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The statistical mechanics of stretched polymers. (English) Zbl 1195.82107

Summary: We describe some recent results concerning the statistical properties of a self-interacting polymer stretched by an external force. We concentrate mainly on the cases of purely attractive or purely repulsive self-interactions, but our results are stable under suitable small perturbations of these pure cases. We provide in particular a precise description of the stretched phase (local limit theorems for the endpoint and local observables, invariance principle, microscopic structure). Our results also characterize precisely the (nontrivial, direction-dependent) critical force needed to trigger the collapsed/stretched phase transition in the attractive case. We also describe some recent progress: first, the determination of the order of the phase transition in the attractive case; second, a proof that a semi-directed polymer in quenched random environment is diffusive in dimensions 4 and higher when the temperature is high enough. In addition, we correct an incomplete argument from D. Ioffe and Y. Velenik [in: Mörters, Peter (ed.) et al., Analysis and stochastics of growth processes and interface models. Selected papers based on the presentations at a regional meeting of the London Mathematical Society and a workshop on ‘Analysis and stochastics of growth processes’, Bath, UK, September 11–15, 2006. Oxford: Oxford University Press. 55–79 (2008; Zbl 1255.60168)].

MSC:

82D60 Statistical mechanics of polymers
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics

Citations:

Zbl 1255.60168
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References:

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