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Table of contents
Remerciements
Introduction
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Index of notations and abbreviations
I Introduction to the Kardar-Parisi-Zhang equation and elements of Random Matrix Theory
1 The Kardar-Parisi-Zhang equation
1.1 Birth of the model
1.2 Some elements around the KPZ universality class
1.3 Some mappings of the KPZ equation
1.4 The Replica Bethe Ansatz
1.5 The full-space problem
1.6 The half-space problem
1.7 Cross-over between fixed points of the KPZ equation
2 Elements of Random Matrix Theory
2.1 Gaussian matrices
2.2 Determinantal and Pfaffian point processes
2.3 From two to one-dimensional kernels
3 Exact solutions to the Kardar-Parisi-Zhang equation
3.1 A brief historical note
3.2 Solutions at all times in full-space
3.3 Solutions at all times in half-space
3.4 A new duality in half-space and general solution to the droplet initial condition .
3.5 Open questions regarding the exact solutions to the KPZ equation
4 Connections and applications of the Kardar-Parisi-Zhang equation
4.1 Hidden connections between RMT and KPZ: the Gorin-Sodin Mapping
4.2 Coincidence of Brownians walkers and exponential moments of KPZ
5 Introduction to the large deviations of the KPZ equation
5.1 Large deviations at short time
5.2 Large deviations at large time
II Short-time height distributions of the solutions to the KPZ equation
6 Perturbative noise rescaling of the KPZ equation: Weak Noise Theory
6.1 Construction of the Weak Noise Theory
6.2 Large deviation function of the Kardar-Parisi-Zhang equation at short time
6.3 Symmetries of the WNT equations in full-space and some considerations in half-space
6.4 From small H to large H and spontaneous symmetry breaking
6.5 Recent applications of the Weak Noise Theory
7 Large deviation solutions at short time: one method to rule them all 83
7.1 The first cumulant approximation of Fredholm determinants at short time
7.2 Large deviations for various initial conditions
7.3 Inverting the Legendre transform
7.4 A hint of universality for the solutions at short time
8 High-precision simulations of the short-time large deviations of the KPZ solutions
8.1 Directed polymer on a lattice
8.2 Introduction to importance sampling
8.3 Comparison of the theoretical predictions with the simulations
8.4 What do the large deviation polymers look like ?
III From the large deviations of KPZ at late time to linear statistics at the edge of Gaussian random matrices.
9 From small times to large times
9.1 How negative can the solution of KPZ be ?
9.2 Systematic time expansion of the edge GUE Fredholm determinant
9.3 Cumulants of the Airy point process: from small times to large times
10 Introduction to the linear statistics at the edge of Gaussian matrices
10.1 The late-time large deviations of KPZ as a microscopic linear statistics
10.2 From macroscopic to microscopic linear statistics
10.3 From the bulk of the Coulomb gas to its edge
11 The four tales of the one tail: solving the linear statistics at the edge
11.1 From the cumulants of the linear statistics to the free energy
11.2 A WKB semi-classical density of states for the Stochastic Airy Operator
11.3 Electrostatic Coulomb gas approach to the linear statistics
11.4 A WKB approximation for the Painlev´e II representation of the linear statistics .
11.5 Solution for monomial walls with parameter
11.6 Where all the physics hides: upper bounds of the excess energy
11.7 Application to non-intersecting Brownian interfaces subject to a needle potential
11.8 Open questions regarding linear statistics at the edge of random matrix spectra
Conclusion and perspectives
Appendix
A Properties of some functions (Airy, Lambert)
B Some technical theorems and lemmas
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