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- W2964343896 abstract "We show that on every Ramanujan graph $${G}$$ , the simple random walk exhibits cutoff: when $${G}$$ has $${n}$$ vertices and degree $${d}$$ , the total-variation distance of the walk from the uniform distribution at time $${t=frac{d}{d-2} log_{d-1} n + ssqrt{log n}}$$ is asymptotically $${{mathbb{P}}(Z > c , s)}$$ where $${Z}$$ is a standard normal variable and $${c=c(d)}$$ is an explicit constant. Furthermore, for all $${1 leq p leq infty}$$ , $${d}$$ -regular Ramanujan graphs minimize the asymptotic $${L^p}$$ -mixing time for SRW among all $${d}$$ -regular graphs. Our proof also shows that, for every vertex $${x}$$ in $${G}$$ as above, its distance from $${n-o(n)}$$ of the vertices is asymptotically $${log_{d-1} n}$$ ." @default.
- W2964343896 created "2019-07-30" @default.
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- W2964343896 date "2016-07-01" @default.
- W2964343896 modified "2023-10-10" @default.
- W2964343896 title "Cutoff on all Ramanujan graphs" @default.
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- W2964343896 doi "https://doi.org/10.1007/s00039-016-0382-7" @default.
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