Correction to: Tightness for the cover time of the two dimensional sphere

1 Correction to: Probability Theory and Related Fields https://doi.org/10.1007/s00440-019-00940-2

Lemma 9.1 in the paper is incorrect as stated. The proof given there yields the following corrected statement.

Lemma 9.1

For any $L-2k\ge k '>k+10\ge 11$ and $n>1$, let $\mathcal {A}_{k'}$ denote an event, measurable on the excursions of the Brownian motion from $\partial B_{d}(y,h_{k'})$ to $\partial B_{d}(y,h_{k'-1})$ during the first n excursions from $\partial B_{d}(y,h_{k })\rightarrow \partial B_{d}(y,h_{k-1})$. Then,

$$\begin{aligned}&\mathbb {P}(\mathcal {A}_{k'} \, | \, T_{y,k'-1\rightarrow k'}^{y,k\overset{n}{\rightarrow }k-1} = m_{k'}, \, \mathcal {G}_k^y)\nonumber \\&\quad = \left( 1+O\left( (k'-k)\frac{h_{k'-1}}{h_{k}}\right) \right) ^{m_{k'}} \mathbb {P}(\mathcal {A}_{k'}\, | \, T_{y,k'-1\rightarrow k'}^{y,k\overset{n}{\rightarrow }k-1} = m_{k'}) \,. \end{aligned}$$

(9.1)

In particular, for all $m_l ;l=k',\ldots L$, and all $y \in \mathbf{S}^2$,

$$\begin{aligned}&\mathbb {P}(T_{y,l-1\rightarrow l}^{y,k\overset{n}{\rightarrow }k-1} = m_l ;l=k'+1,\ldots L \, |\, T_{y,k'-1\rightarrow k'}^{y,k\overset{n}{\rightarrow }k-1} = m_{k'}, \, \mathcal {G}_k^y) \nonumber \\&\quad = \left( 1+O\left( (k'-k)\frac{h_{k'-1}}{h_{k}}\right) \right) ^{m_{k'}}\nonumber \\&\qquad \mathbb {P}(T_{y,l-1\rightarrow l}^{y,k\overset{n}{\rightarrow }k-1} = m_l ;l=k'+1,\ldots L \, |\, T_{y,k'-1\rightarrow k'}^{y,k\overset{n}{\rightarrow }k-1} = m_{k'} ). \end{aligned}$$

(9.2)

In Lemma 9.1 we have $\left| O\left( (k'-k)\frac{h_{k'-1}}{h_{k}}\right) \right| \le 100\left( (k'-k)\frac{h_{k'-1}}{h_{k}}\right) .$

This necessitates numerous corrections, mostly in the order in which estimates are applied. The changes occur in Section 4, as sketched below. Full details are given in the companion posting [2]. Here and in what follows, boldfaced numbers refer to the original numbering in [1].

1. Section 4.1 (first moment estimate) The estimate (4.56) is incorrect. Instead, one needs to use the decoupling estimates from Section 4.5. This necessitates changes starting from the paragraph following (4.47) through the end of the subsection, as follows.

Let

$$\begin{aligned} \mathcal {W}^{\in x}_{y,k}( n)=\left\{ d^1_{{\mathrm{Wa}}}\left( \frac{1}{n} \sum _{i=1}^{ n} \delta _{\theta _{k,i}},\nu _{k}\right) \in \frac{c_{0}}{2\sqrt{n}} I_{ x } \right\} , \end{aligned}$$

(1.1)

so that

$$\begin{aligned} \mathcal {W}^{c}_{y,k}(N_{k,a})\subseteq \cup _{m=\log k}^{\infty } \mathcal {W}^{\in m}_{y,k}( N_{k,a}), \end{aligned}$$

(1.2)

and consequently, setting

$$\begin{aligned} \mathcal {L}_{k,m,p,a}=\mathcal {K}_{k,p,a}\cap \mathcal {W}_{y,k}^{\in m}(N_{k,a}), \end{aligned}$$

(1.3)

we have

$$\begin{aligned} \mathbb {P}\left( \mathcal {K}_{k,p,a}\right) \le \sum _{m=\log k}^{\infty } \mathbb {P}\left( \mathcal {L}_{k,m, p,a}\right) . \end{aligned}$$

(1.4)

Thus to prove (4.47) it suffices to show that for all $m\ge \log k$, all $d^{*}\le k\le L_{-}$, and all $0\le z\le \left( \log L\right) ^{1/4}$,

$$\begin{aligned} \sum _{a=[k^{1/4}]}^{L^{3/4}}\sum _{p=[( k-3)^{1/4}]}^{L^{3/4}}\mathbb {P}\left( \mathcal {L}_{k,m,p,a}\right) \le c(1+z)e^{-2L} e^{ -2z}e^{ -c'\,m^{ 2}}. \end{aligned}$$

(1.5)

Write

$$\begin{aligned} \mathcal {L}'_{k,m,p,a}=\mathcal {H}_{k-3,p}\bigcap \mathcal {H}_{k,a}\bigcap \mathcal {W}_{y,k}^{\in m}(N_{k,a}). \end{aligned}$$

(1.6)

Since the $\theta _{k,i}$ are i.i.d. $\nu _{k}$-distributed random variables, as explained in the paragraph before (4.19), it follows from (4.17) that

$$\begin{aligned} \mathbb {P}\left( \mathcal {W}_{y,k}^{\in m}( N_{k,a})\,\Big |\, T_{k-3}^{y,t_{z}}\right) \le 2e^{ -m^{ 2}/4}. \end{aligned}$$

(1.7)

The following lemma is immediate, details in [2, Lemma 4.6].

Lemma 1.1

There exist constants $c,c'>0$ so that

$$\begin{aligned} \mathbb {P}\left( \mathcal {L}'_{k, m, p,a}\right) \le c\, e^{ -2k-2(z-p)-(z-p)^{2}/4k}e^{ -m^{ 2}/8} e^{-c' (a-p)^{ 2}}. \end{aligned}$$

(1.8)

Let

$$\begin{aligned} k^{+}= & {} k+\lceil 10^{10} \log L\rceil , \quad k^{++}=k+2\lceil 10^{10} \log L\rceil , \end{aligned}$$

(1.9)

$$\begin{aligned} \mathbb {B}_{y,k+3,k^{ +} }^{j' }= & {} \left\{ \gamma \left( l\right) \le \sqrt{2T_{l}^{y,t_{z}}} \text{ for } l=k+3,\ldots ,k^{ +}, \right. \nonumber \\&\qquad \qquad \left. \,\sqrt{2T_{k^{ +}}^{y,t_{z}}} \in I_{\rho _{L}(L-k^{ +})+j'}\right\} , \end{aligned}$$

(1.10)

and

$$\begin{aligned} \widehat{\mathfrak {B}}_{y,k^{ ++}+1,L}= & {} \left\{ \rho _{L}(L-l) \le \sqrt{2T_{l}^{y,t_{z}}} \text{ for } l=k^{ ++}+1,\ldots ,L-1,\right. \nonumber \\&\qquad \qquad \left. \,\sqrt{2T_{L}^{y,t_{z}}}=0\right\} . \end{aligned}$$

(1.11)

We have that,

$$\begin{aligned} \mathbb {P}\left( \mathcal {L}_{k, m, p,a}\right)= & {} \mathbb {P}\left( \mathcal {L}'_{k, m, p,a} \bigcap \mathcal {B}_{y,k+1,L} \right) \le \sum _{j' = \left( k^{ +}\right) ^{1/4}}^{L^{3/4}} \sum _{j'' = \left( k^{ ++}\right) ^{1/4}}^{L^{3/4}} \nonumber \\&\quad \mathbb {P}\left( \mathcal {L}'_{k, m, p,a}\cap \mathbb {B}_{y,k+3,k^{ +} }^{j' }\cap \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L};\,\sqrt{2T_{k^{ ++}}^{y,t_{z}}} \in I_{\rho _{L} (L-k^{ ++})+j''}\right) , \nonumber \\ \end{aligned}$$

(1.12)

up to an error due to the restriction of $j',j''\le L^{3/4}$, which is little o of the right-hand side of [1, (4.43)]. Let

$$\begin{aligned}&\mathcal {G}_{k^{ +}}^y \nonumber \\&\quad = \sigma \text{-algebra } \text{ generated } \text{ by } \text{ the } \text{ excursions } \text{ from } \partial B_{d}(y,h_{k^{ +}-1}) \text{ to } \partial B_{d}(y,h_{k^{ +}}) \end{aligned}$$

(1.13)

(In the definition of $\mathcal {G}_{k^{ +}}^y$, if we start outside $\partial B_{d}(y,h_{k^{ +}-1})$ we include the initial excursion to $\partial B_{d}(y,h_{k^{ +}-1})$. Do not confuse with (3.38).) Note that

$$\begin{aligned} \mathcal {A}_{j'}:= \mathcal {L}'_{k, m, p,a} \cap \mathbb {B}_{y,k+3,k^{ +} }^{j' }\in \mathcal {G}_{k^{ +}}^y. \end{aligned}$$

(1.14)

Using (1.14) we have

$$\begin{aligned}&\mathbb {P}\left( \mathcal {A}_{j'};\,\sqrt{2T_{k^{ ++}}^{y,t_{z}}} \in I_{\rho _{L} (L-k^{ ++})+j''};\, \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L} \right) \nonumber \\&\quad \le \sup _{s\in I_{\rho (L-k^{ +})+j'},\,\, v\in I_{\rho (L-k^{ ++})+j''}} \mathbb {P}\left( \mathcal {A}_{j'};\,\sqrt{2T_{k^{ ++}}^{y,t_{z}}} \in I_{\rho (L-k^{ ++})+j''};\, \right. \nonumber \\&\qquad \qquad \left. \mathbb {P}\left( \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L}\, | \, T_{y,k^{+ +}-1\rightarrow k^{+ +}}^{y,k^{+}\overset{s^{2}/2}{\rightarrow }k^{+}-1} =v^{2}/2, \, \mathcal {G}_{k^{+}}^y\right) \right) . \end{aligned}$$

(1.15)

By Lemma 9.1 with the $k,k'$ there replaced by $k^{ +}, k^{+ +}$ and using that $(1+10(k^{++}-k^{ +})h_{k^{++}}/h_{k^{ +}})^{4L^2}$ is bounded above uniformly, we have that for some universal $c<\infty$

$$\begin{aligned}&\mathbb {P}\left( \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L}\, | \, T_{y,k^{+ +}-1\rightarrow k^{+ +}}^{y,k^{+}\overset{{s^{2}}/{2}}{\rightarrow }k^{+}-1} =v^{2}/2, \, \mathcal {G}_{k^{+}}^y\right) \nonumber \\&\quad \le c\mathbb {P}\left( \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L}\, | \, T_{y,k^{+ +}-1\rightarrow k^{+ +}}^{y,k^{+}\overset{{s^{2}}/{2}}{\rightarrow }k^{+}-1} =v^{2}/2 \right) . \end{aligned}$$

(1.16)

By the barrier estimate of Lemma 8.4 in Appendix I, we see that for $j''$ in the range of summation in (1.12), uniformly in $s\in I_{\rho (L-k^{ +})+j'}$ and $v\in I_{\rho (L-k^{ ++})+j''}$,

$$\begin{aligned} \mathbb {P}\left( \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L}\, | \, T_{y,k^{+ +}-1\rightarrow k^{+ +}}^{y,k^{+}\overset{{s^{2}}/{2}}{\rightarrow }k^{+}-1} =v^{2}/2 \right) \le c( 1+ j'' )e^{ -2(L-k^{+ +})-2 j'' }. \end{aligned}$$

(1.17)

Thus, uniformly in $[k^{1/4}]\le a \le L^{3/4}$,

$$\begin{aligned} \mathbb {P}\left( \mathcal {L}_{k, m, p,a}\right)= & {} \mathbb {P}\left( \mathcal {L}'_{k, m, p,a}\cap \mathcal {B}_{y,k+3,L} \right) \le c e^{ -2(L-k^{+ +}) } \nonumber \\&\times \sum _{j' = \left( k^{ +}\right) ^{1/4}}^{L^{3/4}} \sum _{j'' = \left( k^{ ++}\right) ^{1/4}}^{L^{3/4}} j'' e^{ -2 j'' } \mathbb {P}\left( \mathcal {L}'_{k, m, p,a}\cap \mathbb {B}_{y,k+3,k^{ +}, k^{ ++} }^{j', j'' }\right) ,\nonumber \\ \end{aligned}$$

(1.18)

where

$$\begin{aligned} \mathbb {B}_{y,k+3,k^{ +}, k^{ ++} }^{j', j'' }= & {} \left\{ \gamma \left( l\right) \le \sqrt{2T_{l}^{y,t_{z}}} \text{ for } l=k+3,\ldots ,k^{ +}, k^{ ++},\right. \nonumber \\&\quad \left. \,\sqrt{2T_{k^{ +}}^{y,t_{z}}} \in I_{\rho _{L}(L-k^{ +})+j'},\,\sqrt{2T_{k^{ ++}}^{y,t_{z}}} \in I_{\rho _{L} (L-k^{ ++})+j''}\right\} .\qquad \end{aligned}$$

(1.19)

From this point, the estimates needed are similar to the argument in [1, Section 4.1], using the barrier estimates of Section 8 according to different values of m. The full details and the division to different cases appear in [2, Section 4.1].

Section 4.4 (Second moment estimate: early branching) The last inequality in (4.87) is incorrect. In order to fix that, one needs to change the events to which one applies decoupling, starting from below (4.83), as follows.

Recall the events $\mathbb {B}_{y,k+3,k^{ +} }^{j' }$ and $\widehat{\mathfrak { B}}_{y,k^{ ++}+1,L}$, and the $\sigma$-algebra $\mathcal {G}_{k^{ +}}^y$. We have that

$$\begin{aligned}&\mathbb {P}\left( \mathcal {W}_{y,k}(N_{k,a} )\cap \mathcal {B}_{y,k+3,L}\cap \mathcal {H}_{k,a} \cap \mathcal {I}_{y',z}\right) \le \sum _{j' = \left( k^{ +}\right) ^{1/4}}^{L^{3/4}} \sum _{j'' = \left( k^{ ++}\right) ^{1/4}}^{L^{3/4}} \nonumber \\&\qquad \mathbb {P}\left( \mathcal {W}_{y,k}(N_{k,a} )\cap \mathbb {B}_{y,k+3,k^{ +} }^{j' }\cap \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L}\cap \mathcal {H}_{k,a} \cap \widehat{\mathcal {I}}_{y', k, z},\right. \nonumber \\&\qquad \qquad \qquad \left. \,\sqrt{2T_{k^{ ++}}^{y,t_{z}}} \in I_{\rho _{L} (L-k^{ ++})+j''}\right) +o(e^{-a} E( k)), \end{aligned}$$

(1.20)

where the error term is coming from the restriction $j', j''\le L^{3/4}$. Next, set

$$\begin{aligned} \mathcal {A}_{j'}:= \mathcal {W}_{y,k}(N_{k,a} ) \bigcap \mathcal {H}_{k,a} \bigcap \widehat{\mathcal {I}}_{y', k, z}\cap \mathbb {B}_{y,k+3,k^{ +} }^{j' }\in \mathcal {G}_{k^{ +}}^y. \end{aligned}$$

(1.21)

Using (1.21 ) we have

$$\begin{aligned}&\mathbb {P}\left( \mathcal {A}_{j'};\,\sqrt{2T_{k^{ ++}}^{y,t_{z}}} \in I_{\rho _{L} (L-k^{ ++})+j''};\, \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L} \right) \nonumber \\&\quad \le \sup _{s\in I_{\rho (L-k^{ +})+j'},\,\, v\in I_{\rho (L-k^{ ++})+j''}} \mathbb {P}\left( \mathcal {A}_{j'};\,\sqrt{2T_{k^{ ++}}^{y,t_{z}}} \in I_{\rho (L-k^{ ++})+j''};\, \right. \nonumber \\&\qquad \qquad \left. \mathbb {P}\left( \widehat{\mathfrak { B}}_{y,k^{+ +}+1,L}\, | \, T_{y,k^{+ +}-1\rightarrow k^{+ +}}^{y,k^{+}\overset{s^{2}/2}{\rightarrow }k^{+}-1} =v^{2}/2, \, \mathcal {G}_{k^{+}}^y\right) \right) . \end{aligned}$$

(1.22)

Recall the estimates (1.16) and (1.17). We then have that

uniformly in $[k^{1/4}]\le a \le L^{3/4}$,

$$\begin{aligned}&\mathbb {P}\left( \mathcal {W}_{y,k}(N_{k,a} )\cap \mathcal {B}_{y,k+3,L}\cap \mathcal {H}_{k,a} \cap \mathcal {I}_{y',z}\right) \nonumber \\&\quad \le c e^{ -2(L-k^{+ +}) } \sum _{j' = \left( k^{ +}\right) ^{1/4}}^{L^{3/4}} \sum _{j'' = \left( k^{ ++}\right) ^{1/4}}^{L^{3/4}}( 1+ j'' )e^{ -2 j'' } \nonumber \\&\qquad \times \mathbb {P}\left( \mathcal {W}_{y,k}(N_{k,a} )\cap \mathbb {B}_{y,k+3,k^{ +}, k^{ ++} }^{j', j'' }\cap \mathcal {H}_{k,a} \cap \widehat{\mathcal {I}}_{y', k, z}\right) , \end{aligned}$$

(1.23)

where $\mathbb {B}_{y,k+3,k^{ +}, k^{ ++} }^{j', j'' }$ is as in (1.19).

The next lemma handles the last term in (1.23) and replaces Lemma 4.7 in [1]. The proof, which is given in detail in [2, Lemma 4.9], is very similar to the proof of Lemma 4.7; it uses the decoupling lemma (Lemma 4.11), barrier estimates and the control on Wasserstein distance contained in (4.17).

Lemma 1.2

For some $M_{0}<\infty$ and $k,a,j'$ in the ranges specified above,

$$\begin{aligned}&\mathbb {P}( \mathcal {W}_{y,k}(N_{k,a} )\bigcap \mathbb {B}_{y,k+3,k^{ +}, k^{ ++} }^{j', j'' } \bigcap \mathcal {H}_{k,a} \bigcap {\widehat{\mathcal {I}}_{y', k, z}} ) \nonumber \\&\quad \le F_{k,a,j', j''}\,\,\mathbb {P}({\widehat{\mathcal {I}}_{y', k, z}})+ e^{-2L-\sqrt{L}}, \end{aligned}$$

(1.24)

where

$$\begin{aligned} F_{k,a,j',j''}= & {} c ae^{-2a}e^{-2 (k^{ ++}-k ) } \sum _{{\mathop {\{\bar{j}'': |\bar{j}''-j''|\le 2M_{0}\log k\}}\limits ^{\{\bar{j}': |\bar{j}'-j'|\le 2M_{0}\log k\}}}}\nonumber \\&\bar{j}'\,\frac{e^{ -(\bar{j}'-a)^{ 2}/(4( k^{ +}-k ) )}}{(k^{ +}-k ) } e^{2 \bar{j}'' } \, \frac{e^{-(\bar{j}'-\bar{j}'')^{2}/2(k^{ +}-k )}}{(k^{ +}-k ) }. \end{aligned}$$

(1.25)

Assuming Lemma 1.2, the rest of the proof is similar to [1, Section 4.4]. Full details appear in [2, Section 4.4].