Random walks and random polynomials.
Published: 2006
Language: eng
Number of pages: 75
ISBN-13: 9780494219157
Description: 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).
Notes: Source: Dissertation Abstracts International, Volume: 68-01, Section: B, page: 0325.

Thesis (Ph.D.)--University of Toronto, 2006.

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