  We consider the $N$-particle noncolliding Bernoulli random walk --- a
discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of
$N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning
that they never collide. We study the asymptotic behavior of local statistics
of this process started from an arbitrary initial configuration on short times
$T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial
configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$
in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is
in the "bulk"), and the initial configuration is balanced in a certain sense,
then the space-time local statistics at $x$ are asymptotically governed by the
extended discrete sine process (which can be identified with a translation
invariant ergodic Gibbs measure on lozenge tilings of the plane). We also
establish similar results for certain types of random initial data. Our proofs
are based on a detailed analysis of the determinantal correlation kernel for
the noncolliding Bernoulli random walk.
  The noncolliding Bernoulli random walk is a discrete analogue of the
$\beta=2$ Dyson Brownian Motion whose local statistics are universality
governed by the continuous sine process. Our results parallel the ones in the
continuous case. In addition, we naturally include situations with
inhomogeneous local particle density on scale $T$, which nontrivially affects
parameters of the limiting extended sine process, and in a particular case
leads to a new behavior.
