### Abstract

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.

Original language | English |
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Pages | 147-157 |

Number of pages | 11 |

Publication status | Published - 2005 |

Externally published | Yes |

Event | 17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05 - Taormina, Italy Duration: 20 Jun 2005 → 25 Jun 2005 |

### Conference

Conference | 17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05 |
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Country | Italy |

City | Taormina |

Period | 20/06/05 → 25/06/05 |

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## Cite this

*Self-avoiding walks crossing a square*. 147-157. Paper presented at 17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05, Taormina, Italy.