  Parallel tempering is popular method for allowing MCMC algorithms to properly
explore a $d$-dimensional multimodal target density. One problem with
traditional power-based parallel tempering for multimodal targets is that the
proportion of probability mass associated to the modes can change for different
inverse-temperature values, sometimes dramatically so. Complementary work by
the authors proposes a novel solution involving auxiliary targets that preserve
regional weight upon powering up the density. This paper attempts to address
the question of how to choose the temperature spacings in an optimal way when
using this type of weight-preserving approach.The problem is analysed in a
tractable setting for computation of the expected squared jumping distance
which can then be optimised with regards to a tuning parameter. The conclusion
is that for an appropriately constructed regionally weight-preserved tempering
algorithm targeting a $d$-dimensional target distribution, the consecutive
temperature spacings should behave as $\mathcal{O}\left(d^{-1/2}\right)$ and
this induces an optimal acceptance rate for temperature swap moves that lies in
the interval $[0,0.234]$.
