Self-avoiding walks crossing a square

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.