  In light of increasing recent attention to political polarization,
understanding how polarization can arise poses an important theoretical
question. While more classical models of opinion dynamics seem poorly equipped
to study this phenomenon, a recent novel approach by H\k{a}z{\l}a, Jin, Mossel,
and Ramnarayan (HJMR) proposes a simple geometric model of opinion evolution
that provably exhibits strong polarization in specialized cases. Moreover,
polarization arises quite organically in their model: in each time step, each
agent updates opinions according to their correlation/response with an issue
drawn at random. However, their techniques do not seem to extend beyond a set
of special cases they identify, which benefit from fragile symmetry or
contractiveness assumptions, leaving open how general this phenomenon really
is.
  In this paper, we further the study of polarization in related geometric
models. We show that the exact form of polarization in such models is quite
nuanced: even when strong polarization does not hold, it is possible for weaker
notions of polarization to nonetheless attain. We provide a concrete example
where weak polarization holds, but strong polarization provably fails. However,
we show that strong polarization provably holds in many variants of the HJMR
model, which are also robust to a wider array of distributions of random issues
-- this indicates that the form of polarization introduced by HJMR is more
universal than suggested by their special cases. We also show that the weaker
notions connect more readily to the theory of Markov chains on general state
spaces.
