On the Unlikeliness of Multi-Field Inflation: Bounded Random Potentials
  and our Vacuum
  Based on random matrix theory, we compute the likelihood of saddles and
minima in a class of random potentials that are softly bounded from above and
below, as required for the validity of low energy effective theories. Imposing
this bound leads to a random mass matrix with non-zero mean of its entries. If
the dimensionality of field-space is large, inflation is rare, taking place
near a saddle point (if at all), since saddles are more likely than minima or
maxima for common values of the potential. Due to the boundedness of the
potential, the latter become more ubiquitous for rare low/large values
respectively. Based on the observation of a positive cosmological constant, we
conclude that the dimensionality of field-space after (and most likely during)
inflation has to be low if no anthropic arguments are invoked, since the
alternative, encountering a metastable deSitter vacuum by chance, is extremely
unlikely.
