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Brownian Polymers in Poissonian Environment: a survey
The self-consistency equation and UI properties in the weak disorder
The compensated Poisson measure and some associated martingales
Moderate and large deviations at all temperature
Connections between the stochastic heat equation(s) and the directed polymers in random environments
Table of contents :
I Introduction
1 Les polymeres diriges en milieu aleatoire
1.1 Origine et histoire du modele
1.2 Les modeles de polymeres
1.3 Resultats generaux sur les polymeres
1.4 Fluctuations de la queue de la fonction de partition normee dans la region L2 entiere, en dimension d
1.5 Polymeres et equation de la chaleur stochastique en dimension d 1
2 L’equation KPZ et l’equation SHE en dimension d = 1
2.1 Un modele d’interface aleatoire a l’equilibre
2.2 L’equation KPZ en dimension 1
2.3 L’equation de la chaleur stochastique avec bruit multiplicatif en dimension d = 1
2.4 Le polymere continu (d = 1) [1]
3 La classe d’universalite KPZ en dimension d = 1
3.1 Proprietes de la classe KPZ
3.2 Le regime de desordre intermediaire des polymeres en dimension d = 1
4 Les equations KPZ et SHE en dimension superieure
4.1 Procedure de regularisation
4.2 Reecriture du probleme en termes de polymeres
4.3 Cas de la dimension d 3
4.4 Fluctuations en dimension d 3
4.5 Commentaires
5 Resultats de la these et guide de lecture des chapitres
II Brownian Polymers in Poissonian Environment: a survey
1 Introduction
2 Free energy and phase transition
2.1 Polymer model
2.2 Some key formulas and notations
2.3 Quenched free energy
2.4 Annealed free energy and hierarchy of moments
2.5 Phase transition
3 Weak Disorder, Strong Disorder
3.1 The normalized partition function
3.2 The self-consistency equation and UI properties in the weak disorder
3.2.1 Proof of the self-consistency equation on W1
3.2.2 Uniform integrability in the weak disorder
3.3 The L2-region
3.4 Relations between the dierent critical temperatures
4 Directional free energy
4.1 Point-to-point partition function
4.2 Free energy does not depend on direction
4.3 Local limit theorem
5 The replica overlap and localization
5.1 The compensated Poisson measure and some associated martingales
5.2 The Doob-Meyer decomposition of ln Zt
5.3 The replica overlap and quenched overlaps
5.4 Endpoint localization
5.5 Favorite path and path localization
6 Formulas for variance and concentration
6.1 The critical exponents
6.2 The Clark-Ocone representation
6.3 The variance formula
7 Cameron-Martin transform and applications
7.1 Tilting the polymer
7.2 Consequences for weak disorder regime
7.3 Moderate and large deviations at all temperature
8 Phase diagram in the (; )-plane, d 3
8.1 Strategy for critical curve estimates
8.2 Main results
8.3 Main steps
9 Complete localization
9.1 A mean eld limit
9.2 The regime of complete localization
10 The Intermediate Regime (d = 1)
10.1 Introduction
10.2 Connections between stochastic heat equation(s) and directed polymers
10.2.1 The continuum case
10.2.2 The Poisson case
10.3 Chaos expansions
10.3.1 The continuum case
10.3.2 The Poisson case
10.4 The intermediate regime
10.5 Convergence in terms of processes of the P2P partition function
III The intermediate disorder regime for Brownian polymers in Poisson environment
1 Introduction
1.1 The model and its context
1.2 KPZ universality for polymers and the intermediate disorder regime
1.3 The KPZ equation and the stochastic heat equation
1.4 Connections between the stochastic heat equation(s) and the directed polymers in random environments
1.5 The KPZ universality class and the KPZ equation
1.6 The intermediate disorder regime for the discrete polymer with i.i.d. weights
2 Main results
3 The Wiener-It^o integrals with respect to Poisson process
3.1 The factorial measures
3.2 Multiple stochastic integral over a Poissonian medium
3.3 A Wiener-It^o Chaos Expansion of the normalized partition function
4 The Wiener integrals
4.1 Stochastic integral over the white noise
4.2 Multiple stochastic integral
4.3 Wiener chaos decomposition
4.4 Construction of the P2P and P2L functions of the continuum polymer
5 Asymptotic study of Wiener-It^o integrals
5.1 The scaling relations
5.2 Gaussian limits of Wiener-It^o integrals
6 Proofs
6.1 Some useful formulas
6.2 Proof of Theorem 1.2 : SHE in the Poisson setting
6.3 Proof of Theorem 2.4 : convergence of the P2L partition function
6.4 Proof of Theorem 2.5 : convergence of the point-to-point partition function
6.5 Proof of Theorem 2.7 : convergence in terms of processes
IV Renormalizing the Kardar-Parisi-Zhang equation in d 3 in weak disorder
1 Introduction and main results.
1.1 KPZ equation and its regularization.
1.2 Main results.
1.3 Literature remarks and discussion.
2 Proof of Theorem 1.1
2.1 General initial condition: Proof of (12).
2.2 Narrow-wedge initial condition: Proof of (13).
3 Proof of Theorem 1.3.
4 Appendix.
V Gaussian uctuations and rate of convergence of the Kardar-Parisi- Zhang equation for d 3
1 Introduction and the result.
1.1 Introduction and summary.
1.2 Main results.
2 Proof of Theorem 1.1 and Theorem 1.2.
2.1 Rate of decorrelation.
2.2 Proof of Theorem 1.2.
2.3 Proof of Theorem 1.1.
3 Proof of proposition 2.2
3.1 The rst moment.
3.2 Second moment.
3.3 Proof of Proposition 3.9.
3.4 Proof of Proposition 3.10.
VI Fluctuations of the tail of the polymer partition function for d 3 in the whole L2-region.
1 Introduction
1.1 The model
1.2 The results
1.3 Comments and connections to other models
2 Idea of the proof
2.1 A central limit theorem for martingales
2.2 Structure of the proof of Theorem 2.2
3 Proof
3.1 Some tools.
3.2 Removing the negligeable terms
3.3 The homogenization result
3.4 Proof of Theorem 2.2
3.5 Proof of condition (b): the Lindeberg condition
3.6 Proof of Corollary 1.3
