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

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

## 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 $\beta$.

Item Type: Article PDF - Published Version (469Kb) Peer-reviewed http://www.math.msu.su/~malyshev/cont03.htm 01 May 2007 20 Feb 2013 14:55

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