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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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