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Table of contents
Remerciements
Introduction, goal and outline of the manuscript
Index of notations and abbreviations
I Disordered elastic systems
I.1 The Hamiltonians
I.1.1 The state space
I.1.2 The elastic Hamiltonian
I.1.3 The disorder Hamiltonian
I.1.4 The confining Hamiltonian
I.2 Static phase diagram and the strong disorder regime
I.2.1 The pure system and the thermal fixed point
I.2.2 Relevance and irrelevance of short-range disorder at the thermal fixed point
I.2.3 Static phase diagram
I.2.4 Early attempts at characterizing the strong disorder regime: Flory arguments and the Larkin model
I.3 Various problems considered in this thesis
I.3.1 Shocks in the statics at zero temperature for elastic interfaces .
I.3.2 Avalanche dynamics at the depinning transition for elastic interfaces
I.3.3 Static problem at finite temperature for directed polymers with SR elasticity and the KPZ universality class
I.4 Experimental realizations
I.4.1 Disordered elastic systems pinned in a quenched random environment
I.4.2 Out-of-equilibrium interface growth
II Avalanches and shocks of disordered elastic interfaces
II.1 Introduction
II.2 Avalanches for a particle
II.2.1 Shocks between ground states for toy models of a particle without disorder
II.2.2 Avalanches in the dynamics of a particle on the real line
II.2.3 Shock process versus avalanche process for a particle
II.3 Avalanches for an interface
II.3.1 Shocks for an interface
II.3.2 Avalanches for an interface
II.4 FRG approach to shocks
II.4.1 The functional renormalization group for the statics of disordered elastic interfaces
II.4.2 Applying the functional renormalization group to shocks
II.5 FRG approach to avalanches
II.5.1 The functional renormalization approach to the depinning transition
II.5.2 Applying the functional renormalization group to avalanches
II.6 Summary of the thesis
II.6.1 Introduction
II.6.2 Presentation of the main results of [1]
II.6.3 Presentation of the main results of [2]
II.6.4 Presentation of the main results of [3]
II.7 Conclusion
IIIExactly solvable models of directed polymer
III.1 The KPZ universality class in 1 + 1d
III.1.1 A few models in the KPZ universality class
III.1.2 The KPZ fixed point – strong universality
III.1.3 Universality of the KPZ equation: notion of weak universality and universal scaling limits of DP on the square lattice
III.2 Exact solvability properties
III.2.1 Symmetries of the continuum KPZ equation
III.2.2 An analytical exact solvability property: the stationary measure
III.2.3 An algebraic exact solvability property: Bethe ansatz integrability of the continuum DP
III.2.4 A few words on other exact solvability properties
III.3 Summary of the thesis
III.3.1 Introduction
III.3.2 Presentation of the main results of [4]
III.3.3 Presentation of the main results of [5]
III.3.4 Presentation of the main results of [6]
III.3.5 Presentation of the main results of [7]
III.4 Conclusion
Conclusion
A Paper: Spatial shape of avalanches in the BFM
A.1 Introduction
A.2 The Brownian force model
A.2.1 Model
A.2.2 Velocity Theory
A.2.3 Avalanche-size observables
A.2.4 The ABBM model
A.3 Derivation of the avalanche-size distribution in the BFM
A.4 Avalanche densities and quasi-static limit
A.4.1 Center of mass: ABBM
A.4.2 BFM
A.5 Fully-connected model
A.6 Spatial shape in small systems N = 2, 3.
A.7 Continuum limit
A.7.1 Avalanche size PDF and density in the continuum limit
A.7.2 Rewriting the probability measure on avalanche sizes
A.7.3 The saddle point for large aspect ratio S/ℓ4
A.7.4 Simulations: Protocol and first results
A.8 Fluctuations around the saddle point
A.8.1 Field theoretic analysis
A.8.2 Generating a random configuration, and importance sampling .
A.8.3 The leading correction to the shape at large sizes
A.8.4 Fluctuations of the shape for large avalanches
A.8.5 Asymmetry of an avalanche
A.8.6 Comparison of the perturbative corrections to the numerics
A.8.7 The optimal shape beyond extreme value statistics
A.9 Application to stationary driving
A.10 Conclusion
A.11 App A
A.12 App B
A.13 App C
A.14 App D
A.15 App E
A.16 App F
A.17 App G
A.18 App H
A.19 App I
B Paper: Universality in the spatial shape of avalanches
B.1 Letter
B.2 Supplemental Material
C Paper: Universal correlations between shocks
C.1 Introduction
C.2 Main results
C.3 Model, shock observables and method
C.3.1 Model
C.3.2 The ground state and the scaling limit
C.3.3 Properties of ˜Δ∗(u) and static universality classes
C.3.4 Shocks observables: Densities
C.3.5 Shocks observables: Probabilities
C.3.6 Relation between avalanche-size moments and renormalized force cumulants: First moment
C.3.7 Generating functions
C.3.8 Relation between avalanche-size moments and renormalized force cumulants: Kolmogorov cumulants and chain rule
C.3.9 Strategy of the calculation and validity of the results
C.3.10 Connected versus non-connected averages and the ǫ-expansion .
C.4 Correlations between total shock sizes
C.4.1 Reminder of the diagrammatic rules and extraction of shock moments
C.4.2 Lowest moments
C.4.3 Generating function for all moments
C.4.4 Results for the densities
C.4.5 Analysis of the results
C.5 Local structure of correlations
C.5.1 Reminder: one-shock case
C.5.2 Two-shock case: Notation and diagrammatic result
C.5.3 First moments: arbitrary sources and kernels
C.5.4 First moment: correlations between the local shock sizes for short-ranged elasticity.
C.5.5 First moment: correlations between the local shock sizes for longranged elasticity.
C.6 Measurement of correlations in simulations of d = 0 toy models.
C.6.1 Models and goals
C.6.2 Numerical Results: RB model
C.6.3 Numerical Results: RF model
C.7 Conclusion
C.8 App A
C.9 App B
C.10 App C
C.10.1 Algebraic derivation of Eq. (C.5.17)
C.10.2 More explicit solution for avalanches measured on parallel hyperplanes
C.11 App D
D Paper: Log-Gamma directed polymer and BA
D.1 Introduction
D.2 Model
D.2.1 Model
D.2.2 Rescaled Potential
D.3 Evolution equation and Brunet Bethe ansatz
D.3.1 Evolution equation
D.3.2 Bethe-Brunet Ansatz
D.4 Time evolution of the moments, symmetric transfer matrix
D.4.1 Symmetric transfer matrix and scalar product
D.4.2 Time-evolution of the moments
D.5 The continuum/Lieb-Liniger limit
D.6 Norm of the eigenstates
D.7 Large L limit
D.7.1 Strings
D.7.2 Eigenvalue of a string: energy
D.7.3 Momentum of a string
D.7.4 Phase space
D.7.5 Norm of the string states
D.8 Formula for the integer moments Zn
D.9 Generating function
D.9.1 Generating function for the moments
D.9.2 Generating function: Laplace transform
D.9.3 Probability distribution
D.10 Limit of very long polymers and universality
D.11 Comparison with other results
D.12 Conclusion
D.13 App A
D.14 App B
D.14.1 finite L
D.14.2 in the limit L → +∞
D.15 App C
D.16 App D
D.17 App E
D.18 App F
D.19 App G
D.20 App H
D.21 App I
D.22 App J
D.23 App K
E Paper: On integrable DP models on the square lattice
E.1 Introduction and main results
E.1.1 overview
E.1.2 Main results and outline of the paper
E.2 DPs on Z2: Replica method and integrability
E.2.1 Definition of the model
E.2.2 The replica method and the coordinate Bethe Ansatz.
E.2.3 The constraint of integrability on integer moments.
E.3 Integrable polymer models
E.3.1 The |q| < 1 case.
E.3.2 The q → 1 limit
E.4 Study of the Inverse-Beta Polymer
E.4.1 Moments Formula and Coordinate Bethe Ansatz
E.4.2 Fredholm determinant formulas and KPZ universality
E.4.3 The large length limit and the KPZ universality.
E.4.4 A low temperature limit.
E.5 Conclusion
E.6 App A
E.7 App B
E.8 App C
E.9 App D
F Paper: Beta polymer
F.1 Introduction and main results
F.1.1 Overview
F.1.2 Main results and outline of the paper
F.2 Model and earlier work
F.2.1 The Beta polymer
F.2.2 Relation to a random walk in a random environment
F.2.3 Relation to the problem and notations of Barraquand-Corwin
F.3 Bethe Ansatz solution of the Beta polymer
F.3.1 Bethe ansatz on a line with periodic boundary conditions
F.3.2 Resolution of the Bethe equations in the large L limit: repulsion and free particles
F.3.3 Bethe ansatz toolbox
F.3.4 A large contour-type moment formula
F.4 Asymptotic analysis in the diffusive regime
F.4.1 The issue of the first site
F.4.2 Cauchy type Fredholm determinant formulas
F.4.3 Asymptotic analysis of the first-moment: definition of the optimal direction and of the asymptotic regimes
F.4.4 Asymptotic analysis in the diffusive vicinity of the optimal direction on the Cauchy-type Fredholm determinant
F.4.5 Multi-point correlations in a diffusive vicinity of the optimal direction
F.5 Asymptotic analysis in the large deviations regime: KPZ universality
F.5.1 Recall of the results of Barraquand-Corwin
F.5.2 An inherent difficulty and a puzzle
F.5.3 A formal formula for the moments of the Beta polymer in terms of strings
F.5.4 A formal Fredholm determinant and KPZ universality
F.5.5 Crossover between Gamma and Tracy-Widom fluctuations
F.6 Nested-Contour integral formulas
F.6.1 Alternative moments formulas
F.6.2 Mellin-Barnes type Fredholm determinant
F.7 Numerical results
F.7.1 In the diffusive regime.
F.7.2 In the large deviations regime.
F.8 Conclusion
F.9 App A
F.10 App B
F.11 App C
F.11.1 Two equivalent formulas for the Strict-Weak polymer
F.11.2 A formal formula for the moments of the Beta polymer.
F.12 App D
G Paper: Stationary measures of DPs on Z2
G.1 Introduction
G.2 Recall: stationary measure of the Log-Gamma polymer
G.3 Overview: definitions, main results and outline
G.3.1 Definitions of the models of directed polymers
G.3.2 Stationarity and reversibility properties
G.3.3 Quenched free-energy in point to point models without boundaries
G.3.4 Convergence of point to point models to their stationary state .
G.3.5 Outline and some additional results not presented here
G.4 Stationary measure of the Inverse-Beta polymer
G.4.1 Stationary property of the model with boundaries
G.4.2 Stationarity property of the model with stationary initial condition
G.4.3 Reversibility of the stationary measure: detailed balance property
G.4.4 Relation to other models
G.5 Stationary measure of the Bernoulli-Geometric polymer
G.5.1 Stationarity properties of the Bernoulli-Geometric polymer
G.5.2 Relation to other models
G.6 Convergence to the stationary measure
G.6.1 Free-energy in models with boundaries
G.6.2 Free-energy in models without boundaries
G.6.3 Convergence to the stationary measures
G.6.4 A remark on optimal paths and energy fluctuations in models with boundaries
G.7 Numerical results for the zero-temperature model
G.8 Conclusion
G.9 App A
G.10 App B
Bibliographie
