The zero set of the Gaussian analytic function fz= n=0infinityan znn! 0.0.1 is invariant in distribution under all isometries of the plane and is amazingly "lattice-like". A recent work of Sodin and Tsirelson (2005) shows that it can be matched to a lattice with Gaussian tails. Moreover, the "hole probability" that a disk of radius R contains no zeros of f decays exponentially in the square of the area of the hole. This asymptotic behavior is also observed in the perturbed lattice model in which lattice points are perturbed by independent complex normal random variables.;We consider a time dependent version of f in which the coefficients an are allowed to evolve as independent Ornstein-Uhlenbeck processes. The study of Gaussian analytic functions as dynamic processes was initiated by Peres and Virag (2005). We show that the zero set Zf(t) of the diffusing analytic function defines a time homogeneous Markov process and the "hole probability" that Zf(t) does not intersect a fixed disk of radius R for all t ∈ [0, T] decays like exp( -TecR2 ). This result sharply differentiates the zero set of f from a number of canonical evolving planar point processes. For example, the hole probability of the perturbed lattice model { p (m + in) + xiam,n : m,n ∈ Z } where am,n are i.i.d. Ornstein-Uhlenbeck processes decays like exp(-cTR4). This stark contrast is also present in the overcrowding probability that a disk of radius R contains at least N zeros for all t ∈ [0, T].;In the last chapter we present joint work with Yuval Peres in which we compute precise asymptotics for the radius Rn of the largest disk centered at the origin covered by simple random walk run for n steps. We prove that almost surely limsupn→infinity logRn 2lognlog 3n=14, 0.0.2 where log3 denotes 3 iterations of the log function. This is motivated by a question of Erdo&huml;s and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner (1994).
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