# Correction to: Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time

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## 1 Correction to: Probab. Theory Relat. Fields (2011) 149:397–415 https://doi.org/10.1007/s00440-009-0258-y

## 2 Correction of [3, Theorem 1.2]

Recall from [3] that \(\mathcal {C}_{\scriptscriptstyle (i)}\) denotes the *i*th largest cluster for percolation on the *d*-dimensional torus \(\mathbb T_{r,d}\), so that \(\mathcal {C}_{\scriptscriptstyle (1)}=\mathcal{C}_\mathrm{max}\) is the largest component and \(|\mathcal {C}_{\scriptscriptstyle (2)}|\le |\mathcal {C}_{\scriptscriptstyle (1)}|\) is the size of the second largest component, etc. Then, the statement of [3, Theorem 1.2] should be replaced by the following (shortened) statement:

### Theorem 1.2

*L*sufficiently large in the spread-out case, or

*d*sufficiently large for nearest-neighbor percolation. For every \(m=1,2,\dots \) there exist constants \(b_1, \dots , b_m>0\), such that for all \(\omega \ge 1\), \(r\ge 1\), and all \(i= 1,\dots ,m\),

In [3, Theorem 1.2], an additional non-concentration result was claimed for \(V^{-2/3}|\mathcal{C}_\mathrm{max}|\). The proof of this result is incorrect. Below we will explain why, and replace this statement by a *conditional* version. Unfortunately, we are not able to prove the required condition.

## 3 Last paragraph of discussion in [3, Section 1.3]

In the last paragraph of [3, Section 1.3], we discuss the non-concentration of \(V^{-2/3}|\mathcal{C}_\mathrm{max}|\), a feature that is highly indicative of the critical behavior. This paragraph needs to be removed.

## 4 Corrections to the proof of Theorem 1.2

The proof of [3, Theorem 1.2] still applies, except for [3, Proposition 3.1], where the non-concentration of \(|\mathcal{C}_\mathrm{max}|V^{-2/3}\) is proved. This statement can be replaced by the following conditional statement:

### Proposition 3.1

An inspection of the proof of the upper bound in [3, (2.4)] (which is originally [1, Theorem 1.3]) shows that \(C_\mathrm{{\scriptscriptstyle \mathcal {C}}}=6\) suffices. Indeed, by [2, Proposition 2.1], \(\mathbb {P}_{{\scriptscriptstyle \mathbb {T}}, p_c({{{\mathbb {Z}}}^d })}(|\mathcal {C}|\ge k)\le \mathbb {P}_{{\scriptscriptstyle \mathbb {Z}}, p_c({{{\mathbb {Z}}}^d })}(|\mathcal {C}|\ge k)\). Further, by [4, (9.2.6)], \(\mathbb {P}_{{\scriptscriptstyle \mathbb {Z}}, p_c({{{\mathbb {Z}}}^d })}(|\mathcal {C}|\ge k)\le \frac{{\mathrm e}}{{\mathrm e}-1}M(p_c({{{\mathbb {Z}}}^d }),1/k)\), while [4, Lemma 9.3] proves that \(M(p_c({{{\mathbb {Z}}}^d }), \gamma )\le \sqrt{12\gamma }\). Thus, we need that \(\omega >6^{2/3}\) to keep the second term in (3.3) strictly positive. For this choice, also [3, (3.19)] is satisfied. As a result, the proof can be repaired when we assume (3.2) for \(\omega >6^{2/3}\). \(\square \)

Mind that [3, (2.4)] implies (3.2) for \(\omega <b_{\scriptscriptstyle \mathcal C}\) for a positive constant \(b_{\scriptscriptstyle \mathcal C}\) (which is the same as \(b_1\) in [1, Theorem 1.3]). The actual value of \(b_{\scriptscriptstyle \mathcal C}\) depends of the position of \(p_c({{{\mathbb {Z}}}^d })\) within the critical window of \(p_c(\mathbb T_{r,d})\). While [3, Theorem 2.1] guarantees that \(p_c({{{\mathbb {Z}}}^d })\) does lie within the critical window, it gives us no control on the precise position.

We believe that (3.2) is correct, in fact, even for *all*\(\omega >0\). For example, for the Erdős–Rényi random graph model, which is the corresponding mean-field model, a corresponding statement is true for all \(\omega >0\), cf. [5, Lemma 2.2], where even a local limit version of (3.2) is proved. However, we have not been able to show this for percolation on the high-dimensional torus.

## Notes

### Acknowledgements

The work of RvdH is supported by the Netherlands Organisation for Scientific Research (NWO), through VICI Grant 639.033.806 and the Gravitation NETWORKS Grant 024.002.003.

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