# Hypercube percolation

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### Uittreksel

We study bond percolation on the Hamming hypercube {0,1}^m around the critical probability p_c. It is known that if p=p_c(1+O(2^{-m/3})), then with high probability the largest connected component C_1 is of size Theta(2^{2m/3}) and that this quantity is non-concentrated. Here we show that for any sequence eps_m such that eps_m=o(1) but eps_m >> 2^{-m/3} percolation on the hypercube at p_c(1+eps_m) has |C_1| = (2+o(1)) eps_m 2^m and |C_2| = o(eps_m 2^m) with high probability, where C_2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].
Originele taal-2 Engels Eindhoven Eurandom 71 Gepubliceerd - 2013

### Publicatie series

Naam Report Eurandom 2013027 1389-2355

### Vingerafdruk

Hypercube
Critical Probability
M-sequence
Connected Components
Resolve

### Citeer dit

Hofstad, van der, R. W., & Nachmias, A. (2013). Hypercube percolation. (Report Eurandom; Vol. 2013027). Eindhoven: Eurandom.
Hofstad, van der, R.W. ; Nachmias, A. / Hypercube percolation. Eindhoven : Eurandom, 2013. 71 blz. (Report Eurandom).
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abstract = "We study bond percolation on the Hamming hypercube {0,1}^m around the critical probability p_c. It is known that if p=p_c(1+O(2^{-m/3})), then with high probability the largest connected component C_1 is of size Theta(2^{2m/3}) and that this quantity is non-concentrated. Here we show that for any sequence eps_m such that eps_m=o(1) but eps_m >> 2^{-m/3} percolation on the hypercube at p_c(1+eps_m) has |C_1| = (2+o(1)) eps_m 2^m and |C_2| = o(eps_m 2^m) with high probability, where C_2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].",
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Hofstad, van der, RW & Nachmias, A 2013, Hypercube percolation. Report Eurandom, vol. 2013027, Eurandom, Eindhoven.

Hypercube percolation. / Hofstad, van der, R.W.; Nachmias, A.

Eindhoven : Eurandom, 2013. 71 blz. (Report Eurandom; Vol. 2013027).

Onderzoeksoutput: Boek/rapportRapportAcademic

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N2 - We study bond percolation on the Hamming hypercube {0,1}^m around the critical probability p_c. It is known that if p=p_c(1+O(2^{-m/3})), then with high probability the largest connected component C_1 is of size Theta(2^{2m/3}) and that this quantity is non-concentrated. Here we show that for any sequence eps_m such that eps_m=o(1) but eps_m >> 2^{-m/3} percolation on the hypercube at p_c(1+eps_m) has |C_1| = (2+o(1)) eps_m 2^m and |C_2| = o(eps_m 2^m) with high probability, where C_2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].

AB - We study bond percolation on the Hamming hypercube {0,1}^m around the critical probability p_c. It is known that if p=p_c(1+O(2^{-m/3})), then with high probability the largest connected component C_1 is of size Theta(2^{2m/3}) and that this quantity is non-concentrated. Here we show that for any sequence eps_m such that eps_m=o(1) but eps_m >> 2^{-m/3} percolation on the hypercube at p_c(1+eps_m) has |C_1| = (2+o(1)) eps_m 2^m and |C_2| = o(eps_m 2^m) with high probability, where C_2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].

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Hofstad, van der RW, Nachmias A. Hypercube percolation. Eindhoven: Eurandom, 2013. 71 blz. (Report Eurandom).