  We consider the load balancing system under Poisson arrivals, exponential
services, and homogeneous servers. Upon arrival, a job is to be routed to one
of the servers, where it is queued until service. We consider the Power-of-$d$
choices routing algorithm, which chooses the queue with minimum length among
$d$ randomly sampled queues. We study this system in the many-server
heavy-traffic regime where the number of servers goes to infinity
simultaneously when the load approaches the capacity. In particular, we
consider a sequence of systems with $n$ servers and the arrival rate is given
by $\lambda=n-n^{1-\gamma}$ for some $\gamma \in (0, 0.5)$, known as the
sub-Halfin-Whitt regime. It was shown by [Liu Ying (2020)] that under
Power-of-$d$ choices routing with $d \geq n^\gamma \log n$, the queue length
behaves similarly to that of JSQ and that there are asymptotically zero
queueing delays.
  The focus of this paper is to characterize the behavior when $d$ is below
this threshold. We obtain high probability bounds on the queue lengths for
various values of $d$ and large enough $n$. In particular, we show that when
$d$ grows polynomially in $n$ but slower than in [Liu Ying (2020)], i.e., if
$d$ is $\Theta\left((n^\gamma\log n)^{1/m})\right)$ for some integer $m>1$,
then the asymptotic queue length is $m$ with high probability. Moreover, if $d$
grows polylog in $n$, i.e., slower than any polynomial, but is at least
$\Omega(\log (n)^3)$, the queue length blows up to infinity asymptotically. We
obtain these results by using an iterative state space collapse approach. We
first establish a weak state-space collapse (SSC) on the queue lengths. Then,
we bootstrap on weak SSC to iteratively narrow down the region of the collapse.
After enough steps, this inductive refinement provides the bounds we seek. We
establish these sequences of collapse using Lyapunov drift arguments.
