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The KPZ fixed point – strong universality
There is various degrees of precision which can be achieved when describing the socalled KPZ fixed point and we refer the reader to [161] for the most complete description. Once the identification of a ‘growing interface’ h(t, x) in a model has been made, the KPZUC hypothesis is that the appropriately rescaled large time fluctuations of the interface are completely universal. More precisely, starting from a given initial condition, one first has to subtract the deterministic (angle-dependent) growth rate of the interface v∞(ϕ) := lim t→∞ h(t, ϕt) t , (III.1.39).
which is generally expected to be well-defined and non random (i.e. the above convergence holds with probability 1). One then defines h(t, x) := h(t, x) − tv∞(x/t) .
Universality of the KPZ equation: notion of weak universality and universal scaling limits of DP on the square lattice
The limiting spatial process (III.1.41) does not obey the KPZ equation: the KPZ equation is not the KPZ fixed point. Under a general rescaling, if h(t, x) solves the KPZ equation (III.1.15), ˜h(˜t, ˜x) := b−αh(t = b˜t, x = bz ˜x) solves (dropping the tildes) ∂th(t, x) = bα+z−2 2 c1(∇xh(t, x))2 + bz−2 2 (∇x)2c2h(t, x) + b(z−2α−1)/2c3ξ(t, x) , (III.1.42).
where we have reintroduced explicit constants in front of each term on the right hand side. There is no way to fix z and α to get a scale invariant equation. The values α = 1/2 and z = 3/2 of the KPZUC, a property of the KPZ FP, are not trivially In all the models in the KPZUC, however, the KPZ equation/the continuum DP plays a peculiar role that is linked with the notion of weak universality [186]. The latter refers to two limits:
(i) The weak asymmetry limit. Symmetric growth models (c1 = 0) are in the Edwards-Wilkinson universality class and are characterized by exponents α = 1/2 and z = 2. If an asymmetry is present c1 6= 0, the large scale properties are those of the KPZ FP. Note that rescaling (III.1.42) with α = 1/2, z = 2 and c1 ∼ b−1/2 leaves the KPZ equation itself invariant. More generally it is conjectured that (under the usual assumptions such as locality, etc…) the KPZ equation itself is the universal scaling limit of weakly asymmetric growth models in 1+1d when rescaling space x ∼ b, t ∼ b2 and the asymmetry as b−1/2. For this reason the KPZ equation is sometimes referred to as implementing the universal crossover between the EW FP and the KPZ FP. An important example of this weak-universality property is provided by the work of Bertini and Giacomin [187] that states that, upon scaling space as x ∼ b, t ∼ b2 and the asymmetry p−q as b−1/2 in the corner growth model previously defined, the corner growth model height profile converges to the Cole-Hopf solution of the KPZ equation.
(ii) On the other hand the weak noise limit is linked with the DP in a disordered medium. At zero disorder, c3 = 0, DPs are equivalent to random walks, are diffusive z = 2 and have no disorder fluctuations α = 0. For c3 6= 0, the large scale properties of DPs are described by the KPZ FP. Noting that the rescaling (III.1.42) with α = 0, z = 2 and c3 ∼ b−1/2 leaves the KPZ equation invariant, it is conjectured that the KPZ/MSHE is the universal scaling limit of weakly disordered DPs. This scaling has also been called the intermediate disorder regime in the literature [188]. Let us illustrate it heuristically on the case of DPs on the square lattice, using the notations of Sec. III.1.1. We start with the discrete version of the MSHE in the variables t, ˆx given in (III.1.23) which we recall here Zt+1(ˆx) = e− 1 T Et+1(ˆx) (Zt(ˆx + 1/2) + Zt(ˆx − 1/2)) .
A partial selection of analytical miracles in models in the KPZ universality class
In the previous section we discussed the notion of weak and strong KPZ universality and introduced a few models in the KPZUC. In this chapter we now present a few exact solvability properties that permitted over the years to build the belief in the remarkable properties of the KPZUC and focus on DPs. We will discuss: (i) in Sec. III.2.1 the symmetries of the KPZ equation; (ii) in Sec. III.2.2 the stationary measure of the the KPZ equation and some models of DPs on Z2; (iii) in Sec. III.2.3 the Betheansatz solvability of the continuum DP; (iv) in Sec. III.2.4 some other exact solvability properties: the RSK and gRSK correspondences and Macdonald processes (briefly discussed).
A few results obtained using Bethe ansatz
The replica Bethe ansatz approach to DP has led to a variety of exact results. Known since the work of Kardar [201], it was first applied for technical reasons, to the study of DP properties that can be deduced more or less from the sole knowledge of the ground state energy, i.e. the limit of DPs of large length t ≫ 1 on a finite cylinder L (the limit L → ∞ being eventually taken afterwards). This was used to already determine the critical exponents [201, 212] or the large deviation function for the fluctuations of the free-energy of the DP on the cylinder [204, 205]. Obtaining the universal distribution of fluctuations for the growth of an interface in an infinite space, however, requires to consider the limit t → ∞ with at least L ≫ t2/3. The study of this limit from BA requires a summation over all excited states. This was only achieved recently, partly thanks to the work of Calabrese and Caux [209] who managed to compute the norm of string states (III.2.65).
Even with this knowledge it is still far from trivial to obtain exact results. Additionally since the method is not rigorous from a mathematical point of view due to the too rapid growth of moments, it requires a large number of tricks. Once a solid recipe to tackle this issue has been devised (a recipe that sometimes appears retrospectively to be the shadow of a rigorous derivation, as e.g. by considering a q−deformed model, see [208]), the replica Bethe ansatz approach has led a variety of new (presumably exact) results. Here we name a few: (i) TW-GUE distribution of fluctuations for the point-to-point free energy [173, 165]; (ii) TW-GOE distribution of fluctuations for the point-to-line free energy [177, 178, 179]; (iii) multi-point correlations for the point-to-point free-energy and the Airy process [175, 213]; (iv) one point (Baik-Rains) and multi-point distributions of fluctuations for the point-to-Brownian (i.e. the DP with stationary initial condition) free-energy [183, 184]; (v) TW-GSE distribution of fluctuations for the point-to-point free energy of a directed polymer in a half-space [169]; (vi) fluctuations of free-energy in the crossover from droplet to stationary initial condition [184]; (vii) fluctuations of free-energy in the crossover from droplet to flat initial condition [214]; (viii) distribution of the endpoint of the polymer [215]; (ix) extension to two-times [166]. Some of these results have been shown rigorously since then (see [34]), giving credit to the replica method.
Table of contents :
Remerciements
Introduction, goal and outline of the manuscript
Index of notations and abbreviations
I Disordered elastic systems
I.1 The Hamiltonians
I.1.1 The state space
I.1.2 The elastic Hamiltonian
I.1.3 The disorder Hamiltonian
I.1.4 The confining Hamiltonian
I.2 Static phase diagram and the strong disorder regime
I.2.1 The pure system and the thermal fixed point
I.2.2 Relevance and irrelevance of short-range disorder at the thermal fixed point
I.2.3 Static phase diagram
I.2.4 Early attempts at characterizing the strong disorder regime: Flory arguments and the Larkin model
I.3 Various problems considered in this thesis
I.3.1 Shocks in the statics at zero temperature for elastic interfaces .
I.3.2 Avalanche dynamics at the depinning transition for elastic interfaces
I.3.3 Static problem at finite temperature for directed polymers with SR elasticity and the KPZ universality class
I.4 Experimental realizations
I.4.1 Disordered elastic systems pinned in a quenched random environment
I.4.2 Out-of-equilibrium interface growth
II Avalanches and shocks of disordered elastic interfaces
II.1 Introduction
II.2 Avalanches for a particle
II.2.1 Shocks between ground states for toy models of a particle without disorder
II.2.2 Avalanches in the dynamics of a particle on the real line
II.2.3 Shock process versus avalanche process for a particle
II.3 Avalanches for an interface
II.3.1 Shocks for an interface
II.3.2 Avalanches for an interface
II.4 FRG approach to shocks
II.4.1 The functional renormalization group for the statics of disordered elastic interfaces
II.4.2 Applying the functional renormalization group to shocks
II.5 FRG approach to avalanches
II.5.1 The functional renormalization approach to the depinning transition
II.5.2 Applying the functional renormalization group to avalanches
II.6 Summary of the thesis
II.6.1 Introduction
II.6.2 Presentation of the main results of [1]
II.6.3 Presentation of the main results of [2]
II.6.4 Presentation of the main results of [3]
II.7 Conclusion
IIIExactly solvable models of directed polymer
III.1 The KPZ universality class in 1 + 1d
III.1.1 A few models in the KPZ universality class
III.1.2 The KPZ fixed point – strong universality
III.1.3 Universality of the KPZ equation: notion of weak universality and universal scaling limits of DP on the square lattice
III.2 Exact solvability properties
III.2.1 Symmetries of the continuum KPZ equation
III.2.2 An analytical exact solvability property: the stationary measure
III.2.3 An algebraic exact solvability property: Bethe ansatz integrability of the continuum DP
III.2.4 A few words on other exact solvability properties
III.3 Summary of the thesis
III.3.1 Introduction
III.3.2 Presentation of the main results of [4]
III.3.3 Presentation of the main results of [5]
III.3.4 Presentation of the main results of [6]
III.3.5 Presentation of the main results of [7]
III.4 Conclusion
Conclusion
A Paper: Spatial shape of avalanches in the BFM
A.1 Introduction
A.2 The Brownian force model
A.2.1 Model
A.2.2 Velocity Theory
A.2.3 Avalanche-size observables
A.2.4 The ABBM model
A.3 Derivation of the avalanche-size distribution in the BFM
A.4 Avalanche densities and quasi-static limit
A.4.1 Center of mass: ABBM
A.4.2 BFM
A.5 Fully-connected model
A.6 Spatial shape in small systems N = 2, 3.
A.7 Continuum limit
A.7.1 Avalanche size PDF and density in the continuum limit
A.7.2 Rewriting the probability measure on avalanche sizes
A.7.3 The saddle point for large aspect ratio S/ℓ4
A.7.4 Simulations: Protocol and first results
A.8 Fluctuations around the saddle point
A.8.1 Field theoretic analysis
A.8.2 Generating a random configuration, and importance sampling .
A.8.3 The leading correction to the shape at large sizes
A.8.4 Fluctuations of the shape for large avalanches
A.8.5 Asymmetry of an avalanche
A.8.6 Comparison of the perturbative corrections to the numerics
A.8.7 The optimal shape beyond extreme value statistics
A.9 Application to stationary driving
A.10 Conclusion
A.11 App A
A.12 App B
A.13 App C
A.14 App D
A.15 App E
A.16 App F
A.17 App G
A.18 App H
A.19 App I
B Paper: Universality in the spatial shape of avalanches
B.1 Letter
B.2 Supplemental Material
C Paper: Universal correlations between shocks
C.1 Introduction
C.2 Main results
C.3 Model, shock observables and method
C.3.1 Model
C.3.2 The ground state and the scaling limit
C.3.3 Properties of ˜Δ∗(u) and static universality classes
C.3.4 Shocks observables: Densities
C.3.5 Shocks observables: Probabilities
C.3.6 Relation between avalanche-size moments and renormalized force cumulants: First moment
C.3.7 Generating functions
C.3.8 Relation between avalanche-size moments and renormalized force cumulants: Kolmogorov cumulants and chain rule
C.3.9 Strategy of the calculation and validity of the results
C.3.10 Connected versus non-connected averages and the ǫ-expansion .
C.4 Correlations between total shock sizes
C.4.1 Reminder of the diagrammatic rules and extraction of shock moments
C.4.2 Lowest moments
C.4.3 Generating function for all moments
C.4.4 Results for the densities
C.4.5 Analysis of the results
C.5 Local structure of correlations
C.5.1 Reminder: one-shock case
C.5.2 Two-shock case: Notation and diagrammatic result
C.5.3 First moments: arbitrary sources and kernels
C.5.4 First moment: correlations between the local shock sizes for short-ranged elasticity.
C.5.5 First moment: correlations between the local shock sizes for longranged elasticity.
C.6 Measurement of correlations in simulations of d = 0 toy models.
C.6.1 Models and goals
C.6.2 Numerical Results: RB model
C.6.3 Numerical Results: RF model
C.7 Conclusion
C.8 App A
C.9 App B
C.10 App C
C.10.1 Algebraic derivation of Eq. (C.5.17)
C.10.2 More explicit solution for avalanches measured on parallel hyperplanes
C.11 App D
D Paper: Log-Gamma directed polymer and BA
D.1 Introduction
D.2 Model
D.2.1 Model
D.2.2 Rescaled Potential
D.3 Evolution equation and Brunet Bethe ansatz
D.3.1 Evolution equation
D.3.2 Bethe-Brunet Ansatz
D.4 Time evolution of the moments, symmetric transfer matrix
D.4.1 Symmetric transfer matrix and scalar product
D.4.2 Time-evolution of the moments
D.5 The continuum/Lieb-Liniger limit
D.6 Norm of the eigenstates
D.7 Large L limit
D.7.1 Strings
D.7.2 Eigenvalue of a string: energy
D.7.3 Momentum of a string
D.7.4 Phase space
D.7.5 Norm of the string states
D.8 Formula for the integer moments Zn
D.9 Generating function
D.9.1 Generating function for the moments
D.9.2 Generating function: Laplace transform
D.9.3 Probability distribution
D.10 Limit of very long polymers and universality
D.11 Comparison with other results
D.12 Conclusion
D.13 App A
D.14 App B
D.14.1 finite L
D.14.2 in the limit L → +∞
D.15 App C
D.16 App D
D.17 App E
D.18 App F
D.19 App G
D.20 App H
D.21 App I
D.22 App J
D.23 App K
E Paper: On integrable DP models on the square lattice
E.1 Introduction and main results
E.1.1 overview
E.1.2 Main results and outline of the paper
E.2 DPs on Z2: Replica method and integrability
E.2.1 Definition of the model
E.2.2 The replica method and the coordinate Bethe Ansatz.
E.2.3 The constraint of integrability on integer moments.
E.3 Integrable polymer models
E.3.1 The |q| < 1 case.
E.3.2 The q → 1 limit
E.4 Study of the Inverse-Beta Polymer
E.4.1 Moments Formula and Coordinate Bethe Ansatz
E.4.2 Fredholm determinant formulas and KPZ universality
E.4.3 The large length limit and the KPZ universality.
E.4.4 A low temperature limit.
E.5 Conclusion
E.6 App A
E.7 App B
E.8 App C
E.9 App D
F Paper: Beta polymer
F.1 Introduction and main results
F.1.1 Overview
F.1.2 Main results and outline of the paper
F.2 Model and earlier work
F.2.1 The Beta polymer
F.2.2 Relation to a random walk in a random environment
F.2.3 Relation to the problem and notations of Barraquand-Corwin
F.3 Bethe Ansatz solution of the Beta polymer
F.3.1 Bethe ansatz on a line with periodic boundary conditions
F.3.2 Resolution of the Bethe equations in the large L limit: repulsion and free particles
F.3.3 Bethe ansatz toolbox
F.3.4 A large contour-type moment formula
F.4 Asymptotic analysis in the diffusive regime
F.4.1 The issue of the first site
F.4.2 Cauchy type Fredholm determinant formulas
F.4.3 Asymptotic analysis of the first-moment: definition of the optimal direction and of the asymptotic regimes
F.4.4 Asymptotic analysis in the diffusive vicinity of the optimal direction on the Cauchy-type Fredholm determinant
F.4.5 Multi-point correlations in a diffusive vicinity of the optimal direction
F.5 Asymptotic analysis in the large deviations regime: KPZ universality
F.5.1 Recall of the results of Barraquand-Corwin
F.5.2 An inherent difficulty and a puzzle
F.5.3 A formal formula for the moments of the Beta polymer in terms of strings
F.5.4 A formal Fredholm determinant and KPZ universality
F.5.5 Crossover between Gamma and Tracy-Widom fluctuations
F.6 Nested-Contour integral formulas
F.6.1 Alternative moments formulas
F.6.2 Mellin-Barnes type Fredholm determinant
F.7 Numerical results
F.7.1 In the diffusive regime.
F.7.2 In the large deviations regime.
F.8 Conclusion
F.9 App A
F.10 App B
F.11 App C
F.11.1 Two equivalent formulas for the Strict-Weak polymer
F.11.2 A formal formula for the moments of the Beta polymer.
F.12 App D
G Paper: Stationary measures of DPs on Z2
G.1 Introduction
G.2 Recall: stationary measure of the Log-Gamma polymer
G.3 Overview: definitions, main results and outline
G.3.1 Definitions of the models of directed polymers
G.3.2 Stationarity and reversibility properties
G.3.3 Quenched free-energy in point to point models without boundaries
G.3.4 Convergence of point to point models to their stationary state .
G.3.5 Outline and some additional results not presented here
G.4 Stationary measure of the Inverse-Beta polymer
G.4.1 Stationary property of the model with boundaries
G.4.2 Stationarity property of the model with stationary initial condition
G.4.3 Reversibility of the stationary measure: detailed balance property
G.4.4 Relation to other models
G.5 Stationary measure of the Bernoulli-Geometric polymer
G.5.1 Stationarity properties of the Bernoulli-Geometric polymer
G.5.2 Relation to other models
G.6 Convergence to the stationary measure
G.6.1 Free-energy in models with boundaries
G.6.2 Free-energy in models without boundaries
G.6.3 Convergence to the stationary measures
G.6.4 A remark on optimal paths and energy fluctuations in models with boundaries
G.7 Numerical results for the zero-temperature model
G.8 Conclusion
G.9 App A
G.10 App B
Bibliographie
