In this talk, we consider the family of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$. We establish a uniform boundary Harnack principle with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $\Delta +b = \Delta^{\alpha/2}$ (or equivalently, the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets.

An old result of Kac-Hammersley says that the complex zeros of a Gaussian
random polynomial \sum_{j = 0}^N a_j z^j with i.i.d. normal coefficients a_j, concentrate
on the unit circle. This seems counter intuitive at first, since the zeros could be anywhere.
We will explain this paradox and show that there is a very general result that empirical measures
of complex zeros tend to `equilibrium measures'. We then give a large deviations principle showing
that the probability of deviation from equilibrium measure is exponentially small.