In Diaconis's book Group Representations in Probability and Statistics, Chapter 3C, there are explicit computations for the mixing time of the random walk on the cycle graph $\mathbb{Z}_{p}$, with $p$ odd to avoid periodicity issues. With $P^{*n}$ the distribution of the walk after $n$ steps and $U$ the uniform distribution on $\mathbb{Z}_{p}$, he shows
$$||P^{*n} - U||_{TV}^{2} \leq \frac{1}{2}\sum_{j=1}^{\infty}e^{-c\pi^{2}j^{2}} + o(1)$$
for $c>0$ constant and $n \sim cp^{2}$ (Remark 2, pg. 27). He then cites an unpublished manuscript of his and Ron Graham's ("Monotonicity properties of random walk") in which it is shown that "a similar theta function is asymptotically equal to the variation distance."
I am interested in knowing what the "similar theta function" is. Does anyone know where I can find a proof of the cited result, or even the manuscript itself?
