An almost sure CLT for stretched polymers. (English) Zbl 1286.60026

Ioffe, Dmitry; Velenik, Yvan

Electron. J. Probab. 18, Paper No. 97, 20 p. (2013).

The authors consider diffusive behaviour in \({\mathbb Z}^{d+1}\), \(d\geq 3\), for the related models of stretched polymers. A stretched path \(\gamma\) can be any nearest-neighbour path on \({\mathbb Z}^{d+1}.\) The disorder is modeled by a collection \(\{V(x)\}_{x\in {\mathbb Z}^{d+1}}\) of i.i.d. non-negative random variables. Each visit of the path to a vertex \(x\) exerts the price \(\exp(-\beta V(x)) \), \(\beta>0\). One (of two) way of introduction of the stretch is that the path \(\gamma\) has a fixed length, but it is subject to a drift, which can be interpreted physically as the effect of a force acting on the polymer’s free end. In the paper, an almost-sure central limit theorem is established for the endpoint of the fixed-length version of the model of stretched polymers with non-zero drifts, also at sufficiently high temperatures and in all dimensions \(d+1\geq 4\).

Reviewer: Utkir A. Rozikov (Tashkent)

MSC:

60F05	Central limit and other weak theorems
60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D60	Statistical mechanics of polymers

Keywords:

polymers; random walk representation; random environment; weak disorder; central limit theorem (CLT)