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
and edge set
Notice that, almost-surely, \(\mathcal {G}\) is a connected and infinite graph with \(0 \in V (\mathcal {G}) \subsetneq {\mathbb {Z}}^{4}\). We write
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
and its heat kernel (transition density) with respect to \(\mu _{{{\mathcal {G}}} }\) by
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
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 \(\mathcal {G}\),
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
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
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 \(r > 0\),
for some subset \({{\mathcal {D}}} ={{\mathcal {D}}} (r)\subseteq V ({{\mathcal {G}}}) \) with no more than M elements.
Proof
For \(\varepsilon > 0\), we let
where to be consistent with [6], \(\psi (n)\) is defined as at (1), and \(b_{n, r}\) is given by
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
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
Also, for each \(1 \le i \le N\), the event \(G_{i}\) is defined by
where we define \(\mathcal {G}_{m,n}\) to be the graph with vertex set
and edge set
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
for all \(1 \le i \le N\). Therefore, letting \(G = \cap _{i=1}^{N} G_{i}\), it holds that
where we have applied the fact that \(N \le 5 \varepsilon ^{-1} ( \log \log n)^{7}\).
Now, suppose that
Then we have that
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
for all \(l=0,1, \cdots , t_{i+1, n} - t_{i, n}\). Thus the event F guarantees that
This implies that
which contradicts (3). So, we have
With this in mind, we suppose that the event \(G \cap F^{c}\) occurs. We set
and also write
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
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
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
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
Moreover, on the event \(F^{c} \cap G \cap H\), we have
Reparameterising this, it holds that with probability \(1 - C ( \log n)^{-1} (\log \log n)^{-1-a}\), we have that
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
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
Since \(\varepsilon >0\) is a constant, we finish the proof. \(\square \)
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Acknowledgements
DC was supported by JSPS Grant-in-Aid for Scientific Research (A), 17H01093, JSPS Grant-in-Aid for Scientific Research (C), 19K03540, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. DS was supported by a JSPS Grant-in-Aid for Early-Career Scientists, 18K13425, JSPS Grant-in-Aid for Scientific Research (B), 17H02849, and JSPS Grant-in-Aid for Scientific Research (B), 18H01123.
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Croydon, D.A., Shiraishi, D. Correction to: Exact value of the resistance exponent for four dimensional random walk trace. Probab. Theory Relat. Fields 185, 699–704 (2023). https://doi.org/10.1007/s00440-022-01160-x
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DOI: https://doi.org/10.1007/s00440-022-01160-x
Keywords
- Random walk
- Range of random walk
- Random environment
Mathematics Subject Classification
- 60K37 (primary)
- 05C81
- 60K35
- 82B41