## Giant vacant component left by a random walk in a random $$d$$-regular graph.(English. French summary)Zbl 1267.05237

The trajectory is considered of a simple random walk on a regular graph of order $$n$$ with tree-like structure for increasing $$n$$. The vacant set is the complement of the trajectory. Its percolative properties as $$n$$ increases are investigated at time $$un$$ where $$u$$ is a fixed positive parameter. It is shown that there exists a threshold for $$u$$ such that the largest component of the vacant set is of order $$n$$ below the threshold and of order $$\log(n)$$ above the threshold. Connections with the random interlacement model are also given.

### MSC:

 05C81 Random walks on graphs 60G50 Sums of independent random variables; random walks 05C80 Random graphs (graph-theoretic aspects) 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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### References:

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