Random geometry and the Kardar–Parisi–Zhang universality class
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Random geometry and the Kardar–Parisi–Zhang universality class

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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2015

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English

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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 shortrange 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 firstpassage 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 onepoint and twopoint correlators converge to the behavior expected for the Airy2 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 extremevalue 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.

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