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# Arithmetic aspects of random walks and methods in definite integration

Arithmetic aspects of random walks and methods in definite integration
Armin Straub — Ph.D. Thesis (Tulane University) — advisor: Victor H. Moll, co-advisor: Jonathan M. Borwein — 2012

## Abstract

In the first part of this thesis, we revisit a classical problem: how far does a random walk travel in a given number of steps (of length $$1$$, each taken along a uniformly random direction)? We study the moments of the distribution of these distances as well as the corresponding probability distributions. Although such random walks are asymptotically well understood, very few exact formulas had been known; we supplement these with explicit hypergeometric forms and unearth general structures.

Our investigation of the moments naturally leads us to consider (multiple) Mahler measures. For several families of Mahler measures we are able to give evaluations in terms of log-sine integrals. Therefore, and because of the connections of log-sine integrals with number theory and mathematical physics, we study generalized log-sine integrals and show that they evaluate in terms of the well-studied polylogarithms. A computer algebra implementation of our results demonstrates that a large body of results on log-sine integrals scattered over the literature is now computer-amenable.

The second part is concerned with developing specific methods for evaluating several families of definite integrals arising in diverse contexts (such as calculations in quantum field theories). We also review and illustrate Ramanujan's Master Theorem and show that it generalizes to the method of brackets, which has its roots in the negative dimensional integration method utilized by particle physicists. We then apply this technique to multiple integrals recently studied in a physical context.

Complementary to these symbolic methods, we present an exponentially fast algorithm for numerically integrating rational functions over the real line. This algorithm operates on the coefficients of the rational function instead of evaluating it.

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