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Short walk adventures. (English) Zbl 1448.33018
Bailey, David H. (ed.) et al., From analysis to visualization. A celebration of the life and legacy of Jonathan M. Borwein, Callaghan, Australia, September 25–29, 2017. Cham: Springer. Springer Proc. Math. Stat. 313, 423-439 (2020).
The paper is a good demonstration that the theory of random walks has beautiful connections to a large number of exotic topics of mathematics. Those who study random walks will eventually encounter topics like arithmetic differential equations, algebraic groups, Mahler measure, special function theory, modular functions, and so on.
The first part of the paper studies uniform random walks. An \(N\)-step uniform planar random walk starts at the origin and consists of \(N\) steps of length 1 each taken into a uniformly random direction. Let \(X_N\) be the distance to the origin after \(N\) steps. The \(s\)-th moment of \(X_N\) is denoted by \(W_N(s)\).
We learn from the third section that \(W_N(s)\) is a Mahler measure of the polynomial \(x_1+\cdots+x_N\) at \(s\), and that \(W_N'(0)\) is the logarithmic Mahler measure of the same polynomial. Based on this theory, one can deduce interesting closed form expressions for \(W_k'(0)\), for small integers \(k\).
The second part studies generic two-step random walks, that is, random walks where \(X=e^{2\pi i\theta_1}X_1+e^{2\pi i\theta_2}X_2\), where \(\theta_i\) are uniformly distributed on \([0,1]\), and \(X_i\) are independent random variables on \([0,\infty[\) \((i=1,2)\). The finding of the \(W_N'(0)\) values is described in details in Sections 4–7.
The last part of the paper studies additional relations between Mahler measures and the \(W_N'(0)\) values for small \(N\).
For the entire collection see [Zbl 1442.00024].
MSC:
33E05 Elliptic functions and integrals
33C20 Generalized hypergeometric series, \({}_pF_q\)
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