  In this note, we investigate the behaviour of suprema for band-limited
spherical random fields. We prove upper and lower bound for the expected values
of these suprema, by means of metric entropy arguments and discrete
approximations; we then exploit the Borell-TIS inequality to establish almost
sure upper and lower bounds for their fluctuations. Band limited functions can
be viewed as restrictions on the sphere of random polynomials with increasing
degrees, and our results show that fluctuations scale as the square root of the
logarithm of these degrees.
