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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.


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
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Date accepted:No date available
Date deposited:20 February 2013
Date of first online publication:January 2004
Date first made open access:No date available

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