Pseudo-Mixing Time of Random Walks

Webpage for a paper by Itai Benjamini and Oded Goldreich


We introduce the notion of pseudo-mixing time of a graph define as the number of steps in a random walk that suffices for generating a vertex that looks random to any polynomial-time observer, where, in addition to the tested vertex, the observer is also provided with oracle access to the incidence function of the graph.

Assuming the existence of one-way functions, we show that the pseudo-mixing time of a graph can be much smaller than its mixing time. Specifically, we present bounded-degree $N$-vertex Cayley graphs that have pseudo-mixing time $t$ for any $t(N)=\omega(\log\log N)$. Furthermore, the vertices of these graphs can be represented by strings of length $2\log_2N$, and the incidence function of these graphs can be computed by Boolean circuits of size $poly(\log N)$.

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