This paper presents necessary and sufficient conditions for the peakedness comparison and convex ordering between two elliptically contoured random fields about their centers. A somewhat surprising finding is that the peakedness comparison for the infinite dimensional case differs from the finite dimensional case. For example, a Student’s t distribution is known to be more heavy-tailed than a normal distribution, but a Student’s t random field and a Gaussian random field are not comparable in terms of the peakedness. In particular, the peakedness comparison and convex ordering are made for isotropic elliptically contoured random fields on compact two-point homogeneous spaces.