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Consider a simple lazy random walk on an $n$-vertex undirected, connected graph: this is the Markov chain which transitions from $i$ to $j$ with probability $p_{ij}=1/(2d(i))$ where $d(i)$ is the degree of node $i$. Note that $p_{ii}=1/2$ for all $i$. Define $C(i)$ be the expected time until a walk starting from node $i$ visits every vertex and let $C = \max_i C(i)$. Let $I(k,l)$ be the expected time until two random walks, starting at vertices $k$ and $l$, intersect (i.e., until they visit the same vertex at the same time). Let $I = \max_{k,l} I(k,l)$.

My question is: how big can $I/C$ get as a function of the number of vertices $n$? Is it true that $I/C$ is upper bounded by a constant which is independent of $n$ or of the graph? If not, is it true that $ \frac{I}{C} \leq k \log^l n$ for some constants $k,l$ independent of $n$ and of the graph?

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    answered here: http://mathoverflow.net/questions/97698/cover-time-and-intersection-time-of-random-walks2013-01-25

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