New pattern theorems for square lattice self-avoiding walks and self-avoiding polygons

Abstract

A general pattern theorem for weighted self-avoiding polygons (SAPs) and selfavoiding walks (SAWs) in Z 2 is obtained. The pattern theorem for SAPs fits into the general framework of the pattern theorem for lattice clusters introduced by Madras (1999 Ann. Comb. 3 357-84) . Note that, unlike other pattern theorems proved for SAPs, this pattern theorem does not rely on first establishing a relationship between SAPs and SAWs. These results are applied to obtain pattern theorems for self-interacting SAPs and self-interacting SAWs.

The statistics of collapsing square lattice trails with a fixed number of vertices of degree 4

“…where the lower bounds hold for all Z d with d 2 and the upper bounds hold for d = 2. In this paper, we use the arguments that lead to the above result along with the new pattern theorems for SAWs and SAPs in Z 2 [9] to establish bounds to O(n k ) for…”

“…We also use the pattern theorem for collapsing n-SAPs [9] to obtain bounds on the average n-SAP contact density, l β /n. For example, we show that there exist 1…”

“…• En(k, β) and for Ȇn (k, β). We will rely on the new pattern theorems for SAPs and SAWs derived recently in [9] and refer the reader to this reference for a definition of the term pattern for SAPs or SAWs (but for examples of proper SAP patterns see figure 2). It is also necessary to introduce some new notation here.…”

“…Zhao and Lookman [8] also studied the collapsing free energy of closed (open) k-trails and found bounds for the free energy in terms of that for self-avoiding circuits (walks). In this paper, we use the arguments in [7] and the pattern theorems in [9] to establish tighter bounds (to O(n k )) between the collapsing free energy of closed (open) k-trails and that of self-avoiding circuits (walks). These results indicate that for trails, the limiting collapsing free energy and the collapse transition temperature, if they exist, are independent of the fixed number of vertices of degree 4 while the collapsing free energy critical exponent, if it exists, is increased by 1 for each vertex of degree 4.…”

“…Soteros and Whittington [10] obtained bounds on the limiting density of contacts for the collapsing free energy of self-avoiding polygons (SAPs). We extend their arguments to k-trails and using the pattern theorem for collapsing SAPs [9], establish new bounds on the contact density for k-trails.…”