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.
Adding a column of numbers produces `carries' along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? (Many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae.) The examples give a gentle introduction to the emerging fields of one-dependent and determinantal point processes. This work is joint with Alexei Borodin and Persi Diaconis.
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.
A random walk is called recurrent if it is sure to return to its starting point and transient otherwise. A famous result of Polya is that simple symmetric random walk on the integer lattice is recurrent in dimensions 1 and 2, and transient in higher dimensions. We study random walks that are small perturbations of simple random walk. Our main result is that if the dimension is high enough then these random walks are transient.