×

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.

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
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Biskup, M. and den Hollander, F. (1999). A heteropolymer near a linear interface. Ann. Appl. Probab. 9 668–687. · Zbl 0971.60098
[2] Bolthausen, E. (1987). Markov process large deviations in \(\tau\)-topology. Stochastic Process. Appl. 25 95–108. · Zbl 0625.60026
[3] Bolthausen, E. and den Hollander, F. (1997). Localization transition for a polymer near an interface. Ann. Probab. 25 1334–1366. · Zbl 0885.60022
[4] Chatelin, F. (1993). Eigenvalues of Matrices . Wiley, Chichester. · Zbl 0783.65031
[5] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications , 2nd ed. Springer, New York. · Zbl 0896.60013
[6] Feller, W. (1968). An Introduction to Probability Theory and its Applications 1 , 3rd ed. Wiley, New York. · Zbl 0155.23101
[7] Giacomin, G. (2003). Localization phenomena for random polymer models. Course lecture notes. Available at http://felix.proba.jussieu.fr/pageperso/giacomin/GBpage.html.
[8] Grosberg, A. Yu., Izrailev, S. and Nechaev, S. (1994). Phase transition in a heteropolymer chain at a selective interface. Phys. Rev. E 50 1912–1921.
[9] Isozaki, Y. and Yoshida, N. (2001). Weakly pinned random walk on the wall : Pathwise descriptions of the phase transition. Stochastic Process. Appl. 96 261–284. · Zbl 1058.60091
[10] Minc, H. (1988). Nonnegative Matrices . Wiley, New York. · Zbl 0638.15008
[11] Monthus, C. (2000). On the localization of random heteropolymers at the interface between two selective solvents. European Phys. J. B 13 111–130.
[12] Monthus, C., Garel, T. and Orland, H. (2000). Copolymer at a selective interface and two dimensional wetting: A grand canonical approach. European Phys. J. B 17 121–130.
[13] Janse van Rensburg, E. J. and Rechnitzer, A. (2001). Exchange relations, dyck paths and copolymer adsorption. · Zbl 1043.05009
[14] Sinai, Ya. G. (1993). A random walk with a random potential. Theory Probab. Appl. 38 382–385. · Zbl 0807.60069
[15] Sinai, Ya. G. and Spohn, H. (1996). Remarks on the delocalization transition for heteropolymers. In Topics in Statistical and Theoretical Physics (R. L. Dobrushin, R. A. Minlos, M. A. Shubin and A. M. Vershik, eds.) 219–223. Amer. Math. Soc., Providence, RI. · Zbl 0879.60114
[16] Sommer, J.-U. and Daoud, M. (1995). Copolymers at selective interfaces. Europhys. Lett. 32 407–412.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.