Introduction The model The proof A Polymer in a Multi-Interface Medium Francesco Caravenna Universit` a degli Studi di Padova LPMA ∼ December 8, 2009 Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 1 / 26
Introduction The model The proof References ◮ [CP1] F. Caravenna and N. P´ etr´ elis A polymer in a multi-interface medium AAP (2009) ◮ [CP2] F. Caravenna and N. P´ etr´ elis Depinning of a polymer in a multi-interface medium EJP (2009) Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 2 / 26
Introduction The model The proof Outline 1. Introduction and motivations Polymer models 2. The model and the main results Definition of the model The free energy Path results 3. Techniques and ideas from the proof Some heuristics A renewal theory approach Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 3 / 26
Introduction The model The proof Outline 1. Introduction and motivations Polymer models 2. The model and the main results Definition of the model The free energy Path results 3. Techniques and ideas from the proof Some heuristics A renewal theory approach Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 4 / 26
Introduction The model The proof Polymer models Copolymer and pinning at a single interface A polymer interacting with two solvents and with the interface that separates them: Oil Water Interface Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26
Introduction The model The proof Polymer models Copolymer and pinning at a single interface A polymer interacting with two solvents and with the interface that separates them: _ _ Copolymer _ + + + + + _ _ _ + + + _ + + + + _ + + + Oil + _ _ _ _ + _ + + + _ _ _ Water _ _ + Interface + + ◮ Copolymer interaction with the solvents Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26
Introduction The model The proof Polymer models Copolymer and pinning at a single interface A polymer interacting with two solvents and with the interface that separates them: Pinning Oil Water Interface ◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26
Introduction The model The proof Polymer models Copolymer and pinning at a single interface A polymer interacting with two solvents and with the interface that separates them: _ _ Pinning + Copolymer _ + + + + + _ _ _ + + + + _ + + _ + + + + Oil + _ _ _ _ + _ + + + _ _ _ Water _ _ + Interface + + ◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface ◮ Both interactions may be inhomogeneous and disordered Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26
Introduction The model The proof Polymer models Copolymer and pinning at a single interface A polymer interacting with two solvents and with the interface that separates them: _ _ Pinning + Copolymer _ + + + + + _ _ _ + + + + _ + + _ + + + + Oil + _ _ _ _ + _ + + + _ _ _ Water _ _ + Interface + + ◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface ◮ Both interactions may be inhomogeneous and disordered Localization vs. delocalization? Phase transitions? Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26
Introduction The model The proof Polymer models Copolymer and pinning at a single interface A polymer interacting with two solvents and with the interface that separates them: _ _ Pinning + Copolymer _ + + + + + _ _ _ + + + + _ + + _ + + + + Oil + _ _ _ _ + _ + + + _ _ _ Water _ _ + Interface + + ◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface ◮ Both interactions may be inhomogeneous and disordered Localization vs. delocalization? Phase transitions? Recent results: very good comprehension (survey: [Giacomin ’07]) Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26
Introduction The model The proof Polymer models Multi-interface media More general environments: a multi-interface medium Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 6 / 26
Introduction The model The proof Polymer models Multi-interface media More general environments: a multi-interface medium Water _ _ _ + + __ + _ + + _ + _ _ + _ _ + Oil + + +++ _ _ _ + + + + _ _ + _ + _ _ _ _ + __ + Water _ + + __ __ + + + T ◮ [den Hollander & W¨ uthrich JSP 04]: Copolymer interaction. Path results for log log N ≪ T N ≪ log N ( N = polymer size) Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 6 / 26
Introduction The model The proof Polymer models Multi-interface media More general environments: a multi-interface medium T ◮ [den Hollander & W¨ uthrich JSP 04]: Copolymer interaction. Path results for log log N ≪ T N ≪ log N ( N = polymer size) ◮ We focus on the pinning case. Homogeneous interaction (attractive or repulsive), general T N − → Path behavior ? Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 6 / 26
Introduction The model The proof Polymer models Polymer in a slit Recent physical literature: Polymer confined between two walls and interacting with them ◮ [Brak et al.; J Phys A 2005] ◮ [Martin et al.; J Phys A 2007] ◮ [Owkzarek et al.; J Phys A 2008] T δ 0 δ Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 7 / 26
Introduction The model The proof Polymer models Polymer in a slit Recent physical literature: Polymer confined between two walls and interacting with them ◮ [Brak et al.; J Phys A 2005] ◮ [Martin et al.; J Phys A 2007] ◮ [Owkzarek et al.; J Phys A 2008] T δ 0 δ Attraction/repulsion of interfaces by polymers Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 7 / 26
Introduction The model The proof Polymer models Polymer in a slit 3 T δ 2 T δ T δ 0 δ Φ T − T δ δ + log 2 T 0 δ + log 2 Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 8 / 26
Introduction The model The proof Outline 1. Introduction and motivations Polymer models 2. The model and the main results Definition of the model The free energy Path results 3. Techniques and ideas from the proof Some heuristics A renewal theory approach Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 9 / 26
Introduction The model The proof Definition of the model Definition Recall the situation we want to model T Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26
Introduction The model The proof Definition of the model Definition Recall the situation we want to model T Polymer configurations ← → Trajectories of a random process Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26
Introduction The model The proof Definition of the model Definition Recall the situation we want to model T Polymer configurations ← → Trajectories of a random process Random polymer model: probability measure P T N ,δ on paths Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26
Introduction The model The proof Definition of the model Definition Recall the situation we want to model T Polymer configurations ← → Trajectories of a random process Random polymer model: probability measure P T N ,δ on paths ◮ (1 + 1)-dimensionale model: { ( i , S i ) } i ≥ 0 Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26
Introduction The model The proof Definition of the model Definition Recall the situation we want to model T Polymer configurations ← → Trajectories of a random process Random polymer model: probability measure P T N ,δ on paths ◮ (1 + 1)-dimensionale model: { ( i , S i ) } i ≥ 0 ◮ P T N ,δ absolutely continuous w.r.t. SRW { S i } i ≥ 0 Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26
Introduction The model The proof Definition of the model Definition Ingredients of P T N ,δ : ◮ Simple symmetric random walk S = { S n } n ≥ 0 on Z : S 0 := 0 , S n := X 1 + . . . + X n , with { X i } i i.i.d. and P ( X i = +1) = P ( X i = − 1) = 1 2 . Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26
Introduction The model The proof Definition of the model Definition Ingredients of P T N ,δ : ◮ Simple symmetric random walk S = { S n } n ≥ 0 on Z : S 0 := 0 , S n := X 1 + . . . + X n , with { X i } i i.i.d. and P ( X i = +1) = P ( X i = − 1) = 1 2 . ◮ Polymer size N ∈ 2 N (number of monomers) Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26
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