  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.
