Two Random Multiplicative Processes: Multiplicative Cascades and Eigenvectors of the Random Schrodinger Operator

In this thesis we focus on two distinct random processes constructed using a sequence of independent identically distributed random variables along with a product structure on the underlying space. The first, a collection of random variables attached to vertices of the rooted binary tree, uses the multiplicative structure of paths along the tree while the second, a collection of two by two random matrices, uses the usual multiplication of matrices.In the first chapter we construct a continuous time version of the multiplicative cascade. The multiplicative cascade can be thought of as a randomization of an initial measure on the boundary of a tree, constructed from an independent, identically distributed collection of random variables attached to the tree vertices. The new random measure is constructed from the old by weighting the measure of any vertex in the the tree by the product of the random variables along the path from the root to the vertex. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. This process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It is also a martingale and, under certain extra conditions, it is also continuous. For Gaussian independent increments processes we develop the infinite-dimensional stochastic calculus that describes the evolution of the measure process, and use it to compute the optimal Holder exponent in the Wasserstein distance on measures.In the second chapter, we focus on the eigenvectors of the one-dimensional discrete random Schrodinger operator. This is the Hamiltonian operator on the lattice of integers given by the discrete Laplacian plus an independent, identically distributed delta potential at each point. We restrict this operator to a finite box, pick an eigenvector uniformly at random and characterize the scaling limit as we take the size of the box to infinity. We show that the envelope of this random eigenvector converges weakly to an exponential Brownian martingale. Our analysis uses the well known transfer matrix formulation of the spectral problem; the spectral information is encapsulated in a process of products of independent, identically distributed two by two random matrices. The limiting diffusion for this product process was developed in (Kritchevski, Valko, Virag - 2011) to study the local eigenvalue point process. Our work makes heavy use of that framework.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2014

Author dc:creator

• Rifkind, Benjamin Amichai

Advisor dc:contributor.advisor

• Balint, Virag

• Point Processes
• Probability
• Random Measures
• Random Schrodinger Operators
• Stochastic Process

Handle dc:identifier.uri

http://hdl.handle.net/1807/68309

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/68309