Publication 15-CNA-025
Sharp $N^{3/4}$ Law for the Minimizers of the
Edge-Isoperimetric Problem on the Triangular Lattice
Elisa Davoli
Department of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
elisa.davoli@univie.ac.at
Paolo Piovano
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern
A-1090 Vienna, Austria
paolo.piovano@univie.ac.at
Ulisse Stefanelli
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern
A-1090 Vienna, Austria
ulisse.stefanelli@univie.ac.at
Abstract: We investigate the Edge-Isoperimetric Problem (EIP) for sets
of $n$ points
in the triangular lattice by emphasizing its relation with the emergence
of the Wulff
shape in the crystallization problem. By introducing a suitable notion
of perimeter
and area, EIP minimizers are characterized as extremizers of an
isoperimetric inequality: they attain maximal area and minimal perimeter
among connected congurations. The maximal area and minimal perimeter
are explicitly
quantied in terms
of $n$. In view of this isoperimetric characterizations EIP minimizers
$Mn$ are seen to
be given by hexagonal congurations with some extra points at their
boundary. By
a careful computation of the cardinality of these extra points,
minimizers Mn are estimated to deviate from such hexagonal congurations
by at most $K_t n^{3/4} +o(n^{3/4})$
points. The constant Kt is explicitly determined and shown to be sharp.
Get the paper in its entirety as 15-CNA-025.pdf
« Back to CNA Publications
