Cover times for random walk on dynamical percolation
Abstract: We study the cover time of random walk on dynamical percolation on the torus $\mathbb{Z}_nd$ in the subcritical regime. In this model, introduced by Peres, Stauffer and Steif, each edge updates at rate $\mu$ to open with probability $p$ and closed with probability $1-p$. The random walk jumps along each open edge with rate $1/(2d)$. We prove matching (up to constants) lower and upper bounds for the cover time, which is the first time that the random walk has visited all vertices at least once. Along the way, we also obtain a lower bound on the hitting time of an arbitrary vertex starting from stationarity, improving on the maximum hitting time bounds by Peres, Stauffer and Steif.
- Brownian Motion on Compact Manifolds: Cover Time and Late Points. Electronic Journal of Probability, 8(none):1 – 14, 2003.
- Cover times for brownian motion and random walks in two dimensions. Annals of Mathematics, 160(2):433–464, 2004.
- Uwe Einmahl. Extensions of results of Komlós, Major, and Tusnády to the multivariate case. Journal of Multivariate Analysis, 28(1):20–68, January 1989.
- J. Hermon and P. Sousi. A comparison principle for random walk on dynamical percolation. Ann. Probab., 48(6):2952–2987, 2020.
- Markov Chains and Mixing Times, volume 107. American Mathematical Soc., 2009.
- P. Matthews. Covering problems for Markov chains. Ann. Probab., 16(3):1215–1228, 1988.
- Quenched exit times for random walk on dynamical percolation. Markov Process. Relat. Fields, 24:715–731, 2018.
- Mixing time for random walk on supercritical dynamical percolation. Prob. Theory Relat. Fields, 176, 04 2020.
- Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times. Probab. Theory Relat. Fields, 162:487–530, 2015.
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