1 Correction to: Probab. Theory Relat. Fields (2012) 153:191–232 https://doi.org/10.1007/s00440-011-0343-x

2 Introduction

In this note, we fill a gap in the proof of [6, Theorem 1.2.2], which gives quenched heat kernel estimates for the random walk on the trace of a four-dimensional random walk.

To explain the problem with [6] precisely, let us start by recalling the framework of the article in question. Let \(S=(S_n)_{n\ge 0}\) be the simple random walk on \({\mathbb {Z}}^4\) started at the origin, built on an underlying probability space \((\Omega , {{\mathcal {F}}}, \mathbf {P})\). We regard \(S_{[0, \infty )}= \{ S_{n} : n \ge 0 \}\) as a random subgraph of \({\mathbb {Z}}^{4}\). Namely, we let \(\mathcal {G} =(V(\mathcal {G}),B(\mathcal {G}))\) be the graph with vertex set

$$\begin{aligned}V(\mathcal {G}):=\left\{ S_n:\,n\ge 0\right\} ,\end{aligned}$$

and edge set

$$\begin{aligned}B (\mathcal {G}):=\left\{ \{S_n,S_{n+1}\}:\,n\ge 0\right\} .\end{aligned}$$

Notice that, almost-surely, \(\mathcal {G}\) is a connected and infinite graph with \(0 \in V (\mathcal {G}) \subsetneq {\mathbb {Z}}^{4}\). We write

$$\begin{aligned} \mu _{{{\mathcal {G}}}} (x) = \big | \big \{ \{ x, y \} \in B ( {{\mathcal {G}}} ) \big \} \big | \end{aligned}$$

for the number of edges that contain x, where |A| stands for the cardinality of A. Given a realisation of \(\mathcal {G} \), we denote the simple random walk on \(\mathcal {G} \) by

$$\begin{aligned}X=\left( (X_n)_{n\ge 0}, \mathbf {P}_x^{\mathcal {G} }, x\in V \big ( \mathcal {G} \big ) \right) ,\end{aligned}$$

and its heat kernel (transition density) with respect to \(\mu _{{{\mathcal {G}}} }\) by

$$\begin{aligned} p_{n}^{{{\mathcal {G}}} } (x, y) = \frac{\mathbf {P}_x^{\mathcal {G}} (X_{n} = y)}{ \mu _{{{\mathcal {G}}} } (y)}. \end{aligned}$$

Furthermore, let \(R_{{{\mathcal {G}}}}\) be the effective resistance on \({{\mathcal {G}}}\) when \({{\mathcal {G}}}\) is regarded as an electrical circuit with unit resistors on each edge (see [1, 2, 5] for background on effective resistance). Finally, we set

$$\begin{aligned} \psi (n) = \frac{ \mathbf {E} \, \big ( R_{{{\mathcal {G}}}} (0, S_{n} ) \big )}{n}. \end{aligned}$$
(1)

In [6, Theorem 1.2.2], it was stated that there exist constants \(c_1,c_2\in (0,\infty )\) such that, for \(\mathbf {P}\)-a.e. realisation of \(\mathcal {G}\),

$$\begin{aligned} c_1n^{-1/2} \psi (n)^{1/2}\le p_{2 n}^{\mathcal {G} }\left( 0,0\right) \le c_2n^{-1/2} \psi (n)^{1/2} \end{aligned}$$
(2)

for all large n. However, the proof in [6] incorporated a misapplication of [4, Proposition 3.2], in that it did not include a verification of the resistance lower bound of [4, Eq. (3.3)]. In the next section, we explain how the estimates of [6] are nonetheless enough for deducing that (2) holds. Throughout the paper, we use C to denote a universal positive constant, which may change from line to line.

3 Verification of (2)

The aim of this section is to complete the proof of the almost-sure heat kernel bound of (2). As discussed at the end of the previous section, to do this it is sufficient to check that the resistance lower bound of [4, Eq. (3.3)] holds almost-surely in the present setting. Precisely, let

$$\begin{aligned} B_{{{\mathcal {G}}}} (x, r) = \{ y \in V ( {{\mathcal {G}}} ) \ : \ R_{{{\mathcal {G}}} } (x, y ) < r \} \end{aligned}$$

be the open ball of radius r in \({{\mathcal {G}}}\) centered at x with respect to the effective resistance metric \(R_{{{\mathcal {G}}}} \). Then the required estimate is readily implied by the following proposition.

Proposition 2.1

For \(\mathbf {P}\)-a.e. realisation of \(\mathcal {G}\), it holds that

$$\begin{aligned} \liminf _{r\rightarrow \infty }\frac{R_\mathcal {G}\left( 0,B_\mathcal {G}(0,r)^c\right) }{r}>0. \end{aligned}$$

Proof

Arguing as in the proof of [3, Lemma 4.1], the result is a straightforward consequence of Lemma 2.2 below. (Note that the proof in [3] used a uniform volume doubling condition, but only to give a bound on the number of small balls needed to cover a larger one; for us, this is given by Lemma 2.2.) \(\square \)

Lemma 2.2

For \(\mathbf {P}\)-a.e. realization of \({{\mathcal {G}}}\), there exists a constant \(M \in {\mathbb {N}}\) such that, for all \(r > 0\),

$$\begin{aligned}B_{{{\mathcal {G}}}} (0, r) \subseteq \bigcup _{x \in {{\mathcal {D}}}} B_{{{\mathcal {G}}}} (x, \tfrac{2}{3}r ),\end{aligned}$$

for some subset \({{\mathcal {D}}} ={{\mathcal {D}}} (r)\subseteq V ({{\mathcal {G}}}) \) with no more than M elements.

Proof

For \(\varepsilon > 0\), we let

$$\begin{aligned}{\tilde{Y}}_{k} = \mathbf{1}_{\{ R_{{{\mathcal {G}}}} (0, S_{k} ) \le 8 \varepsilon n \psi (n) \}},\qquad {\tilde{Y}} = \sum _{k=0}^{2 b_{n, 7}} {\tilde{Y}}_{k},\end{aligned}$$

where to be consistent with [6], \(\psi (n)\) is defined as at (1), and \(b_{n, r}\) is given by

$$\begin{aligned}b_{n, r} = \lfloor n ( \log \log n)^{r} \rfloor .\end{aligned}$$

It is then proved in [6, Proposition 3.1.1] that there exist \(\varepsilon \in (0,1)\), \(C_{0} \in (0, \infty )\) and \(\alpha > 0\) such that

$$\begin{aligned} \mathbf {P} \left( {\tilde{Y}} \ge C_{0} n \right) \le C_{0} (\log n)^{-\frac{3}{2}} (\log \log n)^{\alpha }. \end{aligned}$$
(3)

Since \({\tilde{Y}}_{k}\) is monotonically increasing in \(\varepsilon \), we may assume that \(0 < \varepsilon \le \frac{1}{32}\).

Next, for \(i \ge 1\), we set \(t_{i, n} = {(i-1) \varepsilon n}/{2} \) and \(I_{i, n} = [t_{i, n}, t_{i+1, n}] \). We let N be the integer satisfying \(t_{N, n} \le 2 b_{n, 7} < t_{N+1, n}\); note that \(N \le 5 \varepsilon ^{-1} ( \log \log n)^{7}\). Using the constant \(C_{0} > 0\) as in (3), we define the event F to be

$$\begin{aligned}F = \left\{ \left| \left\{ i\in \{1,\dots ,N \}\ : \ R_{{{\mathcal {G}}}} \left( 0, S_{t_{i, n}} \right) \le 4 \varepsilon n \psi (n) \right\} \right| \ge \frac{4 C_{0}}{\varepsilon } \right\} .\end{aligned}$$

Also, for each \(1 \le i \le N\), the event \(G_{i}\) is defined by

$$\begin{aligned}G_{i} = \left\{ R_{{{\mathcal {G}}}_{t_{i,n}, t_{i+1, n}}} \left( S_{t_{i, n}}, S_{t_{i, n} + l} \right) \le \tfrac{2}{3} \varepsilon n \psi (n) \text { for all } l=0,1, \dots , t_{i+1, n} - t_{i, n} \right\} ,\end{aligned}$$

where we define \(\mathcal {G}_{m,n}\) to be the graph with vertex set

$$\begin{aligned}V(\mathcal {G}_{m,n}):=\left\{ S_k:\, m \le k \le n\right\} ,\end{aligned}$$

and edge set

$$\begin{aligned}B (\mathcal {G}_{m,n}):=\left\{ \{S_k,S_{k+1}\}:\, m \le k \le n-1 \right\} ,\end{aligned}$$

and \(R_{\mathcal {G}_{m,n}}\) to be the associated effective resistance. Combining [6, Lemma 2.1.1] and [6, Remark 2.2.2], we have that

$$\begin{aligned}\mathbf {P} (G_{i} ) \ge 1- C (\log n)^{-\frac{3}{2}}\end{aligned}$$

for all \(1 \le i \le N\). Therefore, letting \(G = \cap _{i=1}^{N} G_{i}\), it holds that

$$\begin{aligned}\mathbf {P} (G) \ge 1 - C \varepsilon ^{-1} (\log n)^{-\frac{3}{2}} ( \log \log n)^{7},\end{aligned}$$

where we have applied the fact that \(N \le 5 \varepsilon ^{-1} ( \log \log n)^{7}\).

Now, suppose that

$$\begin{aligned}\mathbf {P} (F) \ge 2 (C + C_{0}) \varepsilon ^{-1} (\log n)^{-\frac{4}{3}} ( \log \log n)^{\alpha + 7}.\end{aligned}$$

Then we have that

$$\begin{aligned} \mathbf {P} (F \cap G )&= \mathbf {P} (F) - \mathbf {P} (F \cap G^{c} ) \\&\ge 2 (C + C_{0}) \varepsilon ^{-1} (\log n)^{-\frac{4}{3}} ( \log \log n)^{\alpha + 7} - C \varepsilon ^{-1} (\log n)^{-\frac{3}{2}} ( \log \log n)^{7} \\&\ge (C + C_{0}) \varepsilon ^{-1} (\log n)^{-\frac{4}{3}} ( \log \log n)^{\alpha + 7}\\&> C_{0} (\log n)^{-\frac{3}{2}} (\log \log n)^{\alpha }. \end{aligned}$$

On the other hand, we assume that the event \(F \cap G\) occurs. Note that if \(R_{{{\mathcal {G}}}} ( 0, S_{t_{i, n}} ) \le 4 \varepsilon n \psi (n)\), the event \(G_{i}\) ensures that

$$\begin{aligned}R_{{{\mathcal {G}}}} \left( 0, S_{t_{i, n} + l} \right) \le R_{{{\mathcal {G}}}} \left( 0, S_{t_{i, n}} \right) + R_{{{\mathcal {G}}}_{t_{i,n}, t_{i+1, n}}} \left( S_{t_{i, n}}, S_{t_{i, n} + l} \right) \le \tfrac{14}{3} \varepsilon n \psi (n)\end{aligned}$$

for all \(l=0,1, \cdots , t_{i+1, n} - t_{i, n}\). Thus the event F guarantees that

$$\begin{aligned}{\tilde{Y}} \ge \frac{\varepsilon n}{2} \times \frac{4 C_{0}}{\varepsilon } = 2 C_{0} n.\end{aligned}$$

This implies that

$$\begin{aligned} \mathbf {P} \left( {\tilde{Y}} \ge 2 C_{0} n \right) \ge \mathbf {P} (F \cap G ) > C_{0} (\log n)^{-\frac{3}{2}} (\log \log n)^{\alpha },\end{aligned}$$

which contradicts (3). So, we have

$$\begin{aligned}\mathbf {P} (F) \le 2 (C + C_{0}) \varepsilon ^{-1} (\log n)^{-\frac{4}{3}} ( \log \log n)^{\alpha + 7}.\end{aligned}$$

With this in mind, we suppose that the event \(G \cap F^{c}\) occurs. We set

$$\begin{aligned}{{\mathcal {I}}} = \left\{ i\in \{1,\dots ,N\} \ : \ R_{{{\mathcal {G}}}} \left( 0, S_{t_{i, n}} \right) \le 4 \varepsilon n \psi (n) \right\} ,\end{aligned}$$

and also write

$$\begin{aligned}{{\mathcal {J}}} = \left\{ i\in \{1,\dots ,N\} \ : \ \exists k \in [t_{i, n}, t_{i+1, n} ] \text { such that } R_{{{\mathcal {G}}}} \left( 0, S_{k} \right) \le 2 \varepsilon n \psi (n) \right\} .\end{aligned}$$

Take \(i \in {{\mathcal {J}}}\) and we have that \(R_{{{\mathcal {G}}}} ( 0, S_{k} ) \le 2 \varepsilon n \psi (n)\) for some \(k \in [t_{i, n}, t_{i+1, n} ] \). Then the event G ensures that

$$\begin{aligned}R_{{{\mathcal {G}}}} \left( 0, S_{t_{i, n}} \right) \le R_{{{\mathcal {G}}}} \left( 0, S_{k} \right) + R_{{{\mathcal {G}}}} \left( S_{k}, S_{t_{i, n}} \right) \le \tfrac{8}{3} \varepsilon n \psi (n),\end{aligned}$$

which shows \(i \in {{\mathcal {I}}}\) and \({{\mathcal {J}}} \subseteq {{\mathcal {I}}}\). So, the event \(F^{c}\) ensures that \(|{{\mathcal {J}}}| \le \frac{4 C_{0}}{\varepsilon }\). Consequently, on the event \(G \cap F^{c}\), we have that

$$\begin{aligned}B_{{{\mathcal {G}}}} \left( 0, 2 \varepsilon n \psi (n) \right) \cap S_{[0, 2b_{n, 7} ]}\subseteq \bigcup _{i \in {{\mathcal {J}}}} B_{{{\mathcal {G}}}} \left( S_{t_{i, n}}, \tfrac{2}{3} \varepsilon n \psi (n) \right) ,\end{aligned}$$

where we let \(S_{[p, q]} = \{ S_{k} \ : \ p \le k \le q \}\) for \(0 \le p \le q < \infty \).

Finally, we define the event H to be

$$\begin{aligned}H = \left\{ R_{{{\mathcal {G}}}} \left( 0, S_{k} \right) > 2 \varepsilon n \psi (n) \text { for all } k \ge 2 b_{n, 7} \right\} .\end{aligned}$$

It then follows from [6, Proposition 2.3.4] (and [6, Remark 2.1.3]) that there exist \(a, C \in (0, \infty )\) such that

$$\begin{aligned}\mathbf {P} ( H) \ge 1 - C ( \log n)^{-1} (\log \log n)^{-1-a}.\end{aligned}$$

Moreover, on the event \(F^{c} \cap G \cap H\), we have

$$\begin{aligned}B_{{{\mathcal {G}}}} \left( 0, 2 \varepsilon n \psi (n) \right) \subseteq \bigcup _{i \in {{\mathcal {J}}}} B_{{{\mathcal {G}}}} \left( S_{t_{i, n}}, \tfrac{2}{3} \varepsilon n \psi (n) \right) ,\qquad |{{\mathcal {J}}}| \le \frac{4 C_{0}}{\varepsilon }.\end{aligned}$$

Reparameterising this, it holds that with probability \(1 - C ( \log n)^{-1} (\log \log n)^{-1-a}\), we have that

$$\begin{aligned}B_{{{\mathcal {G}}}} \left( 0, 2 \varepsilon n \right) \subseteq \bigcup _{x \in {{\mathcal {D}}}} B_{{{\mathcal {G}}}} \left( x, \tfrac{2}{3} \varepsilon n \right) ,\end{aligned}$$

for some subset \({{\mathcal {D}}} \subseteq V ( {{\mathcal {G}}} )\) with \(|{{\mathcal {D}}}| \le \frac{4 C_{0}}{\varepsilon }\). Hence, applying the Borel-Cantelli lemma, we find that, for \(\mathbf {P}\)-a.e. realization of \({{\mathcal {G}}}\), it holds that, for large k

$$\begin{aligned}B_{{{\mathcal {G}}}} \left( 0, 2 \varepsilon 2^{k} \right) \subseteq \bigcup _{x \in {{\mathcal {D}}}'} B_{{{\mathcal {G}}}} \left( x, \tfrac{2}{3} \varepsilon 2^{k} \right) ,\end{aligned}$$

for some subset \({{\mathcal {D}}}' \subseteq V ( {{\mathcal {G}}} )\) with \(| \mathcal{D}'| \le \frac{4 C_{0}}{\varepsilon }\). Using this, for large r with \(2^{k} \le r <2^{k+1}\), we obtain that

$$\begin{aligned}B_{{{\mathcal {G}}}} (0, \varepsilon r) \subseteq B_{{{\mathcal {G}}}} \left( 0, 2 \varepsilon 2^{k} \right) \subseteq \bigcup _{x \in {{\mathcal {D}}}'} B_{{{\mathcal {G}}}} \left( x, \tfrac{2}{3} \varepsilon 2^{k} \right) \subseteq \bigcup _{x \in {{\mathcal {D}}}'} B_{{{\mathcal {G}}}} \left( x, \frac{2}{3} \varepsilon r \right) .\end{aligned}$$

Since \(\varepsilon >0\) is a constant, we finish the proof. \(\square \)