# zbMATH — the first resource for mathematics

A copolymer near a selective interface: variational characterization of the free energy. (English) Zbl 1330.60116
A 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.

##### 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
Full Text:
##### References:
 [1] Berger, Q., Caravenna, F., Poisat, J., Sun, R. and Zygouras, N. (2014). The critical curves of the random pinning and copolymer models at weak coupling. Comm. Math. Phys. 326 507-530. · Zbl 1320.82074 [2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27 . Cambridge Univ. Press, Cambridge. · Zbl 0617.26001 [3] Birkner, M. (2008). Conditional large deviations for a sequence of words. Stochastic Process. Appl. 118 703-729. · Zbl 1136.60019 [4] Birkner, M., Greven, A. and den Hollander, F. (2010). Quenched large deviation principle for words in a letter sequence. Probab. Theory Related Fields 148 403-456. · Zbl 1243.60027 [5] Birkner, M., Greven, A. and den Hollander, F. (2011). Collision local time of transient random walks and intermediate phases in interacting stochastic systems. Electron. J. Probab. 16 552-586. · Zbl 1228.60054 [6] Biskup, M. and den Hollander, F. (1999). A heteropolymer near a linear interface. Ann. Appl. Probab. 9 668-687. · Zbl 0971.60098 [7] Bodineau, T. and Giacomin, G. (2004). On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117 801-818. · Zbl 1089.82031 [8] Bodineau, T., Giacomin, G., Lacoin, H. and Toninelli, F. L. (2008). Copolymers at selective interfaces: New bounds on the phase diagram. J. Stat. Phys. 132 603-626. · Zbl 1157.82025 [9] Bolthausen, E. and den Hollander, F. (1997). Localization transition for a polymer near an interface. Ann. Probab. 25 1334-1366. · Zbl 0885.60022 [10] Caravenna, F. and Giacomin, G. (2005). On constrained annealed bounds for pinning and wetting models. Electron. Commun. Probab. 10 179-189 (electronic). · Zbl 1136.82328 [11] Caravenna, F. and Giacomin, G. (2010). The weak coupling limit of disordered copolymer models. Ann. Probab. 38 2322-2378. · Zbl 1242.82022 [12] Caravenna, F., Giacomin, G. and Gubinelli, M. (2006). A numerical approach to copolymers at selective interfaces. J. Stat. Phys. 122 799-832. · Zbl 1149.82357 [13] Caravenna, F., Giacomin, G. and Toninelli, F. L. (2012). Copolymers at selective interfaces: Settled issues and open problems. In Probability in Complex Physical Systems. Proceedings in Mathematics 11 289-310. Springer, Berlin. · Zbl 1246.82049 [14] Cheliotis, D. and den Hollander, F. (2013). Variational characterization of the critical curve for pinning of random polymers. Ann. Probab. 41 1767-1805. · Zbl 1281.60083 [15] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications , 2nd ed. Applications of Mathematics ( New York ) 38 . Springer, New York. · Zbl 0896.60013 [16] den Hollander, F. (2009). Random Polymers. Lecture Notes in Math. 1974 . Springer, Berlin. · Zbl 0862.60093 [17] den Hollander, F. (2010). A key large deviation principle for interacting stochastic systems. In Proceedings of the International Congress of Mathematicians. Volume IV 2258-2274. Hindustan Book Agency, New Delhi. · Zbl 1227.60111 [18] den Hollander, F. and Opoku, A. A. (2013). Copolymer with pinning: Variational characterization of the phase diagram. J. Stat. Phys. 152 846-893. · Zbl 1277.82080 [19] Feller, V. (1968). An Introduction to Probability Theory and Its Applications , 3rd ed. Wiley, New York. · Zbl 0155.23101 [20] Garel, T., Huse, D. A., Leibler, S. and Orland, H. (1989). Localization transition of random chains at interfaces. Europhys. Lett. 8 9-13. [21] Giacomin, G. (2007). Random Polymer Models . Imperial College Press, London. · Zbl 1125.82001 [22] Giacomin, G. and Toninelli, F. L. (2005). Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Related Fields 133 464-482. · Zbl 1098.60089 [23] Giacomin, G. and Toninelli, F. L. (2006). Smoothing of depinning transitions for directed polymers with quenched disorder. Phys. Rev. Lett. 96 070602. · Zbl 1113.82032 [24] Giacomin, G. and Toninelli, F. L. (2006). Smoothing effect of quenched disorder on polymer depinning transitions. Comm. Math. Phys. 266 1-16. · Zbl 1113.82032 [25] Giacomin, G. and Toninelli, F. L. (2006). The localized phase of disordered copolymers with adsorption. ALEA Lat. Am. J. Probab. Math. Stat. 1 149-180. · Zbl 1134.82006 [26] Mourrat, J.-C. (2012). On the delocalized phase of the random pinning model. In Séminaire de Probabilités XLIV 401-407. Springer, Heidelberg. · Zbl 1257.82118 [27] Orlandini, E., Rechnitzer, A. and Whittington, S. G. (2002). Random copolymers and the Morita approximation: Polymer adsorption and polymer localization. J. Phys. A 35 7729-7751. · Zbl 1049.82080 [28] Toninelli, F. L. (2008). Disordered pinning models and copolymers: Beyond annealed bounds. Ann. Appl. Probab. 18 1569-1587. · Zbl 1157.60090 [29] Toninelli, F. L. (2009). Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14 531-547. · Zbl 1189.60186
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.