21,118 research outputs found
Note on the number of minimal lattice paths restricted by two parallel lines
AbstractThe paper deals with minimal lattice paths from the origin to a point (n,m) which do not cross both two given parallel lines I and II with an incline k (⩾1), but touch the line I a specified number of times. We get the generating functions for the number of distinct minimal lattice paths satisfying these conditions
Random lattice triangulations: Structure and algorithms
The paper concerns lattice triangulations, that is, triangulations of the
integer points in a polygon in whose vertices are also integer
points. Lattice triangulations have been studied extensively both as geometric
objects in their own right and by virtue of applications in algebraic geometry.
Our focus is on random triangulations in which a triangulation has
weight , where is a positive real parameter, and
is the total length of the edges in . Empirically, this
model exhibits a "phase transition" at (corresponding to the
uniform distribution): for distant edges behave essentially
independently, while for very large regions of aligned edges
appear. We substantiate this picture as follows. For sufficiently
small, we show that correlations between edges decay exponentially with
distance (suitably defined), and also that the Glauber dynamics (a local Markov
chain based on flipping edges) is rapidly mixing (in time polynomial in the
number of edges in the triangulation). This dynamics has been proposed by
several authors as an algorithm for generating random triangulations. By
contrast, for we show that the mixing time is exponential. These
are apparently the first rigorous quantitative results on the structure and
dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Hole motion in an arbitrary spin background: Beyond the minimal spin-polaron approximation
The motion of a single hole in an arbitrary magnetic background is
investigated for the 2D t-J model. The wavefunction of the hole is described
within a generalized string picture which leads to a modified concept of spin
polarons. We calculate the one-hole spectral function using a large string
basis for the limits of a Neel ordered and a completely disordered background.
In addition we use a simple approximation to interpolate between these cases.
For the antiferromagnetic background we reproduce the well-known quasiparticle
band. In the disordered case the shape of the spectral function is found to be
strongly momentum-dependent, the quasiparticle weight vanishes for all hole
momenta. Finally, we discuss the relevance of results for the lowest energy
eigenvalue and its dispersion obtained from calculations using a polaron of
minimal size as found in the literature.Comment: 13 pages, 8 figures, to appear in Phys. Rev.
On Minimal Trajectories for Mobile Sampling of Bandlimited Fields
We study the design of sampling trajectories for stable sampling and the
reconstruction of bandlimited spatial fields using mobile sensors. The spectrum
is assumed to be a symmetric convex set. As a performance metric we use the
path density of the set of sampling trajectories that is defined as the total
distance traveled by the moving sensors per unit spatial volume of the spatial
region being monitored. Focussing first on parallel lines, we identify the set
of parallel lines with minimal path density that contains a set of stable
sampling for fields bandlimited to a known set. We then show that the problem
becomes ill-posed when the optimization is performed over all trajectories by
demonstrating a feasible trajectory set with arbitrarily low path density.
However, the problem becomes well-posed if we explicitly specify the stability
margins. We demonstrate this by obtaining a non-trivial lower bound on the path
density of an arbitrary set of trajectories that contain a sampling set with
explicitly specified stability bounds.Comment: 28 pages, 8 figure
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