Abstract view
Bounds on multiple selfavoiding polygons


Published:20180116
Printed: Sep 2018
Kyungpyo Hong,
National Institute for Mathematical Sciences, Daejeon 34047, Korea
Seungsang Oh,
Department of Mathematics, Korea University, Seoul 02841, Korea
Abstract
A selfavoiding polygon is a lattice polygon consisting of a
closed selfavoiding walk on a square lattice.
Surprisingly little is known rigorously about the enumeration
of selfavoiding polygons,
although there are numerous conjectures that are believed to
be true
and strongly supported by numerical simulations.
As an analogous problem of this study,
we consider multiple selfavoiding polygons in a confined region, as a model for multiple ring polymers in physics.
We find rigorous lower and upper bounds of the number $p_{m \times
n}$
of distinct multiple selfavoiding polygons in the $m \times
n$ rectangular grid on the square lattice.
For $m=2$, $p_{2 \times n} = 2^{n1}1$.
And, for integers $m,n \geq 3$,
$$2^{m+n3}
\left(\tfrac{17}{10}
\right)^{(m2)(n2)} \ \leq \ p_{m \times n} \ \leq \
2^{m+n3}
\left(\tfrac{31}{16}
\right)^{(m2)(n2)}.$$
MSC Classifications: 
57M25, 82B20, 82B41, 82D60 show english descriptions
Knots and links in $S^3$ {For higher dimensions, see 57Q45} Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] Polymers
57M25  Knots and links in $S^3$ {For higher dimensions, see 57Q45} 82B20  Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B41  Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82D60  Polymers
