The KPZ universality class and related topics

Abstract

These notes are based on a talk given at the 2018 Arizona School of Analysis and Mathematical Physics. We give a comprehensive introduction to the KPZ universality class, a conjectured class of stochastic process with local interactions related to random growth processes in 1 + 1 dimensions. We describe some of the characteristic properties of the KPZ universality class such as scaling exponents and limiting statistics. In particular, we aim to extract the characteristic properties of the KPZ universality class by understanding the KPZ stochastic partial differential equation by a special discrete approximation given by the asymmetric simple exclusion process (ASEP). The connection with the ASEP is very important as the process enjoys a rich integrability structure that leads to many exact formulas.Contents arXiv:1904.03319v1 [math-ph] 5 Apr 2019 with W (t, x) a Gaussian space-time white noise random variable (i.e. a δ-correlated stationary, Gaussian process with mean zero and covariance E [W (t , x )W (t, x)] = δ(t − t)δ(x − x)). The first term ν∂ 2x h(t, x) is a diffusion term which tends to smooth out the height function as time evolves, and the term λ 2 (∂ x h(t, x)) 2 is a lateral growth term, which makes the surface grow normal to its interface. The SPDE defined by eqn. (1.1) is now called the KPZ equation.It turns out that solving the KPZ equation is highly non-trivial, starting with making the KPZ equation (1.1) well-defined. For instance, through a local scaling argument, one might expect that the height function behaves similar to a Brownian motion due to the space-time white noise, and this means that, a priori, the term ∂ x h(t, x) is a distribution without a well-defined square (∂ x h(t, x)) 2 making the KPZ equation ill-defined. Under certain initial conditions, this issue has been resolved in the work of [BG97, ACQ11] by introducing the Cole-Hopf transformation (see eqn. 2.2) and an appropriate discrete approximation of the KPZ equation, and also, in the work of [Hai11] through the theory of rough paths. In [BG97], Bertini and Giacomoni gave a solution to the KPZ equation for initial conditions h(0, x) that (in average) grow at most linearly, such as flat initial conditions or two-sided Brownian motion initial conditions. In particular, Bertini and Giacomoni approach the KPZ equation by first smoothing out the noise, solving the smoothed KPZ equation via a discrete approximation using a discrete Cole-Hopf transformation, and finally taking the appropriate limits to a solution of the KPZ equation. Then, in [ACQ11], Amir, Corwin and Quaste were able to extend the results of [BG97] by treating the KPZ equation with the infinite narrow wedge initial condition (i.e. h(0, x) = log(δ x=0 )), which is a natural initial condition under the random polymer interpretation of the KPZ equation. In [ACQ11], the authors still use the idea of considering the Cole-Hopf transformation and were able to treat the wedge initial conditions via the exact formulas, introduced by Tracy and Widom in [TW08, TW09], for the asy...

Riemann surfaces for KPZ with periodic boundaries

“…The expression (6) for sharp wedge initial condition is a consequence of the generating function obtained in [39], equation (9), and given for s > 0 by…”

“…KPZ universality in 1+1 dimension [1,2,3,4,5,6,7,8,9] describes large scale fluctuations appearing in a variety of systems such as growing interfaces [10], disordered conductors [11], one-dimensional classical [12,13,14] and quantum [15,16,17] fluids, or traffic flow [18]. The height field h λ (x, t) characterizing KPZ universality depends on position x ∈ R, time t ≥ 0, and on a parameter λ > 0 quantifying the strength of nonlinear effects and the non-equilibrium character of the dynamics.…”

Riemann surface for TASEP with periodic boundaries

“…At large scales, TASEP belongs to KPZ universality [5,6,7,8,9,10,11]. More precisely, calling L the number of lattice sites and N the number of particles, the statistics of the height function of TASEP at fixed density ρ = N/L converges at large L on the time scale t ∼ L 3/2 to that of the KPZ fixed point in finite volume, describing how Tracy-Widom distributions and Airy processes characteristic of the process on the infinite line [12,13,14,15,16,17,18,19,20,21,22] relax [23,24,25,26] to a Brownian stationary state with non-Gaussian large deviations [27,28,29,30,31].…”

“…We consider two sets of Bethe roots y j and w j solutions of the Bethe equations (1), with respective fugacities γ y and γ w assumed to be distinct. Comparing the definitions (2), ( 3) and ( 10), (11), one has for arbitrary γ…”

Report on scipost_201909_00001v1

“…KPZ universality in 1+1 dimension [1][2][3][4][5][6][7][8][9] describes large scale fluctuations appearing in a variety of systems such as growing interfaces [10], disordered conductors [11], onedimensional classical [12][13][14] and quantum [15][16][17] fluids, or traffic flow [18]. The height field h λ (x, t) characterizing KPZ universality depends on position x ∈ R, time t ≥ 0, and on a parameter λ > 0 quantifying the strength of non-linear effects and the nonequilibrium character of the dynamics.…”