Discrepancy convergence for the Drunkard's walk on the sphere (Q5936783)

The author investigates the random walk on the sphere \(S^2\) that starts at time \(0\) at some point and that makes a jump of fixed spherical size \(\theta\in ]0,\pi[\) with uniform directions at each step of time. For large times \(k\), this walk is approximately uniformly distributed. It is shown that the discrepancy distance \(D(k)\) of the walk after \(k\) steps from the uniform distribution satisfies NEWLINE\[NEWLINEC_1 e^{-(k\sin^2 \theta)/2}\leq D(k)\leq C_2 e^{-(k\sin^2 \theta)/8}NEWLINE\]NEWLINE with explicit constants \(C_1\), \(C_2\). The upper bound is obtained by estimating certain sums of Legendre polynomials. This is closely related with recent and more general Berry-Esseen-type estimates of the reviewer for ultraspherical expansions in [Publ. Math. 54, No. 1/2, 103-129 (1999; Zbl 0931.60002)].