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**Periodic copolymers at selective interfaces: a large deviations approach.**
*(English)*
Zbl 1075.60123

E. Bolthausen and F. den Hollander [Ann. Probab. 25, No. 3, 1334–1366 (1997; Zbl 0885.60022)], considered a model for a heterogeneous polymer consisting of hydrophobic and hydrophylic monomers near an oil-water interface. Each \(N\)-step path of \(S=(S_x; x\in \mathbb N )\), a simple symmetric random walk on \(\mathbb Z\) starting at 0, is weighted by the Hamiltonian
\[
H_N (S)= \lambda\sum_{x=1}^N (\omega_x +h)\Delta_x,
\]
where \(\Delta_x=\text{ sign} (S_x)\) if \(S_x\neq 0\) and \(\Delta_x=\text{ sign} (S_{x-1})\) if \(S_x= 0\). \(\lambda\geq 0\) is the coupling constant which plays the role of inverse temperature, \(h\geq 0\) is a parameter, and \(\omega =(\omega_x)\) is an i.i.d. sequence of random variables taking values \(\pm 1\) with probability \(1/2\). They showed that there is a critical curve \(\lambda\mapsto h_c(\lambda)\) where phase transition occurs between localized and delocalized behavior.

In this paper, the paths of \(S\) are weighted by the same Hamitonian, \(H_N\), but the basic assumption is that \(\omega\) is a fixed nontrivial centered and periodic sequence of \(\pm 1\)’s. The first result is an expression of the free energy in terms of the empirical measure of a Markov chain constructed from the return times to 0 of \(S\). A second expression of the free energy is given as the supremum of a variational problem which evaluates the competition between the energy and the entropy of the Markov chain. To prove this, one first shows, as is E. Bolthausen [Stochastic Processes Appl. 25, 95–108 (1987; Zbl 0625.60026)], that the empirical measure of the Markov chain verifies a full large deviation principle with rate functional given by the entropy. This leads to a fairly explicit formula for the free energy in terms of the Perron-Frobenius maximal eigenvalue. The existence of a critical curve and its precise asymptotic behavior at small and large values of \(\lambda\) then follows.

In this paper, the paths of \(S\) are weighted by the same Hamitonian, \(H_N\), but the basic assumption is that \(\omega\) is a fixed nontrivial centered and periodic sequence of \(\pm 1\)’s. The first result is an expression of the free energy in terms of the empirical measure of a Markov chain constructed from the return times to 0 of \(S\). A second expression of the free energy is given as the supremum of a variational problem which evaluates the competition between the energy and the entropy of the Markov chain. To prove this, one first shows, as is E. Bolthausen [Stochastic Processes Appl. 25, 95–108 (1987; Zbl 0625.60026)], that the empirical measure of the Markov chain verifies a full large deviation principle with rate functional given by the entropy. This leads to a fairly explicit formula for the free energy in terms of the Perron-Frobenius maximal eigenvalue. The existence of a critical curve and its precise asymptotic behavior at small and large values of \(\lambda\) then follows.

Reviewer: Daniel Boivin (Brest)

### MSC:

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

60F10 | Large deviations |

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

### Keywords:

localization-delocalization transition; energy-entropy competition; random walk; Donsker-Varadhan theory.
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\textit{E. Bolthausen} and \textit{G. Giacomin}, Ann. Appl. Probab. 15, No. 1B, 963--983 (2005; Zbl 1075.60123)

### References:

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