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Probability Theory and Related Fields

, Volume 175, Issue 3–4, pp 1183–1185 | Cite as

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

  • Markus Heydenreich
  • Remco van der HofstadEmail author
Correction
  • 291 Downloads

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 ith 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

(Random graph asymptotics of the ordered cluster sizes) Fix \(d>6\) and 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 \(i= 1,\dots ,m\),
$$\begin{aligned} {\mathbb {P}}_{{\scriptscriptstyle \mathbb {T}}, p_c(\mathbb {Z}^d)}\Big (\omega ^{-1}V^{2/3} \le |\mathcal {C}_{\scriptscriptstyle (i)}|\le \omega V^{2/3}\Big ) \ge 1-\frac{b_i}{\omega }. \end{aligned}$$
(1.1)
Consequently, the expected cluster sizes satisfy \(\mathbb {E}_{{\scriptscriptstyle \mathbb {T}},p_c({{{\mathbb {Z}}}^d })}|\mathcal {C}_{\scriptscriptstyle (i)}|\ge b_i'\,V^{2/3}\) for certain constants \(b_i'>0\).

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

(\(|\mathcal{C}_\mathrm{max}|V^{-2/3}\) is not concentrated) Under the conditions of [3, Theorem 1.1], and assuming that there exists \(\omega >6^{2/3}\) such that
$$\begin{aligned} \liminf _{V\rightarrow \infty } V^{1/3} {\mathbb {P}}_{{\scriptscriptstyle \mathbb {T}}, p_c(\mathbb {Z}^d)}\Big (|\mathcal {C}|>\omega V^{2/3}\Big )>0, \end{aligned}$$
(3.2)
the random sequence \(|\mathcal{C}_\mathrm{max}|V^{-2/3}\) is non-concentrated.
The proof of [3, Proposition 3.1] can be followed verbatim, except for the discussion right after [3, (3.19)]. Indeed, [3, (3.19)] reads
$$\begin{aligned} {\mathrm{Var}_{p_c({{{\mathbb {Z}}}^d })}}(Z_{\scriptscriptstyle>\omega V^{2/3}} V^{-2/3})\ge & {} V^{-1/3}\mathbb {P}_{{\scriptscriptstyle \mathbb {T}}, p_c({{{\mathbb {Z}}}^d })}\big (|\mathcal {C}|>\omega V^{2/3})\nonumber \\&\times \big [\omega V^{2/3}- V\,\mathbb {P}_{{\scriptscriptstyle \mathbb {T}}, p_c({{{\mathbb {Z}}}^d })}(|\mathcal {C}|>\omega V^{2/3})\big ]\nonumber \\\ge & {} V^{1/3}\mathbb {P}_{{\scriptscriptstyle \mathbb {T}}, p_c({{{\mathbb {Z}}}^d })}(|\mathcal {C}|>\omega V^{2/3})\big [\omega -C_\mathrm{{\scriptscriptstyle \mathcal {C}}}\omega ^{-1/2}\big ], \end{aligned}$$
(3.3)
and below it, we claim that this remains uniformly positive for \(\omega \ge 1\) sufficiently large, by [3, (2.4)]. The problem is that [3, (2.4)] applies only to \(\omega \) that are not too large, while to keep the second factor in (3.3) positive, we need to take \(\omega >0\) sufficiently large, which we cannot satisfy simultaneously.

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.

References

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of MathematicsVrije Universiteit AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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