# Symmetries of the WNT equations in full-space and some considerations in half-space

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## Determinantal and Pfaffian point processes

In this Section, we shall review the very basic properties of Determinantal and Pfaffian point processes required to understand the recent developments on the exact solutions to the Kardar- Parisi-Zhang equation. In pedestrian words, a determinantal point process  is a set of random points on the real line {xi} such that every correlation function of these points is the determinant of a kernel K called the correlation kernel 8k > 1, k(x1, . . . , xk) = det K(xi, xj)k i,j=1 (2.2.1) Remark 2.2.1. In this Thesis, we shall consider integral linear operators acting on the Lebesgue space L2( ) with a subspace of R represented by a kernel K : (x, y) 7! K(x, y). In addition, let us mention that the correlation kernel is not unique. If K is a correlation kernel, then the conjugation of K with any positive weight ! provides an equivalent correlation kernel K(x, y) = !(x)K(x, y)!(y)−1 (2.2.2).
Determinantal processes are the landmark of exclusion models as the determinantal representation
forbids two points to be equal. They are sometimes referred in the physics literature as Fermionic processes due to the Pauli exclusion principle and the Wick’s theorem in quantum mechanics inducing determinantal structures in various correlation functions.

### Extension of the cumulant expansion to Fredholm Pfaffians

Although initially introduced for Determinantal point processes, we shall extend the use of cumulant expansions to Fredholm Pfaffians. The reason for this is the intrinsic relation between the solutions to the Kardar-Parisi-Zhang equation and algebraic processes which scope is not only limited to Determinantal processes but also to Pfaffian processes. When the set {ai} forms a Pfaffian point process, the linear statistics admits a Fredholm Pfaffian representation EK 2 6 4 exp 0 @− +X1 i=1 ‘(ai) 1 A 3 75 = Pf[J − (1 − e−’)K].

#### From two to one-dimensional kernels

Later in this Thesis, when encountering exact solutions to the Kardar-Parisi-Zhang equation we will need to juggle between matrix-valued kernels and scalar value kernels. For this purpose, we present in this Section an equivalent representation of a class of Fredholm Pfaffians with 2 × 2 block kernels in terms of a Fredholm determinant with a scalar valued kernel. Consider a measure dμ on a contour C 2 C and another measure dz on the real line R, depending on a real parameter z. Consider the quantities Q(z) defined by Q(z) = 1 +0.

Equivalence of different matrix kernels at the level of Fredholm Pfaffians

To conclude this Section of the transformation of matrix valued kernels to scalar kernels, let us mention for completeness a useful proposition which states the equivalence at the level of Fredholm Pfaffians of different kernels. The reason why we mention this result is that some kernels appearing in the literature are not directly suited to the Proposition 2.3.2 but are equivalent to kernels verifying its hypothesis by the use of the following Proposition 2.3.3.

Solutions at all times in full-space

For the completeness of the presentation, we shall recall here the exact expressions for the solutions to the KPZ equation at all times. We will focus on the droplet and Brownian initial conditions in full and half-space and we will not discuss the solution to the flat initial condition, expressed in Ref. , as we will not use its exact expression in this Thesis. In particular, we will be interested in the generating function of the exponential of the KPZ height which exhibits a Fredholm representation for all aforementioned initial conditions .

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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