K. S. Alexander, The Effect of Disorder on Polymer Depinning Transitions, Communications in Mathematical Physics, vol.35, issue.1
DOI : 10.1007/s00220-008-0425-5

K. S. Alexander and V. Sidoravicius, Pinning of polymers and interfaces by random potentials, The Annals of Applied Probability, vol.16, issue.2, pp.636-669, 2006.
DOI : 10.1214/105051606000000015

K. S. Alexander and N. Zygouras, Quenched and Annealed Critical Points in Polymer Pinning Models, Communications in Mathematical Physics, vol.280, issue.4, pp.659-689, 2009.
DOI : 10.1007/s00220-009-0882-5

A. Bar, A. Kabakç??-oglu, and D. , Denaturation of circular DNA: Supercoil mechanism, Physical Review E, vol.84, issue.4, p.41935, 2011.
DOI : 10.1103/PhysRevE.84.041935

A. Bar, A. Kabakç??-oglu, and D. Mukamel, Denaturation of circular DNA: Supercoils and overtwist, Physical Review E, vol.86, issue.6, p.61904, 2012.
DOI : 10.1103/PhysRevE.86.061904

Q. Berger, Bivariate renewal theory: renewal theorems and local large deviation estimates , work in progress

Q. Berger, F. Caravenna, J. Poisat, R. Sun, and N. Zygouras, The Critical Curves of the Random Pinning and Copolymer Models at Weak Coupling, Communications in Mathematical Physics, vol.14, issue.33, pp.507-530, 2014.
DOI : 10.1007/s00220-013-1849-0

Q. Berger and H. Lacoin, Sharp critical behavior for pinning models in a random correlated environment, Stochastic Process, Appl, vol.122, pp.1397-1436, 2012.

Q. Berger and H. Lacoin, The Effect of Disorder on the Free-Energy for??the??Random Walk Pinning Model: Smoothing of??the??Phase Transition and Low Temperature Asymptotics, Journal of Statistical Physics, vol.63, issue.2, pp.42-322, 2011.
DOI : 10.1007/s10955-010-0110-x
URL : https://hal.archives-ouvertes.fr/hal-01426338

Q. Berger and H. Lacoin, PINNING ON A DEFECT LINE: CHARACTERIZATION OF MARGINAL DISORDER RELEVANCE AND SHARP ASYMPTOTICS FOR THE CRITICAL POINT SHIFT, Journal of the Institute of Mathematics of Jussieu, vol.2025, pp.1-42, 2016.
DOI : 10.1063/1.4895760
URL : https://hal.archives-ouvertes.fr/hal-01141691

Q. Berger and H. Lacoin, The high-temperature behavior for the directed polymer in dimension $1+2$, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.53, issue.1
DOI : 10.1214/15-AIHP721
URL : https://hal.archives-ouvertes.fr/hal-01170204

Q. Berger and J. Poisat, On the critical curves of the pinning and copolymer models in correlated Gaussian environment, Electronic Journal of Probability, vol.20, issue.0, p.35, 2015.
DOI : 10.1214/EJP.v20-3514
URL : https://hal.archives-ouvertes.fr/hal-01214672

Q. Berger and F. Toninelli, On the critical point of the Random Walk Pinning Model in dimension d = 3, Electron, J. Probab, vol.15, pp.654-683, 2010.

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, 1987.
DOI : 10.1017/CBO9780511721434

M. Birkner and R. Sun, Annealed vs quenched critical points for a random walk pinning model, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.46, issue.2, pp.414-441, 2010.
DOI : 10.1214/09-AIHP319

M. Birkner and R. Sun, Disorder relevance for the random walk pinning model in dimension 3, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.47, issue.1, pp.259-293, 2011.
DOI : 10.1214/10-AIHP374

R. D. Blake and S. G. Delcourt, Thermal stability of DNA, Thermal stability of DNA, pp.3323-3332, 1998.
DOI : 10.1093/nar/26.14.3323

A. A. Borovkov and K. A. Borovkov, On probabilities of large deviations for random walks. I. Regularly varying distribution tails, Theory Probab, Appl, vol.46, pp.193-213, 2000.

A. A. Borovkov and A. A. , Mogul'ski ? i, The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks, Sibirsk. Mat. Zh, vol.37, issue.4, pp.745-782, 1996.

A. A. Borovkov and A. A. , Mogul'ski ? i, Integro-local limit theorems including large deviations for sums of random vectors II, Theory Probab, Appl, vol.45, pp.3-22, 2001.

A. A. Borovkov and A. A. , Mogul'ski ? i, On large deviations of sums of independent random vectors on the boundary and outside of the cramér zone I, Theory Probab, Appl, vol.53, issue.2, pp.301-311, 2009.

R. Brak, A. L. Owczarek, T. Prellberg, and A. J. Guttmann, Finite-length scaling of collapsing directed walks, Phys. Rev. E, vol.48, pp.2386-2396, 1993.

R. Bundschuh and T. Hwa, Statistical mechanics of secondary structures formed by random RNA sequences, Physical Review E, vol.65, issue.3, p.31903, 2002.
DOI : 10.1103/PhysRevE.65.031903

F. Caravenna and F. Hollander, A general smoothing inequality for disordered polymers, Electronic Communications in Probability, vol.18, issue.0, pp.1-15, 2013.
DOI : 10.1214/ECP.v18-2874

F. Caravenna, G. Giacomin, and L. Zambotti, Sharp asymptotic behavior for wetting models in (1+1)-dimension, Elect, J. Probab, vol.11, pp.345-362, 2006.

F. Caravenna, F. L. Toninelli, and N. Torri, Universality for the pinning model in the weak coupling regime
URL : https://hal.archives-ouvertes.fr/hal-01254304

P. Carmona, N. Pétrélis, and G. B. Nguyen, Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase, The Annals of Probability, vol.44, issue.5
DOI : 10.1214/15-AOP1046
URL : https://hal.archives-ouvertes.fr/hal-01382110

E. Carlon, E. Orlandini, and A. L. Stella, Roles of Stiffness and Excluded Volume in DNA Denaturation, Physical Review Letters, vol.88, issue.19, 2002.
DOI : 10.1103/PhysRevLett.88.198101

D. Cheliotis and F. Hollander, Variational characterization of the critical curve for pinning of random polymers, The Annals of Probability, vol.41, issue.3B, pp.41-1767, 2013.
DOI : 10.1214/11-AOP727

F. Comets, Weak disorder for low dimensional polymers: the model of stable laws. Markov Process, Markov Process, pp.681-696, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00204644

D. Cule and T. Hwa, Denaturation of Heterogeneous DNA, Physical Review Letters, vol.79, issue.12, p.2375, 1997.
DOI : 10.1103/PhysRevLett.79.2375

B. Derrida, G. Giacomin, H. Lacoin, and F. L. Toninelli, Fractional Moment Bounds and Disorder Relevance for Pinning Models, Communications in Mathematical Physics, vol.18, issue.3, pp.867-887, 2009.
DOI : 10.1007/s00220-009-0737-0
URL : https://hal.archives-ouvertes.fr/hal-00202700

B. Derrida, V. Hakim, and J. Vannimenus, Effect of disorder on two-dimensional wetting, Journal of Statistical Physics, vol.8, issue.5-6, pp.1189-1213, 1992.
DOI : 10.1007/BF01054419

B. Derrida and M. Retaux, The Depinning Transition in Presence of Disorder: A Toy Model, Journal of Statistical Physics, vol.18, issue.2, pp.268-290, 2014.
DOI : 10.1007/s10955-014-1006-y

M. Doi and S. F. Edwards, The theory of Polymer dynamics, 1986.

R. A. Doney, An Analogue of the Renewal Theorem in Higher Dimensions, Proc. London Math. Soc. (3), pp.669-684, 1966.
DOI : 10.1112/plms/s3-16.1.669

R. A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probability Theory and Related Fields, vol.107, issue.4, pp.451-465, 1997.
DOI : 10.1007/s004400050093

T. R. Einert, H. Orland, and R. R. Netz, Secondary structure formation of homopolymeric single-stranded nucleic acids including force and loop entropy: Implications for DNA hybridization, The European Physical Journal E, vol.69, issue.6, p.55, 2011.
DOI : 10.1140/epje/i2011-11055-2

M. E. Fisher, Walks, walls, wetting, and melting, Journal of Statistical Physics, vol.32, issue.50, pp.667-729, 1984.
DOI : 10.1007/BF01009436

P. Flory, Principles of Polymer Chemistry, 1953.

G. Forgacs, J. M. Luck, T. M. Nieuwenhuizen, and H. Orland, Wetting of a Disordered Substrate: Exact Critical Behavior in Two Dimensions, Physical Review Letters, vol.57, issue.17, pp.2184-2187, 1986.
DOI : 10.1103/PhysRevLett.57.2184

T. Garel and C. Monthus, Numerical study of the disordered Poland???Scheraga model of DNA denaturation, Journal of Statistical Mechanics: Theory and Experiment, vol.2005, issue.06, p.6004, 2005.
DOI : 10.1088/1742-5468/2005/06/P06004

T. Garel and H. Orland, On the role of mismatches in DNA denaturation, arXiv:cond- mat, p.304080

T. Garel and H. Orland, Generalized Poland-Scheraga model for DNA hybridization, Biopolymers, vol.65, issue.6, pp.453-467, 2004.
DOI : 10.1002/bip.20140

G. Giacomin, Random polymer models, World Scientific, 2007.
DOI : 10.1142/p504
URL : https://hal.archives-ouvertes.fr/hal-00155080

G. Giacomin, Renewal convergence rates and correlation decay for homogeneous pinning models, Elect, J. Probab, vol.13, pp.513-529, 2008.
DOI : 10.1214/ejp.v13-497
URL : http://arxiv.org/abs/0706.0341

G. Giacomin, Disorder and critical phenomena through basic probability models ,Lecture notes from the 40th Probability Summer School held in Saint-Flour, Lecture Notes in Mathematics, 2010.

G. Giacomin and M. Khatib, Generalized Poland Sheraga denaturation model and two dimensional renewal processes

G. Giacomin, H. Lacoin, and F. L. Toninelli, Hierarchical pinning models, quadratic maps and quenched disorder, Probability Theory and Related Fields, vol.18, issue.1-2, pp.185-216, 2010.
DOI : 10.1007/s00440-009-0205-y
URL : https://hal.archives-ouvertes.fr/hal-00469493

G. Giacomin, H. Lacoin, and F. L. Toninelli, Marginal relevance of disorder for pinning models, Communications on Pure and Applied Mathematics, vol.14, issue.20, pp.233-265, 2010.
DOI : 10.1002/cpa.20301
URL : https://hal.archives-ouvertes.fr/hal-00338985

G. Giacomin, H. Lacoin, and F. L. Toninelli, Disorder relevance at marginality and critical point shift, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.47, issue.1, pp.148-175, 2011.
DOI : 10.1214/10-AIHP366
URL : https://hal.archives-ouvertes.fr/hal-00413563

G. Giacomin and F. L. Toninelli, The localized phase of disordered copolymers with adsorption, ALEA-Latin American J. Probab, Math. Stat, vol.1, pp.149-180, 2006.

G. Giacomin and F. L. Toninelli, Smoothing Effect of Quenched Disorder on Polymer Depinning Transitions, Communications in Mathematical Physics, vol.46, issue.1, pp.1-16, 2006.
DOI : 10.1007/s00220-006-0008-2
URL : https://hal.archives-ouvertes.fr/hal-00015489

G. Giacomin and F. L. Toninelli, The localized phase of disordered copolymers with adsorption, ALEA-Latin American Journal of Probability and Mathematical Statistics, vol.1, pp.149-180, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00086236

A. B. Harris, Effect of random defects on the critical behaviour of Ising models, Journal of Physics C: Solid State Physics, vol.7, issue.9, pp.1671-1692, 1974.
DOI : 10.1088/0022-3719/7/9/009

F. Hollander, Random polymers, Lectures from the 37th Probability Summer School held in Saint-Flour, Lecture Notes in Mathematics, 1974.

J. Hunter, Renewal theory in two dimensions: asymptotic results, Advances in Applied Probability, vol.35, issue.03, pp.546-562, 1974.
DOI : 10.1090/S0002-9947-1949-0032114-7

A. Kabakç??-oglu, A. Bar, and D. Mukamel, Macroscopic loop formation in circular DNA denaturation, Physical Review E, vol.85, issue.5, p.51919, 2012.
DOI : 10.1103/PhysRevE.85.051919

Y. Kafri and D. , Griffiths Singularities in Unbinding of Strongly Disordered Polymers, Physical Review Letters, vol.91, issue.5, pp.91-038103, 2003.
DOI : 10.1103/PhysRevLett.91.055502

Y. Kafri, D. Mukamel, and L. Peliti, Why is the DNA Denaturation Transition First Order?, Physical Review Letters, vol.85, issue.23, pp.4988-4991, 2000.
DOI : 10.1103/PhysRevLett.85.4988

L. Kaup and B. Kaup, Holomorphic functions of several variables, Studies in Mathematics, 1983.
DOI : 10.1515/9783110838350

J. F. Kingman, Subadditive Ergodic Theory, The Annals of Probability, vol.1, issue.6, pp.882-909, 1973.
DOI : 10.1214/aop/1176996798
URL : http://projecteuclid.org/download/pdf_1/euclid.aop/1176996798

S. G. Krantz and H. R. Parks, A primer of real analytic functions, 2002.

H. Kunz and R. Livi, DNA denaturation and wetting in the presence of disorder, EPL (Europhysics Letters), vol.99, issue.3, p.30001, 2012.
DOI : 10.1209/0295-5075/99/30001

H. Lacoin, New Bounds for the Free Energy of Directed Polymers in Dimension 1??+??1 and 1??+??2, Communications in Mathematical Physics, vol.134, issue.3/4, pp.471-503, 2010.
DOI : 10.1007/s00220-009-0957-3
URL : https://hal.archives-ouvertes.fr/hal-00361863

H. Lacoin, The Martingale approach to disorder irrelevance for pinning models, Electronic Communications in Probability, vol.15, issue.0, pp.418-427, 2010.
DOI : 10.1214/ECP.v15-1572

A. Litan, A statistical mechanical treatment of the open ends in a double-stranded polynucleotide molecule, Biopolymers, vol.40, issue.3, pp.279-282, 1964.
DOI : 10.1002/bip.1964.360020309

S. T. Lovett, Encoded errors: mutations and rearrangements mediated by misalignment at repetitive DNA sequences, Molecular Microbiology, vol.307, issue.5, p.1243, 2004.
DOI : 10.1111/j.1365-2958.2004.04076.x

R. A. Neher and U. Gerland, Intermediate phase in DNA melting, Physical Review E, vol.73, issue.3, p.30902, 2006.
DOI : 10.1103/PhysRevE.73.030902

G. B. Nguyen and N. Pétrélis, A Variational Formula for the Free Energy of the Partially Directed Polymer Collapse, Journal of Statistical Physics, vol.48, issue.6, pp.1099-1120, 2013.
DOI : 10.1007/s10955-013-0748-2

C. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, F. Sciortino et al., Long-range correlations in nucleotide sequences, Nature, vol.356, issue.6365, pp.168-170, 1992.
DOI : 10.1038/356168a0

I. F. Pinelis, A problem on large deviations in a space of trajectories, Theory Probab, Appl, vol.26, pp.69-84, 1981.

D. Poland and H. A. Schegara, Phase transitions in one dimension and the HelixCoil transition in polyamino acids, J. Chem. Phys, pp.45-1456, 1966.

D. Poland and H. A. Schegara, Occurrence of a Phase Transition in Nucleic Acid Models, The Journal of Chemical Physics, vol.45, issue.5, pp.1464-1469, 1966.
DOI : 10.1063/1.1727786

D. Poland and H. A. Scheraga, Theory of helix-coil transitions in biopolymers;: Statistical mechanical theory of order-disorder transitions in biological macromolecules, 1970.

C. Richard and A. J. Guttmann, Poland???Scheraga Models and the DNA Denaturation Transition, Journal of Statistical Physics, vol.115, issue.3/4, pp.925-947, 2004.
DOI : 10.1023/B:JOSS.0000022370.48118.8b

E. Rouchka, RNA secondary structure prediction, Lecture notes, 2003.

S. Shneer and V. Wachtel, A unified approach to the heavy-traffic analysis of the maximum of random walks, Theory Probab, Appl, vol.55, pp.332-341, 2011.

K. Sture, J. Nordholm, and S. A. Rice, Phase transitions and end effects in models of biopolymers, J. Chem. Phys, pp.59-5605, 1973.

M. V. Tamm and S. K. Nechaev, Unzipping of two random heteropolymers: Ground-state energy and finite-size effects, Physical Review E, vol.78, issue.1, p.11903, 2008.
DOI : 10.1103/PhysRevE.78.011903
URL : https://hal.archives-ouvertes.fr/hal-00351856

F. Tanaka and H. Takahashi, Elastic theory of supercoiled DNA, The Journal of Chemical Physics, vol.83, issue.11, p.6017, 1985.
DOI : 10.1063/1.449637

R. Thomas, The denaturation of deoxyribonucleic acids, Trans. Faraday Soc, vol.50, p.304, 1954.

I. Tobias, B. D. Coleman, and W. K. Olson, The dependence of DNA tertiary structure on end conditions: Theory and implications for topological transitions, The Journal of Chemical Physics, vol.101, issue.12, pp.10990-10996, 1994.
DOI : 10.1063/1.467849

F. L. Toninelli, Critical Properties and Finite-Size Estimates for the Depinning Transition of Directed Random Polymers, Journal of Statistical Physics, vol.37, issue.4-5, pp.1025-1044, 2007.
DOI : 10.1007/s10955-006-9123-x

F. L. Toninelli, A Replica-Coupling Approach to Disordered Pinning Models, Communications in Mathematical Physics, vol.163, issue.1, pp.389-401, 2008.
DOI : 10.1007/s00220-008-0469-6

F. L. Toninelli, Disordered pinning models and copolymers: Beyond annealed bounds, The Annals of Applied Probability, vol.18, issue.4, pp.1569-1587, 2008.
DOI : 10.1214/07-AAP496
URL : http://arxiv.org/abs/0709.1629

T. Vojta and R. Sknepnek, Critical points and quenched disorder: From Harris criterion to rare regions and smearing, physica status solidi (b), vol.837, issue.9, pp.2118-2127, 2004.
DOI : 10.1002/pssb.200404798

F. Watbled, Sharp asymptotics for the free energy of 1+1 dimensional directed polymers in an infinitely divisible environment, Electronic Communications in Probability, vol.17, issue.0, pp.1-9, 2012.
DOI : 10.1214/ECP.v17-2221
URL : https://hal.archives-ouvertes.fr/hal-00723044

J. D. Watson and F. H. Crick, MOLECULAR STRUCTURE OF NUCLEIC ACIDS, JAMA, vol.269, issue.15, pp.737-738, 1953.
DOI : 10.1001/jama.1993.03500150078030

J. H. White and W. R. Bauer, Calculation of the twist and the writhe for representative models of DNA, Journal of Molecular Biology, vol.189, issue.2, pp.329-341, 1986.
DOI : 10.1016/0022-2836(86)90513-9

R. Wei, On the Long-Range Directed Polymer Model, Journal of Statistical Physics, vol.138, issue.3???4
DOI : 10.1007/s10955-016-1612-y

E. Yeramian, Complexity and Tractability. Statistical Mechanics of Helix-Coil Transitions in Circular DNA as a Model-Problem, Europhysics Letters (EPL), vol.25, issue.1, pp.49-55, 1964.
DOI : 10.1209/0295-5075/25/1/009