For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.
References
K. S. Alexander and N. Zygouras. Quenched and annealed critical points in polymer pinning models. arXiv:0805.1708. 0805.1708 MR2534789 1188.82154 10.1007/s00220-009-0882-5K. S. Alexander and N. Zygouras. Quenched and annealed critical points in polymer pinning models. arXiv:0805.1708. 0805.1708 MR2534789 1188.82154 10.1007/s00220-009-0882-5
T. Bodineau and G. Giacomin. On the localization transition of random copolymers near selective interfaces J. Statist. Phys. 117 (2004), 801–818. 1089.82031 10.1007/s10955-004-5705-7T. Bodineau and G. Giacomin. On the localization transition of random copolymers near selective interfaces J. Statist. Phys. 117 (2004), 801–818. 1089.82031 10.1007/s10955-004-5705-7
T. Bodineau, G. Giacomin, H. Lacoin and F. L. Toninelli. Copolymers at selective interfaces: new bounds on the phase diagram. J. Statist. Phys. 132 (2008), 603-626. MR2429695 1157.82025 10.1007/s10955-008-9579-yT. Bodineau, G. Giacomin, H. Lacoin and F. L. Toninelli. Copolymers at selective interfaces: new bounds on the phase diagram. J. Statist. Phys. 132 (2008), 603-626. MR2429695 1157.82025 10.1007/s10955-008-9579-y
E. Bolthausen, F. Caravenna and B. de Tilière. The quenched critical point of a diluted disordered polymer model. Stochastic Process. Appl., to appear. arXiv:0711.0141v1 [math.PR]. 0711.0141v1 MR2513116 10.1016/j.spa.2008.07.008E. Bolthausen, F. Caravenna and B. de Tilière. The quenched critical point of a diluted disordered polymer model. Stochastic Process. Appl., to appear. arXiv:0711.0141v1 [math.PR]. 0711.0141v1 MR2513116 10.1016/j.spa.2008.07.008
E. Bolthausen and F. den Hollander. Localization transition for a polymer near an interface. Ann. Probab. 25 (1997), 1334–1366. 0885.60022 10.1214/aop/1024404516 euclid.aop/1024404516E. Bolthausen and F. den Hollander. Localization transition for a polymer near an interface. Ann. Probab. 25 (1997), 1334–1366. 0885.60022 10.1214/aop/1024404516 euclid.aop/1024404516
F. Caravenna, G. Giacomin and M. Gubinelli. A Numerical Approach to Copolymers at Selective Interfaces. J. Statist. Phys. 122 (2006), 799–832. 1149.82357 10.1007/s10955-005-8081-zF. Caravenna, G. Giacomin and M. Gubinelli. A Numerical Approach to Copolymers at Selective Interfaces. J. Statist. Phys. 122 (2006), 799–832. 1149.82357 10.1007/s10955-005-8081-z
B. Derrida, G. Giacomin, H. Lacoin and F. L. Toninelli. Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys., to appear. arXiv:0712.2515 [math.PR]. 0712.2515 1226.82028 10.1007/s00220-009-0737-0B. Derrida, G. Giacomin, H. Lacoin and F. L. Toninelli. Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys., to appear. arXiv:0712.2515 [math.PR]. 0712.2515 1226.82028 10.1007/s00220-009-0737-0
R. A. Doney. One-sided local large deviations and renewal theorems in the case of infinite mean. Probab. Theory Rel. Fields 107 (1997), 451–465. 0883.60022 10.1007/s004400050093R. A. Doney. One-sided local large deviations and renewal theorems in the case of infinite mean. Probab. Theory Rel. Fields 107 (1997), 451–465. 0883.60022 10.1007/s004400050093
W. Feller. An introduction to probability theory and its applications, Vol. I, Second Edition, John Wiley & Sons, 1966. 0138.10207W. Feller. An introduction to probability theory and its applications, Vol. I, Second Edition, John Wiley & Sons, 1966. 0138.10207
G. Giacomin. Random Polymer Models. Imperial College Press, London (2007). 1125.82001G. Giacomin. Random Polymer Models. Imperial College Press, London (2007). 1125.82001
G. Giacomin, H. Lacoin and F. L. Toninelli. Hierarchical pinning models, quadratic maps and quenched disorder. Probab. Theory Rel. Fields, to appear. arXiv:0711.4649 [math.PR]. 0711.4649 1190.60093 10.1007/s00440-009-0205-yG. Giacomin, H. Lacoin and F. L. Toninelli. Hierarchical pinning models, quadratic maps and quenched disorder. Probab. Theory Rel. Fields, to appear. arXiv:0711.4649 [math.PR]. 0711.4649 1190.60093 10.1007/s00440-009-0205-y
G. Giacomin and F. L. Toninelli. Estimates on path delocalization for copolymers at selective interfaces. Probab.Theory Rel. Fields 133 (2005), 464–482. 1098.60089 10.1007/s00440-005-0439-2G. Giacomin and F. L. Toninelli. Estimates on path delocalization for copolymers at selective interfaces. Probab.Theory Rel. Fields 133 (2005), 464–482. 1098.60089 10.1007/s00440-005-0439-2
F. L. Toninelli. Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18 (2008), 1569-1587. 1157.60090 10.1214/07-AAP496 euclid.aoap/1216677132F. L. Toninelli. Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18 (2008), 1569-1587. 1157.60090 10.1214/07-AAP496 euclid.aoap/1216677132