  Towards formulating quantum gravity, we present a novel mechanism for the
emergence of spacetime geometry from randomness. In [arXiv:1705.06097], we
defined for a given Markov stochastic process "the distance between
configurations," which enumerates the difficulty of transition between
configurations. In this article, we consider stochastic processes of large-$N$
matrix models, where we regard the eigenvalues as spacetime coordinates. We
investigate the distance for the effective stochastic process of
one-eigenvalue, and argue that this distance can be interpreted in noncritical
string theory as probing a classical geometry with a D-instanton. We further
give an evidence that, when we apply our formalism to a tempered stochastic
process of $U(N)$ matrix, where the 't Hooft coupling is treated as another
dynamical variable, a Euclidean AdS$_2$ geometry emerges in the extended
configuration space in the large-$N$ limit, and the horizon corresponds to the
Gross-Witten-Wadia phase transition point.
