We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at (L, L), and are entirely contained in the square [0, L] × [0,L] on the square lattice ℤ 2. The number of distinct walks is known to grow as λ L2 +o(L2). We give a precise estimate for A as well as obtaining upper and lower bounds. We give exact results for the number of SAW of length 2L + 2K for K = 0, 1, 2 and asymptotic results for K = o(L 1/3). We also consider the model in which a weight or fugacity x is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For x < 1/μ the average length of a SAW is proportional to L, while for x >1/μ it is proportional to L 2. Here μ is the growth constant of unconstrained SAW in ℤ 2. For x =1/μ we provide numerical evidence, but no proof, that the average walk length is 0(L 4/3). We also consider Hamiltonian walks under the same restrictions. These grow as τ L2 +o(L2) on the same L × L lattice. We give precise estimates for τ, as well as upper and lower bounds, and prove τ < λ.
|Number of pages||11|
|Publication status||Published - 2005|
|Event||17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05 - Taormina, Italy|
Duration: 20 Jun 2005 → 25 Jun 2005
|Conference||17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05|
|Period||20/06/05 → 25/06/05|