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The full-space problem
For the rest of this Chapter, let us introduce and discuss some initial conditions of the Kardar-Parisi-Zhang equation. In this Section we shall focus on the full-space problem and after that we shall turn to the half-space one. One of the most general initial condition studied so far for the Kardar-Parisi-Zhang equation in full-space is the following:
where BL and BR are independent one-sided unit Brownian motions with BL(0) = BR(0) = 0. The two point correlators are given by E BR(x)BR(x0) = min(x, x0). The parameters wL and wR measure the slope of the initial profile on each side of zero and the parameters σL and σR describe the variance of the randomness on each side of zero. At the time of completion of this manuscript, all solvable cases of the Kardar-Parisi-Zhang equation in full-space can be considered by choosing the parameters σL and σR in {0, 1}2. Indeed, the list of solvable cases are the following:
1. The wedge initial condition where we choose σR = σL = 0. In this case, (i) wL = wR = ω → ∞ describes the droplet initial condition, (ii) wL = ∞, wR = 0 (resp. wR = ∞ and wL = 0), describes the half-flat initial condition and (iii) wL = wR = 0 describes the flat initial condition. Taking wL = wR = ω enables to study a cross-over between the flat and droplet initial condition.
2. The Brownian initial condition where we choose σL = σR = 1. In this case, wL = wR = 0 describes the stationary initial condition. Taking wL = wR = w enables to study a cross-over between the stationary and droplet initial condition.
3. The Flat-Brownian initial condition where we choose σL = 0. Taking wL → ∞ describes the half-Brownian and taking wR → ∞ describes the half-flat. Note that by symmetry we can study a symmetric situation with L ↔ R.
As indicated during the introduction to the Replica Bethe Ansatz method, the generating function of the solution to the KPZ equation can be expressed in terms of Fredholm determinants which provide a direct connection between random matrix ensembles and the KPZ equation. Anticipating the rest of this Thesis, the droplet initial condition is related to the Tracy-Widom distribution for the Gaussian Unitary Ensemble (GUE), the flat initial condition to the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) and the stationary initial condition to the so-called Baik Rains distribution.
The full-space problem with droplet initial condition was originally the first one to be solved exactly. Although the KPZ equation appeared in 1986, its first solution appeared in 2010 high-lighting its complexity and the effort engaged both in the physics and mathematics community. In addition, recent impressive experiments on liquid crystals from Takeuchi and Sano have been able to validate the large-time random matrix statistics of the KPZ height fluctuation [25–30].
The half-space problem
Although most of these results have been obtained on the full space, it is interesting for appli-cations to study also half-space models, e.g. defined only on the half line x ∈ R+. Recently indeed experiments were able to access the half-space geometries [31]. Moreover there are strong theoretical motivations since integrability properties are sometimes preserved by going to the half-space, with the proper boundary conditions. Progress started with discrete models, notably in mathematics. Indeed, the half-space problem has been addressed for some models in the KPZ universality class. In a pioneering paper, Baik and Rains [107] studied the longest increasing sub-sequences (LIS) of symmetrized random permutations which maps to a discrete zero temper-ature model of a directed polymer in a half-space, with a tunable parameter α which makes the boundary more attractive as α increases. They found, in the limit of large polymer length t, a transition when α reaches the critical value αc = 1. For α < αc the PDF of the fluctuations of the directed polymer energy is given by the Tracy-Widom distribution for the Gaussian Symplectic Ensemble (GSE) of random matrices [61] on the characteristic KPZ scale t1/3. For α > αc the PDF is Gaussian on the scale t1/2, as the directed polymer paths are bound to the diagonal line. At the critical point, α = αc the PDF is given by the GOE Tracy-Widom distribution on the t1/3 scale. A similar transition was found for the height distribution in the discrete PNG growth model on a half-line, with a source at the origin, as the nucleation rate at the origin is increased above a threshold [108]. Finally, some results were also obtained for the TASEP in a half-space [45] and for the (finite temperature) log-gamma DP with symmetric weights [109,110] and in half-quadrant geometries [111].
For the coming presentation of the KPZ equation on a half-line, we will borrow some elements from Ref. [112]. The KPZ equation (1.1.5) on the half-line considers the space variable x ∈ R+ along with a Neumann boundary condition (b.c.)
A is a real parameter which describes the interaction with the boundary (a wall at x = 0). The wall is repulsive for A > 0 and attractive for A < 0, in addition the case A = +∞ imposes Z(x = 0, t) = 0, i.e. an infinitely repulsive or absorbing wall. On the contrary, A = 0 is seen as a reflecting wall. For the half-space problem, we need in addition to regularize the initial condition when it is not properly defined at x = 0, as it is the case for a hard-wall with boundary coefficient A = +∞. In this case, the droplet initial condition for instance reads h x, t |x − κ| η , η + Z x, t δ x κ .
Figure 1.4: Top. The directed polymer in random environment with a wall on the left and fixed endpoints along the wall. The attraction to the wall is parametrized by the parameter A. At A = −1/2 there is a phase transition, for A < −1/2 the polymer spends most of the next to the wall, while at A > −1/2 the polymer is unpinned. Depending on the value of A, the statistics of the fluctuations of the corresponding free energy, equivalent to the KPZ height, changes from Gaussian (A < −1/2), to GOE Tracy-Widom (A = −1/2) and GSE Tracy-Widom (A > −1/2). Bottom. The same transition is observed for the ground state eigenstate of the attractive Lieb-Liniger model for bosons on the half-line (the wall is at the left of the origin). The wave function changes from being delocalized (A > −1/2) to being localized at the boundary (A < −1/2) where all particles are bounded to the wall. Courtesy of J. De Nardis in Ref. [112].
The half-space problem was considered by Kardar in Ref. [113] in the equivalent representa-tion in terms of a directed polymer in a half space bounded by a wall. A binding transition to the wall, that was termed depinning by quenched randomness, was predicted for A = −1/2 from heuristic considerations on the ground state of the delta Bose gas in the presence of a wall. The phase transition, which we represent in Fig. 1.4, separates a phase where the directed polymer is bound to the wall for A < −12 to an unbound phase for A > −12 , as the attraction to the wall is de-creased. Numerical studies also addressed the half-line problem: the convergence to the GSE was explored [31] in a half-space geometry aiming to open the way for an experimental confirmation. In addition, connections to conductance fluctuations in Anderson localization were explored [114].
More recently, exact solutions to the KPZ equation were obtained for three specific values of A, i.e. A = +∞, 0, −1/2 [71, 79, 80] for the droplet initial condition. In all three cases the solution can be expressed in terms of a Fredholm Pfaffian [6, 71, 79, 80]. For A = +∞, the in-finitely repulsive wall, it was found [71] that the PDF of the scaled height, H(t)/t1/3, converges at large t to the GSE Tracy-Widom distribution [61, 104, 105]. For A = 0, it was also found that the large-time limit of the PDF corresponds to the GSE Tracy-Widom distribution [79]. Both cases used the mapping to the delta Bose gas, with use of respectively the RBA for A = +∞ and nested contour integral representations of the moments for A = 0. The critical case A = −12 was solved instead using a continuum limit from the ASEP model with an open boundary [80, 115]. It was found that at large time the PDF converges to the GOE Tracy-Widom distribution which thus describes the large-time behavior at the transition.
It is natural to conjecture that the transition for the KPZ equation at A = −1/2 is in the same universality class, in the large-time limit, as the one discovered by Baik and Rains in [107] and that this universality is common to the full KPZ universality class, see Ref. [115]. Baik and Rains performed a detailed analysis on a scale α − 1 = wt−1/3 around the transition. They found that the PDF depends continously on w and interpolates between the GSE/GOE/Gaussian distributions as w is increased. This transition PDF was obtained as a solution of Painlev´ type system of equations. Further results were obtained recently using Pfaffian-Schur processes, for variants of TASEP models and last passage percolation in a half-quadrant [116,117]. Not only the one-point, but also the multi-point height distributions were studied and for arbitrary positions with respect to the wall. A Fredholm Pfaffian was obtained with an explicit expression around the GSE/GOE/Gaussian transition, hence we may conjecture that it is compatible with the Painlev´ system of [107].
Cross-over between fixed points of the KPZ equa-tion
So far we have explored the Kardar-Parisi-Zhang universality class but it is in fact possible to go beyond and to define subclasses of universality in terms of initial or boundary conditions. As a concrete example, let us recall that for the KPZ equation in a half-space, we discussed the fact that the Robin boundary condition A controls the statistics of the fluctuations of the one-point KPZ height at large time. Therefore, varying this parameter allows to realize a cross-over between different large-time distributions, also seen as basin of attraction of the dynamics.
More generally, for the family of initial conditions previously introduced, it is possible to de-fine by dimensional analysis other cross-over, or relevant, parameters [2, 6, 118, 119]. Indeed, the slope of the wedge initial condition or the drift of the Brownian initial condition play a cross-over role both at short time and at large time in terms of the distribution of the solution to the KPZ equation. We represent in Table 1.1 the relevant parameters at short time and at large time found in the literature describing various cross-overs and indicate explicitly the related interpolation at large time. The cross-overs are realized when the relevant parameter is increased from 0 to +∞.
This Table has to be understood in the following way:
• At short time, all boundary conditions with finite A will behave as the reflecting wall A = 0. All initial conditions with finite w or ω will behave as the stationary initial condition w = 0 or the flat initial condition ω = 0. The singular cases will be the hard-wall A = +∞ and the droplet initial condition w = w˜ = +∞.
• At late time, all boundary conditions with A > −1/2 will behave as the hard-wall A = +∞. All initial conditions with finite w or ω will behave as the droplet initial condition. The singular cases will be the critical wall A = −1/2, the stationary initial condition w = 0 and the flat initial condition ω = 0.
Table of contents :
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
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
