Self-Avoiding Polygons: Sharp Asymptotics of Canonical Partition Functions Under the Fixed Area Constraint

O. Hryniv, D. Ioffe

2004, v.10, Issue 1, 1-64


We study the self-avoiding polygons (SAP) connecting the
vertical and the horizontal semi-axes of the
positive quadrant of $\Z^2$. For a fixed $\be>0$, assign to
each such polygon $\om$ the weight $\exp\{-\beta|\om|\}$, $|\om|$
denoting the length of~$\om$, and consider
the sum $\calZ_{Q,+}$ of these weights for all SAP enclosing area
$Q>0$. We study the statistical properties of such SAP and, in
particular, derive the exact asymptotics for the partition function
$\calZ_{Q,+}$ as $Q\to\infty$.
The results are valid for any $\be>\be_c$, $\be_c$ being the
critical value for the 2D self-avoiding walks.

Keywords: self-avoiding random walks, phase boundaries, sharp local limit theorems, equilibrium crystal shapes


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