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 ${\mathbb{R}}^2$, 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 $R^{\chi }$, with a fluctuation exponent $\chi \simeq 1/3$, while the lateral spread of the minimizing geodesic between two points at a distance $L$ grows as $L^{\xi }$, with wandering exponent value $\xi \simeq 2/3$. Results on related first-passage percolation (FPP) problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar-Parisi-Zhang (KPZ) universality class of surface kinetic roughening, with $\xi$ and $\chi$ 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.

References in corpus (28)

• Shape Fluctuations and Random Matrices
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• The one-dimensional KPZ equation: an exact solution and its universality
• Discrete polynuclear growth and determinantal processes
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• Growing interfaces uncover universal fluctuations behind scale invariance
• On the Numerical Evaluation of Fredholm Determinants
• An exact solution for the KPZ equation with flat initial conditions
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Effects of Particle Shape on Growth Dynamics at Edges of Evaporating Colloidal Drops
• Statistical Self-Similarity of One-Dimensional Growth Processes
• Local behavior and hitting probabilities of the Airy1 process
• Continuum statistics of the Airy2 process
• Anisotropic Interface Depinning - Numerical Results
• Finite time corrections in KPZ growth models
• Universal fluctuations in radial growth models belonging to the KPZ universality class
• Statistics of circular interface fluctuations in an off-lattice Eden model
• Universal fluctuations in KPZ growth on one-dimensional flat substrates
• Non-universal parameters, corrections and universality in Kardar-Parisi-Zhang growth
• The circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer
• Short paths for first passage percolation on the complete graph
• A simplified proof of the relation between scaling exponents in first-passage percolation
• Intrinsic geometry approach to surface kinetic roughening
• Extreme Value Statistics Distributions in Spin Glasses
• A Shape Theorem for Riemannian First-Passage Percolation
• Geodesics of Random Riemannian Metrics
• Extreme-Value Distributions and the Freezing Transition of Structural Glasses

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