Large scale behavior of semiflexible heteropolymers
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 1, p. 97-118

We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the disorder) are modeled in terms of random rotations. We focus on the quenched regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative Fourier analysis, we establish the brownian character of the model on large scales and we obtain an expression for the diffusion constant. We moreover give conditions yielding quantitative mixing properties.

On considère un modèle discret pour un polymère semi-flexible et hétérogène. Le bruit thermique et le caractère hétérogène du polymère (le désordre) sont modélisés en termes de rotations aléatoires. Nous nous concentrons sur le régime de désordre gélé, c'est-à-dire, l'analyse est effectuée pour une réalisation fixée du désordre. Les modèles semi-flexibles diffèrent sensiblement des marches aléatoires à petite échelle, mais à grande échelle un comportement brownien apparaît. En exploitant des techniques de calcul tensoriel et d'analyse de Fourier non-commutative, nous établissons le caractère brownien du modèle à grande échelle et nous obtenons une expression pour la constante de diffusion. Nous donnons aussi des conditions qui entraînent des propriétés quantitatives de mélange.

DOI : https://doi.org/10.1214/08-AIHP310
Classification:  60K37,  82B44,  60F05,  43A75
Keywords: heteropolymer, semiflexible chain, disorder, persistence length, large scale limit, tensor analysis, non-commutative Fourier analysis
@article{AIHPB_2010__46_1_97_0,
     author = {Caravenna, Francesco and Giacomin, Giambattista and Gubinelli, Massimiliano},
     title = {Large scale behavior of semiflexible heteropolymers},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {1},
     year = {2010},
     pages = {97-118},
     doi = {10.1214/08-AIHP310},
     zbl = {1192.82041},
     mrnumber = {2641772},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_1_97_0}
}
Caravenna, Francesco; Giacomin, Giambattista; Gubinelli, Massimiliano. Large scale behavior of semiflexible heteropolymers. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 1, pp. 97-118. doi : 10.1214/08-AIHP310. http://www.numdam.org/item/AIHPB_2010__46_1_97_0/

[1] D. Bensimon, D. Dohmi and M. Mézard. Stretching a heteropolymer. Europhys. Lett. 42 (1998) 97-102.

[2] K. F. Freed. Polymers as self-avoiding walks. Ann. Probab. 9 (1981) 537-556. | MR 624681 | Zbl 0468.60097

[3] P. G. De Gennes. Scaling Concepts in Polymer Physics. Cornell Univ. Press, Ithaca, NY, 1979.

[4] P. R. Halmos. Measure Theory. Graduate Texts in Mathematics 18. Springer, Berlin, 1974. | Zbl 0283.28001

[5] E. Hewitt and K. A. Ross. Abstract Harmonic Analysis I. Grundlehren der Mathematischen Wissenschaften 115. Springer, Berlin, 1963. | Zbl 0115.10603

[6] E. Hewitt and K. A. Ross. Abstract Harmonic Analysis II. Grundlehren der Mathematischen Wissenschaften 152. Springer, Berlin, 1970. | MR 262773 | Zbl 0830.43001

[7] L. G. Gorostiza. The central limit theorem for random motions for d dimensional Euclidean space. Ann. Probab. 1 (1973) 603-612. | MR 353408 | Zbl 0263.60010

[8] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin, 2003. | MR 1943877 | Zbl 1018.60002

[9] J. F. Marko and E. Siggia. Bending and twisting elasticity of DNA. Macromolecules 27 (1994) 981-988.

[10] J. D. Moroz and P. Nelson. Torsional directed walks, entropic elasticity, and DNA twist stiffness. Proc. Natl. Acad. Sci. USA 94 (1997) 14418-14422.

[11] P. H. Roberts and H. D. Ursell. Random walk on a sphere and on a Riemannian manifold. Philos. Trans. R. Soc. London Ser. A Math. Phys. Eng. Sci. 252 (1960) 317-356. | MR 117795 | Zbl 0094.31901

[12] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales 2. Cambridge Mathematical Library. Cambridge Univ. Press, 2000. (Itô Calculus, 2nd edition.) | MR 1780932 | Zbl 0977.60005

[13] J. S. Rosenthal. Random rotations: Characters and random walks on SO(N). Ann. Probab. 22 (1994) 398-423. | MR 1258882 | Zbl 0799.60007

[14] B. Roynette. Théorème central-limite pour le groupe des déplacements de ℝd. Ann. Inst. H. Poincaré Probab. Statist. 10 (1974) 391-398. | Numdam | MR 375422 | Zbl 0324.60026

[15] C. Vaillant, B. Audit, C. Thermes and A. Arnéodo. Formation and positioning of nucleosomes: Effect of sequence-dependent long-range correlated structural disorder. Eur. Phys. J. E 19 (2006) 263-277.

[16] P. A. Wiggins and P. C. Nelson. Generalized theory of semiflexible polymers. Phys. Rev. E 73 (2006) 031906.

[17] Y. Zhou and G. S. Chirikjian. Conformational statistics of semiflexible macromolecular chains with internal joints. Macromolecules 39 (2006) 1950-1960. | MR 2755422