  Do cool-core (CC) and noncool-core (NCC) clusters live in different
environments? We make novel use of H$\alpha$ emission lines in the central
galaxies of redMaPPer clusters as proxies to construct large (1,000's) samples
of CC and NCC clusters, and measure their relative assembly bias using both
clustering and weak lensing. We increase the statistical significance of the
bias measurements from clustering by cross-correlating the clusters with an
external galaxy redshift catalog from the Sloan Digital Sky Survey III, the
LOWZ sample. Our cross-correlations can constrain assembly bias up to a
statistical uncertainty of 6%. Given our H$\alpha$ criteria for CC and NCC, we
find no significant differences in their clustering amplitude. Interpreting
this difference as the absence of halo assembly bias, our results rule out the
possibility of having different large-scale (tens of Mpc) environments as the
source of diversity observed in cluster cores. Combined with recent
observations of the overall mild evolution of CC and NCC properties, such as
central density and CC fraction, this would suggest that either the cooling
properties of the cluster core are determined early on solely by the local
(<200 kpc) gas properties at formation or that local merging leads to
stochastic CC relaxation and disruption in a periodic way, preserving the
average population properties over time. Studying the small-scale clustering in
clusters at high redshift would help shed light on the exact scenario.
