  Arrow's Theorem concerns a fundamental problem in social choice theory: given
the individual preferences of members of a group, how can they be aggregated to
form rational group preferences? Arrow showed that in an election between three
or more candidates, there are situations where any voting rule satisfying a
small list of natural "fairness" axioms must produce an apparently irrational
intransitive outcome. Furthermore, quantitative versions of Arrow's Theorem in
the literature show that when voters choose rankings in an i.i.d.\ fashion, the
outcome is intransitive with non-negligible probability.
  It is natural to ask if such a quantitative version of Arrow's Theorem holds
for non-i.i.d.\ models. To answer this question, we study Arrow's Theorem under
a natural non-i.i.d.\ model of voters inspired by canonical models in
statistical physics; indeed, a version of this model was previously introduced
by Raffaelli and Marsili in the physics literature. This model has a parameter,
temperature, that prescribes the correlation between different voters. We show
that the behavior of Arrow's Theorem in this model undergoes a striking phase
transition: in the entire high temperature regime of the model, a Quantitative
Arrow's Theorem holds showing that the probability of paradox for any voting
rule satisfying the axioms is non-negligible; this is tight because the
probability of paradox under pairwise majority goes to zero when approaching
the critical temperature, and becomes exponentially small in the number of
voters beyond it. We prove this occurs in another natural model of correlated
voters and conjecture this phenomena is quite general.
