  Q-exchangeable ergodic distributions on the infinite symmetric group were
classified by Gnedin-Olshanski (2012). In this paper, we study a specific
linear combination of the ergodic measures and call it the Mallows product
measure. From a particle system perspective, the Mallows product measure is a
reversible stationary blocking measure of the infinite-species ASEP and it is a
natural multi-species extension of the Bernoulli product blocking measures of
the one-species ASEP. Moreover, the Mallows product measure can be viewed as
the universal product blocking measure of interacting particle systems coming
from random walks on Hecke algebras.
  For the random infinite permutation distributed according to the Mallows
product measure we have computed the joint distribution of its neighboring
displacements, as well as several other observables. The key feature of the
obtained formulas is their remarkably simple product structure. We project
these formulas to ASEP with finitely many species, which in particular recovers
a recent result of Adams-Balazs-Jay, and also to ASEP(q,M).
  Our main tools are results of Gnedin-Olshanski about ergodic Mallows measures
and shift-invariance symmetries of the stochastic colored six vertex model
discovered by Borodin-Gorin-Wheeler and Galashin.
