A copolymer near a selective interface: variational characterization of the free energy.

*(English)*Zbl 1330.60116A polymer is a chain of bonds (called monomers) that forms a path which describe a molecule. In the study of polymers there is a function \(H_{n}\), called the Hamiltonian, which associates an energy value to each path of \(n\)-monomers. Such a function also gives rise to a probability measure on the space of paths of the polymer. In this paper, a polymer is studied where each monomer has additionally associated another random value related to the environment, which itself is divided into two different substances; such a model is called a copolymer. For example, one such model is if there are two types of solvents in the environment, and each monomer is compulsory allergic to one of the substances. Thus, in one path of the polymer, there would be monomers living or not in the substance where they are allergic. In the model considered here, the Hamiltonian penalizes the situation where there are monomers living in their allergic substance.

One of the main questions is studying the so-called energy of the polymer when the length of the chain goes to infinity.

Since the environment is random, there are two kinds of probability measure, the quenched case when the measure is conditioned to the given environment, and the annealed case when it is not conditioned. The results of the paper are inequalities relating the energy of the polymer between the quenched and the annealed case. There are also inequalities for the so-called critical curve, which describes precisely where the energy function is zero.

The analysis is elaborated and it makes use of the so-called large deviation principles.

One of the main questions is studying the so-called energy of the polymer when the length of the chain goes to infinity.

Since the environment is random, there are two kinds of probability measure, the quenched case when the measure is conditioned to the given environment, and the annealed case when it is not conditioned. The results of the paper are inequalities relating the energy of the polymer between the quenched and the annealed case. There are also inequalities for the so-called critical curve, which describes precisely where the energy function is zero.

The analysis is elaborated and it makes use of the so-called large deviation principles.

Reviewer: Carlos Gabriel Pacheco (México D. F.)

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60K37 | Processes in random environments |

60F10 | Large deviations |

82B27 | Critical phenomena in equilibrium statistical mechanics |

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

##### Keywords:

copolymer; selective interface; free energy; critical curve; random environment; large deviation principles
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\textit{E. Bolthausen} et al., Ann. Probab. 43, No. 2, 875--933 (2015; Zbl 1330.60116)

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