  We investigate the effect of propagation of cosmic-ray electrons (CRE) on the
nonthermal (synchrotron) - far-infrared correlations in M31 and M33. The
thermal (TH) and nonthermal (NTH) emission components of the radio continuum
emission at 1.4 GHz and one higher frequency are compared with dust emission
from M31 and M33 using Spitzer data. In both galaxies the TH emission is
linearly correlated with the emission from warm dust (24 \mu m, 70 \mu m), but
the power laws of the NTH-FIR correlations have exponents b < 1 that increase
with increasing frequency. Furthermore, the values of b for M33 are
significantly smaller (b ~ 0.4) than those for M31 (b ~ 0.6). We interpret the
differences in b as differences in the diffusion length of the CRE. We estimate
the diffusion length in two ways: (1) by smoothing the NTH emission at the
higher frequency until the correlation with NTH emission at 1.4 GHz has b = 1,
and (2) by smoothing the TH emission until the correlation with the NTH
emission at the same frequency has b = 1, assuming that the TH emission
represents the source distribution of the CRE. Our smoothing experiments show
that M31 only has a thin NTH disk with a scale height of h = 0.3-0.4 kpc at 1.4
GHz, whereas M33 has a similar thin disk as well as a thick disk with scale
height h_thick ~ 2 kpc. In the thin disks, the (deprojected) diffusion length
at 1.4 GHz is ~ 1.5 kpc, yielding a diffusion coefficient of ~ 2 10^28 cm^2/s.
The structure, strength and regularity of the magnetic field in a galaxy as
well as the existence of a thick disk determine the diffusion of the CRE, and
hence, the power-law exponent of the NTH-FIR correlations.
