Abstract
We introduce a class of statistical mechanics non-disordered models – the homogeneous pinning models – starting with the particular case of random walk pinning. We solve the model in the sense that we compute the precise asymptotic behavior of the partition function of the model. In particular, we obtain a formula for the free energy and show that the model exhibits a phase transition, in fact a localization/delocalization transition. We focus in particular on the critical behavior, that is on the behavior of the system close to the phase transition. The approach is then generalized to a general class of Markov chain pinning, which is more naturally introduced in terms of (discrete) renewal processes. We complete the chapter by introducing the crucial notion of correlation length and by giving an overview of the applications of pinning models. Ising models are presented at this stage because pinning systems appear naturally as limits of two dimensional Ising models with suitably chosen interaction potentials. In spite of the fact that these lecture notes may be read focusing exclusively on pinning, the physical literature on disordered systems and Ising models cannot be easily disentangled. So a full appreciation of some physical arguments/discussions in these notes does require being acquainted with Ising models.
Keywords
- Partition Function
- Random Walk
- Correlation Length
- Ising Model
- Renewal Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
eferences
D.B. Abraham, Surface structures and phase transitions, exact results, in Phase Transitions and Critical Phenomena, vol. 10 (Academic, London, 1986), pp. 1–74
K.S. Alexander, Excursions and local limit theorems for Bessel-like random walks. Electron. J. Probab. 16, 1–44 (2011)
S. Asmussen, Applied Probability and Queues, 2nd edn. (Springer, New York, 2003)
R.J. Baxter, Exactly solved models in statistical mechanics (Academic, London, 1982)
J. Bertoin, Subordinators: examples and applications. Lectures on probability theory and statistics (Saint-Flour, 1997), Lecture Notes in Mathematics, vol. 1717 (1999), pp. 1–91
R. Blossey, E. Carlon, Reparametrizing the loop entropy weights: effect on DNA melting curves. Phys. Rev. E 68, 061911 (2003) (8 pages)
T.W. Burkhardt, Localization-delocalization transition in a solid-on-solid model with a pinning potential. J. Phys. A Math. Gen. 14, L63–L68 (1981)
P. Caputo, F. Martinelli, F.L. Toninelli, On the approach to equilibrium for a polymer with adsorption and repulsion. Electron. J. Probab. 13, 213–258 (2008)
F. Caravenna, J.-D. Deuschel, Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction. Ann. Probab. 36, 2388–2433 (2008)
F. Caravenna, G. Giacomin, L. Zambotti, Sharp asymptotic behavior for wetting models in (1+1)-dimension. Electron. J. Probab. 11, 345–362 (2006)
F. Caravenna, G. Giacomin, L. Zambotti, A renewal theory approach to periodic copolymers with adsorption. Ann. Appl. Probab. 17, 1362–1398 (2007)
F. Caravenna, N. Pétrélis, A polymer in a multi-interface medium. Ann. Appl. Probab. 19, 1803–1839 (2009)
J.B. Conway, Functions of One Complex Variable, 2nd edn. Graduate Texts in Mathematics, vol. 11 (Springer, New York, 1978)
M. Cranston, L. Koralov, S. Molchanov, B. Vainberg, Continuous model for homopolymers. J. Funct. Anal. 256, 2656–2696 (2009)
D. Cule, T. Hwa, Denaturation of heterogeneous DNA. Phys. Rev. Lett. 79, 2375–2378 (1997)
W. Feller, An Introduction to Probability Theory and Its Applications, vol. I, 3rd edn. (Wiley, New York, 1968)
W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. (Wiley, New York, 1971)
R. Fernández, J. Frölich, A.D. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Texts and Monographs in Physics (Springer, New York, 1992)
M.E. Fisher, Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–729 (1984)
M. Fixman, J.J. Freire, Theory of DNA melting curves. Biopolymers 16, 2693–2704 (1977)
A. Garsia, J. Lamperti, A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37, 221–234 (1963)
G. Giacomin, Random Polymer Models (Imperial College Press, London, 2007)
G. Giacomin, Renewal convergence rates and correlation decay for homogeneous pinning models. Electron. J. Probab. 13, 513–529 (2008)
G. Giacomin, Renewal sequences, disordered potentials, and pinning phenomena, in Spin Glasses: Statics and Dynamics, Progress in Probability, vol. 62 (2009), pp. 235–270
G. Giacomin, F.L. Toninelli, Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)
C.R. Heathcote, Complete exponential convergence and some related topics. J. Appl. Probab., 4, 217–256 (1967)
Y. Kafri, D. Mukamel, L. Peliti, Why is the DNA denaturation transition first order? Phys. Rev. Lett. 85, 4988–4991 (2000)
D.G. Kendall, Unitary dilations of Markov transition operators and the corresponding integral representation of transition probability matrices, in Probability and Statistics, ed. by U. Grenander (Almqvist and Wiksell, Stockholm, 1959), pp. 138–161
J.M.J. van Leeuwen, H.J. Hilhorst, Pinning of rough interface by an external potential. Phys. A 107, 319–329 (1981)
D. Marenduzzo, A. Trovato, A. Maritan, Phase diagram of force-induced DNA unzipping in exactly solvable models. Phys. Rev. E 64, 031901 (2001) (12 pages)
B. Roynette, M. Yor, Penalising Brownian Paths. Lecture Notes in Mathematics, vol. 1969 (Springer, New York, 2009)
J. Sohier, Finite size scaling for homogeneous pinning models. ALEA Lat. Am. J. Probab. Math. Stat. 6, 163–177 (2009)
J. Sohier, Phénomènes d’accrochage et théorie des fluctuations, PhD thesis, Univ. Paris Diderot, November 2010
F.L. Toninelli, Critical properties and finite-size estimates for the depinning transition of directed random polymers. J. Stat. Phys. 126, 1025–1044 (2007)
F.L. Toninelli, Correlation lengths for random polymer models and for some renewal sequences. Electron. J. Probab. 12, 613–636 (2007)
F.L. Toninelli, Localization transition in disordered pinning models. Effect of randomness on the critical properties, in Methods of Contemporary Mathematical Statistical Physics, Lecture Notes in Mathematics, vol. 1970, 129–176 (2009)
Y. Velenik, Localization and delocalization of random interfaces. Probab. Surv. 3, 112–169 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Giacomin, G. (2011). Homogeneous Pinning Systems: A Class of Exactly Solved Models. In: Disorder and Critical Phenomena Through Basic Probability Models. Lecture Notes in Mathematics(), vol 2025. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21156-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-21156-0_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21155-3
Online ISBN: 978-3-642-21156-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
