that shows Chazelle's lower bound was not tight for
`low space': we prove that we must have . Our lower bound does not close the gap to the existing data
structures, however, our second result is that our analysis is tight. Thus, we
believe the gap is in fact natural since lower bounds are proven for idempotent
semigroups while the data structures are built for general semigroups and thus
they cannot assume (and use) the properties of an idempotent semigroup. As a
result, we believe to close the gap one must study lower bounds for
non-idempotent semigroups or building data structures for idempotent
semigroups. We develope significantly new ideas for both of our results that
could be useful in pursuing either of these directions
Quantum Crystals and Spin Chains
In this note, we discuss the quantum version of the melting crystal corner in
one, two, and three dimensions, generalizing the treatment for the quantum
dimer model. Using a mapping to spin chains we find that the two--dimensional
case (growth of random partitions) is integrable and leads directly to the
Hamiltonian of the Heisenberg XXZ ferromagnet. The three--dimensional case of
the melting crystal corner is described in terms of a system of coupled XXZ
spin chains. We give a conjecture for its mass gap and analyze the system
numerically.Comment: 34 pages, 26 picture
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