Around a Theorem of Pólya

Title: Around a Theorem of Pólya

Speaker: Louis Arenas

Date/Time/Location: Friday, October 21st and 28th at 12:00pm, NES 102

Abstract: Random walks have cemented themselves as relevant objects of study in modeling diverse phenomenon ranging from Brownian motion to the prediction of European stock options through Black-Scholes. In a paper published in 1921 by George Polya, he investigates walks on ${\mathbb{Z}}^d$ and asks when the walks are recurrent or transient. Meaning if there’s probability 1 of picking a random walk which returns to the origin after finitely many steps. In this talk we present a recent (published in 2014) paper by Jonothan Novak which proves Pólya’s theorem using tools that a ${19}^{th}$ century Mathematician would recognize.

Although Novak’s proof is beautiful in its own right, our motivation will be to extend his proof techniques to a non-commutative setting. Specifically, we pose the problem of recurrence of random walks on the Cayley graph of the free group generated by d letters, $F_d$. Random walks on non-commutative groups were of interest to people like Harry Kesten who under guidance of his advisor Mark Kac wanted criteria of amenability of finitely generated groups. Kesten would go on to formulate a criterion of amenability in his 1958 Ph.D. thesis.

Our focus will be to motivate the tools of free probability theory to answer the question of recurrence on $F_d$.