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.