In Part 2, we consider an n-step simple symmetric random walk {Sk} on Z2 with the final point Sn= (pn, q n), which is motivated by group theory. When n & rarr; infinity, we prove that with probability tending to 1 there exists a line l whose slope is qnpn such that S0, S 1 ..., Sn meet l once at a unique point. This answers an open conjecture from group theory, which is given by Sapir. In the last part, we consider the real random power series fU (z) = Sinfinityi=0 bizi with i.i.d. standard real normal coefficients {bn} and U = ( -l, 1). With a very simple proof, we obtain concise analytical expressions for n-point correlations between real zeros of fU (z) in the unit interval U = ( -1, 1). Consider the zero set of a Gaussian analytic function f(z) which is an at least 3-dimensional polynomial in C (its values form an at least 3 dimensional vector space as random variables). Virag conjectures that there are always two points z1 and z2 such that p(z1, z2)> p(z1)p(z2), where p(z) is the intensity of the zero process at z and p(z1, z2) is the joint intensity. In the first part, we prove that the above conjecture is true for f(z) = Snk=0 akbkzk where {an} are i.i.d. standard complex Gaussian coefficients and {bn} are non-random constants. We consider more general cases f(z) = A 0 + A1z + A 2z2 where (A0,A1,A2) are jointly Gaussian random variables, and prove that the above conjecture is also true. Furthermore, we consider f(z) = Snk=0 akzk. We get the rates of Convergence for hole probability (there is no zero of the polynomial in this disk) and full probability (all zeros of the polynomial are contained in this disk).

Topics in Random Polynomials presents a rigorous and comprehensive treatment of the mathematical behavior of different types of random polynomials. These polynomials-the subject of extensive recent research-have many applications in physics, economics, and statistics. The main results are presented in such a fashion that they can be understood and used by readers whose knowledge of probability incorporates little more than basic probability theory and stochastic processes.

This book is devoted exclusively to a very special class of random processes, namely, to random walk on the lattice points of ordinary Euclidian space. The author considers this high degree of specialization worthwhile because the theory of such random walks is far more complete than that of any larger class of Markov chains. Almost 100 pages of examples and problems are included.

Comprehensive and interdisciplinary text covering the interplay between random walks and structure theory.