17 research outputs found
Convex lattice polygonal lines with a constrained number of vertices
A detailed combinatorial analysis of planar convex lattice polygonal lines is
presented. This makes it possible to answer an open question of Vershik
regarding the existence of a limit shape when the number of vertices is
constrained
Convex cones, integral zonotopes, limit shape
This paper is about integral zonotopes. It is proven that large zonotopes in
a convex cone have a limit shape, meaning that, after suitable scaling, the
overwhelming majority of the zonotopes are very close to a fixed convex set.
Several combinatorial properties of large zonotopes are established
Non-crossing frameworks with non-crossing reciprocals
We study non-crossing frameworks in the plane for which the classical
reciprocal on the dual graph is also non-crossing. We give a complete
description of the self-stresses on non-crossing frameworks whose reciprocals
are non-crossing, in terms of: the types of faces (only pseudo-triangles and
pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a
geometric condition on the stress vectors at some of the vertices.
As in other recent papers where the interplay of non-crossingness and
rigidity of straight-line plane graphs is studied, pseudo-triangulations show
up as objects of special interest. For example, it is known that all planar
Laman circuits can be embedded as a pseudo-triangulation with one non-pointed
vertex. We show that if such an embedding is sufficiently generic, then the
reciprocal is non-crossing and again a pseudo-triangulation embedding of a
planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation
embedding of a planar Laman circuit, the reciprocal is still non-crossing and a
pseudo-triangulation, but its underlying graph may not be a Laman circuit.
Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal
arise as the reciprocals of such, possibly singular, stresses on
pseudo-triangulation embeddings of Laman circuits.
All self-stresses on a planar graph correspond to liftings to piece-wise
linear surfaces in 3-space. We prove characteristic geometric properties of the
lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure
Universality of the limit shape of convex lattice polygonal lines
Let be the set of convex polygonal lines with
vertices on and fixed endpoints and .
We are concerned with the limit shape, as , of "typical"
with respect to a parametric family of probability
measures on , including the uniform
distribution () for which the limit shape was found in the early 1990s
independently by A. M. Vershik, I. B\'ar\'any and Ya. G. Sinai. We show that,
in fact, the limit shape is universal in the class , even though
() and are asymptotically singular. Measures are
constructed, following Sinai's approach, as conditional distributions
, where are suitable product measures on the
space , depending on an auxiliary "free"
parameter . The transition from to
is based on the asymptotics of the probability
, furnished by a certain two-dimensional local limit
theorem. The proofs involve subtle analytical tools including the M\"obius
inversion formula and properties of zeroes of the Riemann zeta function.Comment: Published in at http://dx.doi.org/10.1214/10-AOP607 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Fluctuations of lattice zonotopes and polygons
Following Barany et al., who proved that large random lattice zonotopes
converge to a deterministic shape in any dimension after rescaling, we
establish a central limit theorem for finite-dimensional marginals of the
boundary of the zonotope. In dimension 2, for large random convex lattice
polygons contained in a square, we prove a Donsker-type theorem for the
boundary fluctuations, which involves a two-dimensional Brownian bridge and a
drift term that we identify as a random cubic curve