Random geometry and the Kardar–Parisi–Zhang universality class

Abstract

We consider a model of a quenched disordered geometry in which a random metric is defined on  2 , which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius R scales as χ R , with a fluctuation exponent χ ≃ 1 3, while the lateral spread of the minimizing geodesic between two points at a distance L grows as ξ L , with wandering exponent value ξ ≃ 2 3. Results on related first-passage percolation problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar-Parisi-Zhang universality class of surface kinetic roughening, with ξ and χ relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy-Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW-GUE statistics with good accuracy in arrival times.

Kardar–Parisi–Zhang universality in first-passage percolation: the role of geodesic degeneracy

“…Two different scaling regimes indicated by the broken lines are clearly observed. For most cases there is an initial regime of the form σ 2 T ∼ d, which is followed by the asymptotic scaling σ 2 T ∼ d 2β with β = 1 3 , in agreement with the expected KPZ universality class [7]. The pre-asymptotic regime can be arbitrarily large and exceed the lattice limits, as in the upper curve corresponding to LogN(0.1, 0.0002), or arbitrarily short so that it can not be observed, e.g.…”

“…For the square lattice we will assume that path lengths (denoted by l) and distances between lattices sites (given by d) will be given in units of the lattice spacing. It is worth noticing that times of arrival can be regarded as distances in a different metric, as it is done in [7]. Dis-ordered lattices will be addressed in section VI and their construction will be discussed there.…”

“…Results for case U(0.1, 9.9) were obtained for L = 2000 and from an ensemble of 1.5 · 10 5 points. Horizontal broken lines stand for the TW-GUE values for skewness (−0.224) and kurtosis (0.0934)[7].…”

A KPZ Cocktail-Shaken, not Stirred...

“…We mention that, aside from some early works [101,102,103], and more recently [104], there has been little direct effort on numerical integration of 2d KPZ equation in polar coordinates; see, too [105,106,107]. Indeed, all work on this subclass, aside from radial Eden model simulations [108,109], have resorted to difficult, somewhat frustrating pt-pt simulations of various KPZ/DPRM models [110,65] in what is, effectively, constrained wedge geometries.…”

Gaussian statistics as an emergent symmetry of the stochastic scalar Burgers equation

“…where ν, D > 0 and λ are parameters, r ∈ R d , and η is non- * Electronic address: enrodrig@math.uc3m.es † Electronic address: cuerno@math.uc3m.es conserved, zero-mean, uncorrelated Gaussian noise. Having been seminally put forward [10] right at the crossroads among important domains of non-equilibrium phenomena -like randomly stirred fluids, polymer dynamics in disordered media, and surface kinetic roughening-, the KPZ equation is recently being found to describe the universal behavior of a surprisingly wide range of strongly correlated systems [11], like bacterial range expansion [12], diffusion-limited growth [13], turbulent liquid crystals [14], classical non-linear oscillators [15], stochastic hydrodynamics [16], reaction-limited growth [17], random geometry [18], superfluid exciton polaritons [19], or incompressible polar active fluids [20].…”