The KPZ scaling theory

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Table of contents

Avant-propos
Foreword
1 Denitions and Main results
1.1 KPZ universality
1.2 Integrable exclusion processes
1.2.1 General description
1.2.2 The q-TASEP
1.2.3 Introduction to q-analogues
1.2.4 The q-Hahn TASEP and the q-Hahn distribution
1.2.5 The asymmetric q-Hahn exclusion process
1.2.6 Multi-particle asymmetric diusion model
1.3 Directed lattice paths
1.3.1 Random walk in space-time i.i.d. Beta environment
1.3.2 Beta polymer
1.3.3 Bernoulli-Exponential directed rst passage percolation
1.4 Exact solvability: origins and methods
1.4.1 Macdonald processes
1.4.2 The duality method
1.4.3 Fredholm determinant formulas
1.5 Limit Theorems
1.5.1 KPZ scaling theory
1.5.2 Tracy-Widom distribution and the BBP phase transition
1.5.3 q-TASEP with slower particles
1.5.4 Beyond KPZ scaling theory ?
1.5.5 Second order corrections to the large deviation principle for the Beta-RWRE
1.5.6 Bernoulli-Exponential FPP Open questions
2 Asymptotic analysis of the q-TASEP
2.1 Introduction and main result
2.1.1 The q-TASEP
2.1.2 Main result
2.2 Asymptotic analysis
2.2.1 Case > q, Tracy-Widom uctuations
2.2.2 Case = q, critical value
2.2.3 Case < q, Gaussian uctuations
3 The q-Hahn asymmetric exclusion process
3.1 Introduction
3.2 Preliminaries on q-analogues and the q-Hahn distribution
3.2.1 Useful q-series
3.2.2 A symmetry identity for the q-Hahn distribution
3.3 An asymmetric exclusion process solvable via Bethe ansatz
3.3.1 General case
3.3.2 Two-sided generalizations of q-TASEP
3.3.3 Degenerations to known systems
3.4 Predictions from the KPZ scaling theory
3.4.1 Hydrodynamic limit
3.4.2 Magnitude of uctuations
3.4.3 Critical point Fredholm determinant asymptotics
3.5 Asymptotic analysis
3.5.1 Proof of Theorem 3.5.2
3.5.2 Proof of Theorem 3.5.4
3.5.3 Proofs of Lemmas about properties of f0
4 Beta Random walk in random environment
4.1 Denitions and main results
4.1.1 Random walk in space-time i.i.d. Beta environment
4.1.2 Denition of the Beta polymer
4.1.3 Bernoulli-Exponential directed rst passage percolation
4.1.4 Exact formulas
4.1.5 Limit theorem for the random walk
4.1.6 Localization of the paths
4.1.7 Limit theorem at zero-temperature
4.2 From the q-Hahn TASEP to the Beta polymer
4.2.1 The q-Hahn TASEP
4.2.2 Convergence of the q-Hahn TASEP to the Beta polymer
4.2.3 Equivalence Beta-RWRE and Beta polymer
4.3 Rigorous replica method for the Beta polymer
4.3.1 Moment formulas
4.3.2 Second proof of Theorem 4.1.12
4.4 Zero-temperature limit
4.4.1 Proof of Proposition 4.1.10
4.4.2 Proof of Theorem 4.1.18
4.5 Asymptotic analysis of the Beta RWRE
4.5.1 Fredholm determinant asymptotics
4.5.2 Precise estimates and steep-descent properties
4.5.3 Relation to extreme value theory
4.6 Asymptotic analysis of the Bernoulli-Exponential FPP
4.6.1 Statement of the result
4.6.2 Deformation of contours
4.6.3 Limit shape of the percolation cluster for xed t
References