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High-precision determination of the critical exponent γ for self-avoiding walks

“…Thus if use the value of γ given by [17], see (1), we get a value for β c in agreement with their estimate (13). However if we use our estimate of γ, we get a β c which is off by five to twenty times the stated error bars.…”

“…A simulation [14] with walks of length up to N = 80000, using a data analysis taking careful account of the corrections to scaling, gave ν = 0.5877±0.0006 in good agreement with the renormalization-group estimates. Also universality is well satisfied: a simulation [13] in a slightly different geometry provided ν = 0.5867 ± 0.0013 (68% confidence limit) while a recent high-statistics simulation [15] for the off-lattice Kratky-Porod model with excluded volume gave ν = 0.5880 ± 0.0018. One may ask if the same phenomenon occurs for the other critical exponents.…”

Universality of subleading corrections for self-avoiding walks in the presence of one-dimensional defects

“…Let us notice that our estimate of ∆ is somewhat lower than the estimate of [1], ∆ ≈ 0.24 and of [2], ∆ ≈ 0.217 ± 0.013. The origin of these discrepancies is probably in the neglected additional corrections to scaling: the results of [1] are obtained from an exact enumeration and thus probe only very short walks, while the estimate of [2] comes from walks which have mainly N ≈ 1000 −4000. Here we use longer SAWs and a much higher statistics and thus we can do a much more detailed study of the role of the next subleading terms.…”

“…In this paper we have studied the role played by one-dimensional vacancies in the critical behaviour of the three-dimensional SAW. As already pointed out in [1,2] a new critical exponent arises. We have carefully checked that the exponent depends only on the dimensionality of the vacancies by verifying its independence from the shape of the excluded region: in particular it does not depend on the discrete symmetry which the lattice has after the introduction of the excluded region.…”

Random walks with short-range interaction and mean-field behavior

“…This result is quite puzzling; it is exact for D =1, 2 and 4, but it is definitely wrong in D = 3 [8,9,10,11]. Moreover in 4 − ǫ dimensions the exact results and Flory's one differ at first order in ǫ [3].…”