Is the magnitude of the Peccei-Quinn scale set by the landscape?
  Rather general considerations of the string theory landscape imply a mild
statistical draw towards large soft SUSY breaking terms tempered by the
requirement of proper electroweak symmetry breaking where SUSY contributions to
the weak scale are not too far from m(weak)~ 100 GeV. Such a picture leads to
the prediction that m_h~ 125 GeV while most sparticles are beyond current LHC
reach. Here we explore the possibility that the magnitude of the Peccei-Quinn
(PQ) scale f_a is also set by string landscape considerations within the
framework of a compelling SUSY axion model. First, we examine the case where
the PQ symmetry arises as an accidental approximate global symmetry from a more
fundamental gravity-safe Z(24)^R symmetry and where the SUSY mu parameter
arises from a Kim-Nilles operator. The pull towards large soft terms then also
pulls the PQ scale as large as possible. Unless this is tempered by rather
severe (unknown) cosmological or anthropic bounds on the density of dark
matter, then we would expect a far greater abundance of dark matter than is
observed. This conclusion cannot be negated by adopting a tiny axion
misalignment angle theta_i because WIMPs are also overproduced at large f_a.
Hence, we conclude that setting the PQ scale via anthropics is highly unlikely.
Instead, requiring soft SUSY breaking terms of order the gravity-mediation
scale m_{3/2}~ 10-100 TeV places the mixed axion-neutralino dark matter
abundance into the intermediate scale sweet zone where f_a~ 10^{11}-10^{12}
GeV. We compare our analysis to the more general case of a generic SUSY DFSZ
axion model with uniform selection on theta_i but leading to the measured dark
matter abundance: this approach leads to a preference for f_a~ 10^{12} GeV.
