Self{attractive random polymers
TL;DR: In this article, the authors considered a repulsion-attraction model for a random polymer of flnite length in Z d, and they showed that for ∞ > fl the attraction dominates the repulsion, i.e., with high probability the polymer is contained in a flnnite box whose size is independent of the length of the polymer.
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Abstract: We consider a repulsion{attraction model for a random polymer of flnite length in Z d . Its law is that of a flnite simple random walk path in Z d receiving a penalty e i2fl for every self-intersection, and a reward e ∞=d for every pair of neighbouring monomers. The non-negative parameters fl and ∞ measure the strength of repellence and attraction, respectively. We show that for ∞ > fl the attraction dominates the repulsion, i.e., with high probability the polymer is contained in a flnite box whose size is independent of the length of the polymer. For ∞ < fl the behaviour is difierent. We give a lower bound for the rate at which the polymer extends in space. Indeed, we show that the probability for the polymer consisting of n monomers to be contained in a cube of side length †n 1=d tends to zero as n tends to inflnity.
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References
The Configuration of Real Polymer Chains
TL;DR: In this article, the average linear dimension of a polymer chain is taken proportional to a power of the chain length, that power must be greater than the value 0.50 previously deduced in the conventional ''random flight'' treatment of molecular configuration.
1.1K
Self-Avoiding Walk in 5 or More Dimensions
David C. Brydges,Thomas Spencer +1 more
TL;DR: In this paper, it was shown that for a T step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT→∞.
Self-avoiding walk in five or more dimensions I. The critical behaviour
Takashi Hara,Gordon Slade +1 more
TL;DR: In this article, the authors used the lace expansion to study the standard self-avoiding walk in the d-dimensional hypercubic lattice, ford ≥ 5, and proved that the number of n-step selfavoiding walks satisfiescn~Aμn, where μ is the connective constant.
The lace expansion for self-avoiding walk in five or more dimensions
TL;DR: In this paper, it was proved that the standard model of self-avoiding walk in five or more dimensions has the same critical behaviour as the simple random walk, assuming convergence of the lace expansion.
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