Spectral Problems on Lamplighter Groups and Free Probability

The main focus of present project proposal lies on the crossroads of geometric group theory, percolation theory, free probability and mathematical physics. We will study computational and qualitative aspects of the spectral theory of free lamplighter groups in the light of a recently discovered identity between the spectral measures of percolation clusters and lamplighter groups. With this background we want to attack the following questions. 1. Find a connection between percolation on free product groups and Voiculescu`s free probability 2. Compute more examples of kernel dimensions of convolution operators on lamplighter groups 3. Determine the asymptotics of kernel dimensions of uniform random trees 4. Find applications of Mourre`s method to free probability. All these questions are first steps on the way to a better understanding of the nature of the spectra of percolation clusters, which is of some interest in mathematical physics. As a second complementary theme, we want to investigate so-called spreadable cumulants, which we introduced recently as a combinatorial tool to study certain kinds of noncommutative probability theories like Muraki`s monotone independence.

The subjects of the project are related to several aspects of Free Probability. The main results open a new algorithmic approach to the arithmetics of the noncommutative free field. Besides this, some noncommutative statistical characterization problems are solved, problems on random matrices are studied and first steps are taken towards tropical free probability.Free Probability is the most successful instance of a noncommutative probability theory. Apart from its origins in the theory of operator algebras, it exhibits strong analogies to classical probability and features a rich set of connections and applications to random matrix theory, combinatorics of noncrossing partitions, harmonic analysis on free groups, representation theory of the symmetric group and quantum information theory.The present project covers several aspects of Free Probability.1. The main results are motivated by harmonic analysis on free groups. Recently a new method was developed to compute spectra of convolution operators. To this end a method known as linearization was rediscovered, which encodes noncommutative polynomials and rational expressions in terms of linear matrix pencils. These are amenable to the machinery of operator valued free convolution and numerical computation of spectra of arbitrary elements of the free groups algebra is possible. To this end it is essential that the matrix pencils have small and if possible minimal dimension, which is called the rank of the element. From an abstract point of view this can be seen as finding minimal representations of elements of the noncommutative free field, which is a difficult open problem. Certain aspects of this problem are solved, namely a full solution the word problem, i.e., the answer to the question whether two minimal matrix pencils represent the same element, finding minimal representations of inverses and minimal representations of sums and products of elements of small rank.2. The distribution of quadratic forms of noncommutative free random variables are investigated. It turns out that certain quadratic forms like the sample variance exhibit the phenomenon of cancellation of odd cumulants, which has no analog in classical probability. Using this phenomenon we fully characterize the distributions whose free sample variance coincide with the free sample invariance of a semicircular random variable. The answer to the corresponding question in classical probability is still open.3. Recently Marcus, Spielman and Srivastava investigated certain combinatorial convolutions of polynomials which arose in connection with the Kadison-Singer conjecture and indicated an asymptotic relation to free probability. We explicitly computed analogs of this convolutions in the setting of max-plus algebra, a variant of idempotent (also known as tropical) mathematics. This can be seen as a first step into tropical free probability.