Depinning of a polymer in a multi-interface medium. (English) Zbl 1192.60105

Caravenna, Francesco; Pétrélis, Nicolas

Electron. J. Probab. 14, 2038-2067 (2009).

Summary: We consider a model which describes a polymer chain interacting with an infinity of equi-spaced linear interfaces. The distance between two consecutive interfaces is denoted by \(T = T_{N}\) and is allowed to grow with the size \(N\) of the polymer. When the polymer receives a positive reward for touching the interfaces, its asymptotic behavior has been derived by F. Caravenna and N. Pétrélis [Ann. Appl. Probab. 19, No. 5, 1803–1839 (2009; Zbl 1206.60089)], showing that a transition occurs when \(T_{N}\) grows like \(\log N\). In the present paper, we deal with the so-called depinning case, i.e., the polymer is repelled rather than attracted by the interfaces. Using techniques from renewal theory, we determine the scaling behavior of the model for large \(N\) as a function of \({T_{N}}_{N}\), showing that two transitions occur, when \(T_{N}\) grows like \(N^{1/3}\) and when \(T_{N}\) groes like \(N^{1/2}\) respectively.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F05	Central limit and other weak theorems
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics

Keywords:

polymer model; pinning model; random walk; renewal theory; localization/Delocalization transition

Citations:

Zbl 1206.60089