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We study hitting times in simple random walks on graphs, which measure thetime required to reach specific target vertices. Our main result establishes asharp lower bound for the variance of hitting times. For a simple random walkon a graph with $n$ vertices, we prove that the variance of the hitting timefrom a vertex $x$ to a vertex $y$, denoted $\tau_y$, is at least of the order$\mathbb{E}_x(\tau_y)^2 / \log n$. When the graph is a tree, we show that $n$can be replaced by the graph's distance between vertices $x$ and $y$.

Optimal lower bound for the variance of hitting times for simple random walks on graphs