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In this note, we investigate the behaviour of suprema for band-limitedspherical random fields. We prove upper and lower bound for the expected valuesof these suprema, by means of metric entropy arguments and discreteapproximations; we then exploit the Borell-TIS inequality to establish almostsure upper and lower bounds for their fluctuations. Band limited functions canbe viewed as restrictions on the sphere of random polynomials with increasingdegrees, and our results show that fluctuations scale as the square root of thelogarithm of these degrees.

A Note on Global Suprema of Band-Limited Spherical Random Functions