Critical exponents for square lattice trails with a fixed number of vertices of degree 4

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

“…In [7] it was established that there exist positive constants , C, D, C, D 0 , D, N such that for all n N and for any k 0,…”

“…Recently [7], we focused on the number of n-step trails with a fixed number, k, of vertices of degree 4 (k-trails) and derived combinatorial bounds relating the number of such trails to the number of n-step self-avoiding walks or circuits. In particular, we established that the number of square lattice n-edge closed (open) k-trails can be bounded above and below (to O(n k )) by the number of n-step self-avoiding circuits (walks).…”

“…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.…”

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

“…One such theorem, due to Kesten [4], is for self-avoiding walks in Z d , d 2, and states: for an appropriately defined pattern P, there exists an > 0 such that all but exponentially few sufficiently long n-step self-avoiding walks contain the pattern P at least n times. Kesten's pattern theorem has been used, for example, to establish results about knotting probabilities for self-avoiding polygons in Z 3 [5] and to help establish relationships between entropic critical exponents for various lattice models [6].…”

“…The new pattern theorem for SAPs has the advantage that its proof does not rely on establishing any relationships between weighted SAPs and weighted SAWs, in contrast to previous approaches for establishing pattern theorems for SAPs [1,3]. For example, the previously cited applications of Kesten's pattern theorem to self-avoiding polygons [5,6] each rely on the fact that the limiting entropy per site for self-avoiding polygons (or exponential growth rate of the number of self-avoiding polygons) is equal to that for self-avoiding walks. Hence the pattern theorem for SAPs established herein can be applied to cases for which it is not known whether the limiting free energies for SAWs and SAPs are equal, as is the case for example, when the interaction energy is proportional to the number of nearest-neighbour contacts [3,8], i.e.…”