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

1 Correction to: Probab. Theory Relat. Fields (2011) 149:397–415

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 \(i= 1,\dots ,m\),
$$\begin{aligned} {\mathbb {P}}_{{\scriptscriptstyle \mathbb {T}}, p_c(\mathbb {Z}^d)}\Big (\omega ^{-1}V^{2/3} \le |\mathcal {C}_{\scriptscriptstyle (i)}|\le \omega V^{2/3}\Big ) \ge 1-\frac{b_i}{\omega }. \end{aligned}$$
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
$$\begin{aligned} \liminf _{V\rightarrow \infty } V^{1/3} {\mathbb {P}}_{{\scriptscriptstyle \mathbb {T}}, p_c(\mathbb {Z}^d)}\Big (|\mathcal {C}|>\omega V^{2/3}\Big )>0, \end{aligned}$$
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
$$\begin{aligned} {\mathrm{Var}_{p_c({{{\mathbb {Z}}}^d })}}(Z_{\scriptscriptstyle>\omega V^{2/3}} V^{-2/3})\ge & {} V^{-1/3}\mathbb {P}_{{\scriptscriptstyle \mathbb {T}}, p_c({{{\mathbb {Z}}}^d })}\big (|\mathcal {C}|>\omega V^{2/3})\nonumber \\&\times \big [\omega V^{2/3}- V\,\mathbb {P}_{{\scriptscriptstyle \mathbb {T}}, p_c({{{\mathbb {Z}}}^d })}(|\mathcal {C}|>\omega V^{2/3})\big ]\nonumber \\\ge & {} V^{1/3}\mathbb {P}_{{\scriptscriptstyle \mathbb {T}}, p_c({{{\mathbb {Z}}}^d })}(|\mathcal {C}|>\omega V^{2/3})\big [\omega -C_\mathrm{{\scriptscriptstyle \mathcal {C}}}\omega ^{-1/2}\big ], \end{aligned}$$
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.



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.


  1. 1.
    Borgs, C., Chayes, J., van der Hofstad, R., Slade, G., Spencer, J.: Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Struct. Algorithms 27(2), 137–184 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Heydenreich, M., van der Hofstad, R.: Random graph asymptotics on high-dimensional tori. Commun. Math. Phys. 270(2), 335–358 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Heydenreich, M., van der Hofstad, R.: Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time. Probab. Theory Relat. Fields 149(3–4), 397–415 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Heydenreich, M., van der Hofstad, R.: Progress in High-Dimensional Percolation and Random Graphs. CRM Short Courses Series. Springer, Cham (2017)CrossRefGoogle Scholar
  5. 5.
    van der Hofstad, R., Kager, W., Müller, T.: A local limit theorem for the critical random graph. Electron. Commun. Probab. 14, 122–131 (2009)MathSciNetCrossRefGoogle Scholar

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© 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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