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Copolymer at selective interfaces and pinning potentials: weak coupling limits. (English) Zbl 1172.82318
Summary: We consider a simple random walk of length $$N$$, denoted by $$(S_i)_{i\in\{1,\dots,N\}}$$, and we define $$(w_i)_{i\geq1}$$ a sequence of centered i.i.d. random variables. For $$K\in\mathbb N$$ we define $$((\gamma_i^{-K},\dots, \gamma_i^K))_{i\geq1}$$ an i.i.d sequence of random vectors. We set $$\beta\in\mathbb R$$, $$\lambda\geq0$$ and $$h\geq0$$, and transform the measure on the set of random walk trajectories with the Hamiltonian $$\lambda \sum_{i=1}^N (w_i+h) \operatorname{sign}(S_i)+ \beta \sum_{j=-K}^K \sum_{i=1}^N \gamma_i^j{\mathbf 1}_{\{S_i=j\}}$$. This transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width $$2K$$ around an interface between oil and water.
In the present article we prove the convergence in the limit of weak coupling (when $$\lambda$$, $$h$$ and $$\beta$$ tend to 0) of this discrete model towards its continuous counterpart. To that aim we further develop a technique of coarse graining introduced by E. Bolthausen and F. den Hollander [Ann. Probab. 25, No. 3, 1334–1366 (1997; Zbl 0885.60022)]. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.

MSC:
 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K37 Processes in random environments 82D60 Statistical mechanical studies of polymers
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