No Evidence for Millimeter Continuum Source Overdensities in the
  Environments of $z\gtrsim6$ Quasars
  Bright high-redshift quasars ($z>6$), hosting supermassive black holes ($\rm
M_{BH} > 10^8 M_{\odot}$), are expected to reside in massive host galaxies
embedded within some of the earliest and most massive galaxy overdensities. We
analyze 1.2\,mm ALMA dust continuum maps of 35 bright quasars at $6<z<7$ and
search the primary beam for excess dust continuum emission from sources with
L$_{\rm IR} \gtrsim{ 10^{12}}$\,L$_{\odot}$ as evidence for early
protoclusters. We compare the detection rates of continuum sources at
$\geq5\sigma$ significance in the fields surrounding the quasars (A$_{\rm eff}$
= 4.3\,arcmin$^2$) with millimeter number counts in blank field surveys. We
discover 15 millimeter sources in the fields excluding the quasars themselves,
corresponding to an overdensity $\delta_{\rm gal} \equiv (N_{\rm gal} - N_{\rm
exp})/N_{\rm exp} = -0.07\pm0.56$, consistent with no detected overdensity of
dusty galaxies within 140 physical kpc of the quasars. However, the apparent
lack of continuum overdensity does not negate the hypothesis that quasars live
in overdense environments, as evidenced by strong [CII] overdensities found on
the same scales to similarly-selected quasars. The small field of view of ALMA
could miss a true overdensity if it exists on scales larger than 1\,cMpc, if
the quasar is not centered in the overdensity, or if quasar feedback plays a
role close to the quasar, but it is most likely that the large line of sight
volume probed by a continuum survey will wash out a true overdensity signal. We
discuss the necessary factors in determining the bias with which dusty
star-forming galaxies trace true dark matter overdensities in order to improve
upon overdensity searches in dust continuum.
