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This note is an addendum to 'Critical values of random analytic functions oncomplex manifolds, Indiana Univ. Math. J. 63 No. 3 (2014), 651-686.' by R.Fengand S. Zelditch (arXiv:1212.4762). In this note, we give the formula of thelimiting distribution of the critical values at critical points of randomanalytic functions of fixed Morse index and we compute explicitly suchlimitings for local maxima and saddle values in the case of Riemann surfaces;we also prove that the second order term in the full expansion of thedistribution of the critical values is topologically invariant on Riemannsurfaces.

Critical values of fixed Morse index of random analytic functions on Riemann surfaces