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## Exact solvability: origins and methods

In order to step back on the denitions given in the previous sections, it is essential to introduce the reader to the theory of Macdonald processes. However, since we do not use this theory so much, we explain in Section 1.4.2 another approach leading to similar results. We call this approach

the duality method. We then explain the outcome of these methods: Fredholm determinantal formulas. Most of the concepts and methods that we present in this section are introduced in [BC14, BCS14], but have a wider range of applicability than Macdonald processes and exclusion processes.

### Fredholm determinant formulas

All the models that we have dened previously admit closed formulas for the Laplace transform { or its q-deformed counterpart { of meaningful observables, in terms of Fredholm determinants. As we explain in Section 1.5, these formulas are convenient to prove limit theorems towards the Tracy-Widom distribution. However, we do not need to use any of the properties of the theory of Fredholm determinants. The only thing that we need is an explicit expression, in the form of a Fredholm determinant. We provide here a supercial introduction to Fredholm determinants, following [BC14, Denition 3.2.6], and we only explain how to compute them numerically. We refer to the chapter 3 of [Sim79] for a more complete introduction of Fredholm determinants.

Denition 1.4.1. Fix a Hilbert space L2(X; ), where X is a measure space and a measure on X. In this manuscript, X is often a contour in the complex plane. When X = is a simple positively oriented smooth contour in C, we write only L2() and is implicitly understood to be the path measure along divided by 2i.

#### KPZ scaling theory

The KPZ scaling theory constitutes an educated guess to compute exactly the expressions of all constants arising in limit theorems for exclusion processes in the KPZ universality class. The range of applicability of this theory is actually larger than exclusion processes. However, it comes from the physics literature, in particular the works of Krug, Meakin and Halpin-Healy [KMHH92], and the results are so far highly conjectural. In this section, we present these heuristic claims in the case of exclusion processes, following the approach of Spohn [Spo12]. We try to focus on intuitions, and warn the reader that some statements would need more precise denitions to be rigorous.

Consider an exclusion process as presented in Section 1.2. Assume that the translation invariant stationary measures are precisely labelled by the density of particles , where = lim n!1 1 2n + 1 #fparticles between n and ng.

**Tracy-Widom distribution and the BBP phase transition**

Almost all the results that we state in this section are limit theorems towards the Tracy- Widom distribution. Two dierent approaches [TW94] can be used in order to dene this distribution and both can be used for proving limit theorems 11. On one hand, one can give an exact formula for the probability distribution function, in terms of the Painleve II nonlinear ODE. On the other hand, one can dene the Tracy-Widom distribution as the scaling limit of extreme points of determinantal point processes.

We follow the second approach. The Tracy-Widom distribution is dened as the distribution of the right-most point of a point process in R dened by its n point correlation functions n(x1; : : : ; xn) = det K(xi; xj) n i;j=1 .

**q-TASEP with slower particles**

For the q-TASEP, the KPZ scaling conjecture has been shown by Ferrari and Vet}o in [FV13] for one-point uctuations, under a restriction on the parameters. Actually, the result is proved only in the part of the rarefaction fan where the speed of particles is below 1=2 (it means that a macroscopic part of the rarefaction fan was missing). In [Bar15], we extend 12 Ferrari-Vet}o’s proof to remove this restriction, and also study the case when all but nitely many particles have speed 1 (i.e. they jump at rate 1 qgap) and a nite number of particles have a slower speed (they jump at rate (1 qgap) for some < 1). The case of a faster particle is not interesting, since it has no in uence on the macroscopic scale. According to the parallel with the TASEP, it is reasonable to expect to see the BBP transition, and this is proved in [Bar15]. More precisely, the largest eigenvalue of the deformed GUE corresponds to the inverse of the speed , whereas the multiplicity of this eigenvalue corresponds to the number of particles having the minimal speed .

To understand this intuitively, let us examine the case where only the rst particle is slower and has rate . It is quite evident that the rst particle will create a trac jam: by coupling the slowed-down process with a usual qTASEP, one can see that the second particle (and more generally the nth particle) would like to move faster than in average, but it cannot since the rst particle blocks. Hence, in the presence of a slower particle, the next particles are slowed down, as in a trac jam. How long is the trac jam ? Is it macroscopic ? One can provide heuristic answers to these questions. The particles concerned by the trac jam are those which, in absence of a slower particle, would have an average speed greater than . This concerns a macroscopic quantity of particles. What the BBP transition says then { and this cannot be inferred from simple probabilistic arguments 13 { is that inside the trac jam, particles have a Gaussian behaviour 14, outside the trac jam, they behave exactly as in absence of a slower particle, and at the border of the trac jam, the positions of particles uctuate according to the BBP distribution of rank 1 (because there is only one slow particle in this example). The Figure 1.11 summarizes these explanations.

**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 **