Abstract
We consider a model of a quenched disordered geometry in which a random metric is defined on , 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 , with a fluctuation exponent , while the lateral spread of the minimizing geodesic between two points at a distance L grows as , with wandering exponent value . 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.
Highlights
Random geometry is a branch of mathematics [1] with deep connections to physics, ranging from statistical mechanics to quantum gravity [2, 3]
We have shown evidence of KPZ scaling in a purely geometric model, in which the role of the evolving interface is played by balls of increasing radii in a random manifold
If the radius of the balls is thought of as time, we show that the growth of the Euclidean roughness of the ball isW ∼ t χ, with χ = 1 3
Summary
Random geometry is a branch of mathematics [1] with deep connections to physics, ranging from statistical mechanics to quantum gravity [2, 3]. The so-called wandering and fluctuation exponents, ξ and χ, for the geodesic and ball fluctuations, respectively, denote a certain universal fractal nature of straight lines and circles on a random geometry They occur for FPP on a lattice, where they are known to correspond, through an appropriate interpretation [15], to those characterizing the dynamics of a growing interface. Evidence has gathered, showing that systems in the KPZ universality class do share the values of the scaling exponents β and z, and the full probability distribution of the interface fluctuations [21], being accurately described by an universal, stationary, stochastic process that goes by the name of Airy process [22, 23].
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