Self-avoiding polygons: sharp asymptotics of canonical partition functions under the fixed area constraint

Hryniv, O.; Ioffe, D.

Abstract

The paper considers the ensemble of self-avoiding paths in $Z^2$ which join the positive vertical axis with the positive horizontal axis, and take value on the first quadrant of the plane. To each such path $\omega$ is associated its length $|\omega|$ and the area $A_+(\omega)$ enclosed by the path and the axes. For every fixed area $Q>0$, the partition function $$ Z_{Q,+}\coloneq\sum_{\omega\colon A_+(\omega)=Q}e^{-\beta|\omega|} $$ is considered for all $\beta>\beta_c$, where the critical value $\beta_c$ is defined by the property that the sum of $e^{-\beta |\omega|}$ over all self-avoiding paths on $Z^2$ is finite if and only if $\beta>\beta_c$. The main result of the paper is a sharp estimate, up to order one and with explicitly determined constants, of $Z_{Q,+}$ in the limit as $Q\uparrow+\infty$. Unlike many related results concerning phase segregation for lattice models, this result does not rely on cluster expansion and holds for every supercritical $\beta$.

Citation

Hryniv, O., & Ioffe, D. (2004). Self-avoiding polygons: sharp asymptotics of canonical partition functions under the fixed area constraint. Markov processes and related fields, 10(1), 1-64

• Journal Article Type: Article
• Publication Date: Jan 1, 2004
• Deposit Date: May 1, 2007
• Publicly Available Date: Feb 20, 2013
• Journal: Markov processes and related fields.
• Print ISSN: 1024-2953
• Publisher: Polymat
• Peer Reviewed: Peer Reviewed
• Volume: 10
• Issue: 1
• Pages: 1-64
• Public URL: https://durham-repository.worktribe.com/output/1560644
• Publisher URL: http://www.math.msu.su/~malyshev/cont03.htm