Abstract
Consider a fixed “end pattern” (a short self-avoiding walk) that can occur as the first few steps of an arbitrarily long self-avoiding walk on ℤd. It is a difficult open problem to show that asN→ ∞, the fraction ofN-step self-avoiding walks beginning with this pattern converges. It is shown that asN → ∞, this fraction is bounded away from zero, and that the ratio of the fractions forN andN+2 converges to one. Similar results are obtained when patterns are specified at both ends, and also when the endpoints are fixed.
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Madras, N. End patterns of self-avoiding walks. J Stat Phys 53, 689–701 (1988). https://doi.org/10.1007/BF01014220
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DOI: https://doi.org/10.1007/BF01014220