Density of states for the Anderson model on nested fractals

Abstract

We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians $H^\omega =H_0+V^\omega$ on fractal spaces of infinite diameter. The kinetic term $H_0$ is given by $\phi (-{\mathcal {L}}),$ where ${\mathcal {L}}$ is the Laplacian on the fractal and $\phi$ is a completely monotone function satisfying some mild regularity conditions. The random potential $V^\omega$ is of alloy-type.

Appendices

Appendix A: Technical lemmas on subordinate processes

We need the following technical results.

Lemma A.1

Let ${\mathcal {K}}^{\langle \infty \rangle }$ be an USNF with the GLP and let the assumption (B) hold. For every $t>0$ and $a>1$ we have

$$\begin{aligned} \sum _{M=1}^{\infty } \sup _{x \in {\mathcal {K}}^{\langle \infty \rangle }} {\textbf{P}}^x \left[ \sup _{s\le t} |X_s - x| > a^M \right] <\infty . \end{aligned}$$

Proof

The proof follows the lines of that of [28, Lemma 2.3]. The only difference is that now we use the Euclidean distance instead of the geodesic one, and apply the sub-Gaussian estimates from [19, Lemma 5.6, Remark 3.7]. At the end, we also use the summation property (2.20) as before. $\square$

Let us recall that $\Delta _{M,i}$ for $1 \le i \le N$ denote the M-complexes in ${\mathcal {K}}^{\langle M+1 \rangle }$, see Definition 2.1 (9).

Lemma A.2

Let ${\mathcal {K}}^{\langle \infty \rangle }$ be an USNF with the GLP and let the assumption (B) hold. We have the following.

• (a) For

  $$\begin{aligned} C(M,t):= \sup _{x,y \in {\mathcal {K}}^{\langle M \rangle }} \sum _{\begin{array}{c} y^{\prime } \in \pi _{M}^{-1}(y) \\ y' \notin {\mathcal {K}}^{\left\langle M+1 \right\rangle } \end{array}} p(t,x,y'), \quad t>0, \quad M \in {\mathbb {Z}} \end{aligned}$$

  (A.1)

  we have

  $$\begin{aligned} \sum _{M=1}^{\infty } C(M,t) < \infty , \quad t>0; \end{aligned}$$

  in particular, for every $t>0$, $C(M,t) \rightarrow 0$ as $M \rightarrow \infty$.

• (b) For every $t>0$,

  $$\begin{aligned} \sum _{M=1}^{\infty } \frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} \left| p(t,x,x) - p_M(t,x,x) \right| \mu (\textrm{d}x) < \infty ; \end{aligned}$$

  (A.2)

  in particular

  $$\begin{aligned} \frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} \left| p(t,x,x) - p_M(t,x,x) \right| \mu (\textrm{d}x) \rightarrow 0, \quad as \ M \rightarrow \infty .\nonumber \\ \end{aligned}$$

  (A.3)

Proof

By Tonelli’s theorem and [49, formula (3.3) and Lemma 3.5], we get

$$\begin{aligned} C(M,t)&= \sup _{x,y \in {\mathcal {K}}^{\langle M \rangle }} \int _0^{\infty } \sum _{\begin{array}{c} y^{\prime } \in \pi _{M}^{-1}(y) \\ y' \notin {\mathcal {K}}^{\left\langle M+1 \right\rangle } \end{array}} g(t,x,y') \eta _t(\textrm{d}u) \\&\le c_1 \int _0^{\infty } L^{-d M} \left( \frac{L^M}{t^{1/d_w}} \vee 1\right) ^{d - \frac{d_w}{d_J-1}} \exp \left( -c_2 \left( \frac{L^M}{t^{1/d_w}} \vee 1\right) ^{\frac{d_w}{d_J-1}}\right) \eta _t(\textrm{d}u) \\&\le c_1 \int _0^{L^{d_{w}M}} L^{-d M} \left( \frac{L^M}{u^{1/d_w}}\right) ^{d - \frac{d_w}{d_J-1}} \exp {\left( -c_2 \left( \frac{L^M}{u^{1/d_w}}\right) ^{\frac{d_w}{d_J-1}}\right) } \eta _t(\textrm{d}u) \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ + c_3 L^{-d M} \eta _{t} \left( L^{d_{w}M}, \infty \right) . \end{aligned}$$

Further steps of the proof follow exactly the reasoning in the proof of [28, Lemma 2.5 (b)].

The proof of (b) is similar to that of [28, Lemma 2.7], i.e. we start with

$$\begin{aligned}&\frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} \left| p(t,x,x) - p_M(t,x,x) \right| \mu (\textrm{d}x) \\&\qquad \le \frac{N}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} \sup _{\begin{array}{c} x^{\prime } \in \pi _{M}^{-1}(x) \\ x' \in {\mathcal {K}}^{\langle M+1 \rangle } \backslash {\mathcal {K}}^{\langle M \rangle } \end{array}} p(t,x,x') \mu (\textrm{d}x)\\&\qquad \quad + \frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} \sum _{\begin{array}{c} x^{\prime } \in \pi _{M}^{-1}(x) \\ x' \notin {\mathcal {K}}^{\langle M+1 \rangle } \end{array}} p(t,x,x') \mu (\textrm{d}x)\\&\qquad =: {\mathcal {A}}_M(t) + {\mathcal {B}}_M(t) \end{aligned}$$

and first observe that ${\mathcal {B}}_M(t)$ is a term of convergent series by part (a). To prove that the same is true for ${\mathcal {A}}_M(t)$, we only need to modify the estimate in the proof of the quoted lemma. The difference is that the domain of the integration in ${\mathcal {A}}_M(t)$ has to be divided into two different sets $E_M^1$ and $E_M^2$. In the present general case, we cannot use geodesic balls. We will apply the graph distance here.

Let us denote $V_M^{\langle M \rangle } = \{v_1,\ldots , v_k\}$ and

$$\begin{aligned} E_M^1 = \bigcap _{i=1}^{k} \left\{ y \in {\mathcal {K}}^{\langle M \rangle }: d_{\lfloor M/2\rfloor } (y, v_i) > 1\right\} , \quad E_M^2 = {\mathcal {K}}^{\langle M \rangle } \backslash E_M^1. \end{aligned}$$

(A.4)

Recall that $\sup _{(x,y) \in {\mathcal {K}}^{\langle \infty \rangle } \times {\mathcal {K}}^{\langle \infty \rangle }} p(t,x,y) < \infty$ by Lemma 2.1 (a). Since $E_M^2$ consists of k $\lfloor M/2 \rfloor$-complexes (each one attached to one of the vertices from $V_M^{\langle M \rangle }$), we have that

$$\begin{aligned} \frac{\mu \left( E_M^2\right) }{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } = \frac{k \cdot N^{\lfloor M/2 \rfloor }}{N^M}. \end{aligned}$$

(A.5)

In consequence, the part of ${\mathcal {A}}_M(t)$ including the integral over $E_M^2$ is a term of a convergent series.

When $x \in E_M^1$ and $x^{\prime } \in \pi _{M}^{-1}(x) \backslash {\mathcal {K}}^{\langle M \rangle }$, then $d_{\lfloor M/2 \rfloor } (x, x') >2$, so from [27, Lemma A.2] we have $|x-x'|> c_4 L^{M/2}$. Using the subordination formula for the density p(t, x, y) and the subgaussian estimates from [39] for the density g(u, x, y), we then get

$$\begin{aligned} \frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{E_M^1} \sup _{\begin{array}{c} x^{\prime } \in \pi _{M}^{-1}(x) \\ x' \in {\mathcal {K}}^{\langle M+1 \rangle } \backslash {\mathcal {K}}^{\langle M \rangle } \end{array}} p(t,x,x') \mu (\textrm{d}x) \le c_5 \int _0^{\infty } u^{-d_s / 2} \textrm{e}^{-c_6 \left( \frac{L^{M/2}}{u^{1/d_w}}\right) ^{\frac{d_w}{d_J-1}}} \eta _t(\textrm{d}u). \end{aligned}$$

Collecting these estimates, we obtain

$$\begin{aligned} {\mathcal {A}}_M(t) \le c_7 \left( N^{-M/2}+ \int _0^{\infty } u^{-d_s / 2} \textrm{e}^{-c_6 \left( \frac{L^{M/2}}{u^{1/d_w}}\right) ^{\frac{d_w}{d_J-1}}} \eta _t(\textrm{d}u)\right) , \end{aligned}$$

with the constants $c_6, c_7$ independent of M, and from this point the proof can be continued in the same manner as that of [28, Lemma 2.7]. $\square$

Appendix B: Proofs of the statements from Sect. 3

Proof of Proposition 3.1

Fix $t>0$. For $\mu$-almost all $x \in {\mathcal {K}}^{\langle M+1\rangle },$ by Lemmas 2.2(a),  3.1, and the inclusion $\pi _{M+1}^{-1}(x) \subset \pi _M^{-1}\left( \pi _M(x)\right)$, we may write

$$\begin{aligned}&p_{M+1}(t,x,x) {\textbf{E}}_{M+1,t}^{x,x} [{\mathbb {E}}_{{\mathbb {Q}}} \textrm{e}^{-\int _0^t V^{\omega }_{M+1}(X^{M+1}_s)\textrm{d}s} ]\\&\quad = \sum _{x' \in \pi _{M+1}^{-1} (x)} p(t,x,x') {\textbf{E}}_{t}^{x,x'} \left[ {\mathbb {E}}_{{\mathbb {Q}}} \textrm{e}^{-\int _0^t V^{\omega }_{M+1}(\pi _{M+1}(X_s))\textrm{d}s} \right] \\&\quad \le \sum _{x' \in \pi _{M}^{-1} \left( \pi _M(x)\right) } p(t,x,x') {\textbf{E}}_{t}^{x,x'} \left[ {\mathbb {E}}_{{\mathbb {Q}}} \textrm{e}^{-\int _0^t V^{\omega }_{M+1}(\pi _{M+1}(X_s))\textrm{d}s} \right] \\&\quad \le \sum _{x' \in \pi _{M}^{-1} \left( \pi _M(x)\right) } p(t,x,x') {\textbf{E}}_{t}^{x,x'} \left[ {\mathbb {E}}_{{\mathbb {Q}}} \textrm{e}^{-\int _0^t V^{\omega }_{M}(\pi _{M}(X_s))\textrm{d}s} \right] . \end{aligned}$$

Using this estimate, the definition of $\Lambda _{M^*}^{N}(t,\omega )$ and Lemma 2.2(b), we may now follow the argument in the second part of the proof of [28, Theorem 3.1], getting that

$$\begin{aligned} {\mathbb {E}}_{{\mathbb {Q}}}[\Lambda _{M+1}^{N, V_{M+1}^\omega }(t)] \le {\mathbb {E}}_{{\mathbb {Q}}}[ \Lambda _{M}^{N, V_M^\omega }(t)], \quad M \in \mathbb {Z_+}. \end{aligned}$$

Since ${\mathbb {E}}_{{\mathbb {Q}}}\Lambda _{M}^{N, V_M^\omega }(t) \ge 0$, it converges to a finite limit $\Lambda (t)$ as $M \rightarrow \infty$. This completes the proof. $\square$

Proofs of Lemmas 3.2 and 3.3 given below are of technical nature. They follow the steps and ideas from the proofs of Proposition 3.1 and Lemmas 3.1$-$3.2 in [28]. The main difference here is that we now work with different type of random potentials and the state space is now a general USNF (let us emphasize that the Sierpiński triangle which was studied in the quoted paper is one of the simplest planar nested fractals with high regularity). This causes some extra geometric issues, which are solved by using the graph distance (the geodesic metric may not be defined at all) and the comparison principle from [27, Lemma A.2]. Therefore, we will follow only the main steps of the proofs which are affected by these changes, focusing on the most critical differences.

Proof of Lemma 3.2

We first show (a). By following the argument at the beginning of [28, Proposition 3.1] we obtain that for each fixed $t>0$ there exists a constant $c=c(t)$ such that

$$\begin{aligned} {\mathbb {E}}_{{\mathbb {Q}}} \left( \Lambda _{M}^{D, V^\omega }(t) - \Lambda _{M}^{N, V^\omega } (t) \right) ^2 \le c \left( R_{1,M}(t) + R_{2,M}(t)\right) , \quad M \in {\mathbb {Z}}_{+}, \end{aligned}$$

where

$$\begin{aligned} R_{1,M}(t)&= \frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} p(t,x,x) {\textbf{E}}_{t}^{x,x} \left[ t\ge \tau _{{\mathcal {K}}^{\langle M\rangle }} \right] \mu (\textrm{d}x)\\ R_{2,M}(t)&= \frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} \sum _{x' \in \pi _M^{-1}(x), x' \ne x} p(t,x,x') \mu (\textrm{d}x) \\&= \frac{1}{\mu \left( {\mathcal {K}}^{\langle M \rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} (p_M(t,x,x) - p(t,x,x)) \mu (\textrm{d}x). \end{aligned}$$

Note that the above bound does not depend on $\omega$. By Lemma A.2 (b), $R_{2,M}(t)$ is the term of a convergent series, so we only need to estimate $R_{1,M}(t)$.

Denote the vertices from ${\mathcal {V}}_M^{\langle M \rangle }$ as $v_i$, $1\le i\le k$, and let $\Delta _{\left\lfloor M/2 \right\rfloor ,v_i} \subset {\mathcal {K}}^{\langle M \rangle }$, be the $\left\lfloor M/2 \right\rfloor$-complex attached to $v_i$. If the process starts from $x \in {\mathcal {D}}_M:={\mathcal {K}}^{\langle M \rangle } \backslash \bigcup _{i=1}^{k} \Delta _{\left\lfloor M/2 \right\rfloor ,v_i}$, then, using [27, Lemma A.2], we have

$$\begin{aligned} \left\{ t\ge \tau _{{\mathcal {K}}^{\langle M \rangle }}\right\} \subseteq \left\{ \sup _{0<s\le t} d_{\left\lfloor M /2 \right\rfloor }(x,X_s)>2\right\} \subseteq \left\{ \sup _{0<s\le t} |x-X_s| > c_{1} L^{M/2}\right\} , \end{aligned}$$

with a constant $c_1$, independent of M. We also have

$$\begin{aligned}{} & {} \left\{ \sup _{0<s\le t} |x-X_s|> c_{1} L^{M/2} \right\} \subseteq \left\{ \sup _{0<s\le t/2} |x-X_s|> c_{1} L^{M/2} \right\} \\{} & {} \qquad \cup \left\{ \sup _{t/2<s\le t} |x-X_s| > c_{1} L^{M/2} \right\} . \end{aligned}$$

Recall that $\mu \left( {\mathcal {K}}^{\langle M \rangle } \backslash {\mathcal {D}}_M \right) = k N^{\left\lfloor M/2 \right\rfloor }$. Then, by using the upper bound in Lemma 2.1(a), the formula (2.25) and the symmetry of the bridge measure, we get

$$\begin{aligned} \int _{{\mathcal {K}}^{\langle M \rangle }} p(t,x,x)&{\textbf{P}}^{t}_{x,x} \left[ t\ge \tau _{{\mathcal {K}}^{\langle M\rangle }} \right] \mu (\textrm{d}x)\\&\le c_{2} \mu \left( {\mathcal {K}}^{\langle M \rangle } \backslash {\mathcal {D}}_M \right) + \int _{{\mathcal {D}}_M} p(t,x,x) {\textbf{P}}^{x,x}_{t}\left[ t\ge \tau _{{\mathcal {K}}^{\langle M \rangle }}\right] \mu (\textrm{d}x)\\&\le c_3 \left( N^{M/2} + N^M \sup _{x \in {\mathcal {K}}^{\langle M \rangle }} {\textbf{P}}^{x}\left[ \sup _{0<s\le t/2} |x-X_s|>c_{1} L^{M/2}\right] \right) . \end{aligned}$$

Therefore

$$\begin{aligned} R_{1,M}(t) \le c_3 \left( N^{-M/2} + \sup _{x \in {\mathcal {K}}^{\langle M \rangle }} {\textbf{P}}^{x}\left[ \sup _{0<s\le t/2} |x-X_s|>c_{1} L^{M/2}\right] \right) , \end{aligned}$$

which is, by Lemma A.1, a term of a convergent series. The proof of (a) is completed.

The proof of the first convegence in (b) follows the lines of that of (a) and it is omitted. We only show the second convergence.

First observe that

$$\begin{aligned}{} & {} \left| {\mathbb {E}}_{{\mathbb {Q}}} \left( \Lambda _{M}^{D, V^\omega } (t) - \Lambda _{M}^{D, V_M^\omega } (t)\right) \right| \\{} & {} \quad \le \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {K}}^{\langle M \rangle }} p(t,x,x){\textbf{E}}^{x,x}_{t} \left[ \left| F(t,M)\right| ; t<\tau _{{\mathcal {K}}^{\langle M \rangle }}\right] \mu (\textrm{d}x), \end{aligned}$$

where

$$\begin{aligned} F(t,M)&= {\mathbb {E}}_{{\mathbb {Q}}} \bigg ( \exp \bigg ( -\int _0^t \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle }} \xi _y(\omega ) W\left( X_s, y \right) \ \textrm{d}s \bigg ) \\ {}&- \exp \bigg ( -\int _0^t \sum _{y \in {\mathcal {V}}_{0}^{\langle M\rangle }}\xi _y(\omega ) \sum _{y' \in \pi _{M}^{-1}(y)} W\left( X_s, y' \right) \ \textrm{d}s \bigg ) \bigg ) . \end{aligned}$$

By using exactly the same notation and argument as in the proof of part (a), we may write

$$\begin{aligned} \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) }&\int _{{\mathcal {K}}^{\langle M \rangle }} p(t,x,x) {\textbf{E}}^{x,x}_{t} \left[ \left| F(t,M)\right| ; t<\tau _{{\mathcal {K}}^{\langle M \rangle }}\right] \mu (\textrm{d}x) \\&\le \frac{c_4 \mu \left( {\mathcal {K}}^{\langle M\rangle } \backslash {\mathcal {D}}_M \right) }{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } + \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {D}}_M} p(t,x,x){\textbf{E}}^{x,x}_{t} \left[ \left| F(t,M)\right| ; t<\tau _{{\mathcal {K}}^{\langle M \rangle }}\right] \mu (\textrm{d}x) \\&= c_4 \frac{k N^{\left\lfloor M/2\right\rfloor }}{N^M} + \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {D}}_M} p(t,x,x){\textbf{E}}^{x,x}_{t} \left[ \left| F(t,M)\right| ; t<\tau _{{\mathcal {K}}^{\langle M \rangle }}\right] \mu (\textrm{d}x). \end{aligned}$$

We see that the first term converges to 0 as $M \rightarrow \infty$. It is sufficient to show that the second term goes to zero as well; denote it by I(t, M). We have

$$\begin{aligned} I(t,M)&\le \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {D}}_M} p(t,x,x){\textbf{E}}^{x,x}_{t} \left[ \left| F(t,M)\right| ; t<\tau _{{\mathcal {D}}_M }\right] \mu (\textrm{d}x) \\&\quad + \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {D}}_M} p(t,x,x){\textbf{P}}^{x,x}_{t} \left[ t\ge \tau _{{\mathcal {D}}_M } \right] \mu (\textrm{d}x) \\&=: I_1(t,M) + I_2(t,M). \end{aligned}$$

To estimate $I_1(t,M)$ we will use the inequality $|\textrm{e}^{-x}-\textrm{e}^{-y}| \le |x-y|, x,y>0$, the fact that W and $\xi _y$ are nonnegative.

$$\begin{aligned}&\left| F(t,M)\right| \\&\le {\mathbb {E}}_{{\mathbb {Q}}} \left| \exp \left( -\int _0^t \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle }} \xi _y(\omega ) W\left( X_s, y \right) \ \textrm{d}s \right) \right. \\&\quad \left. - \exp \left( -\int _0^t \sum _{y \in {\mathcal {V}}_{0}^{\langle M\rangle }}\xi _y(\omega ) \sum _{y' \in \pi _{M}^{-1}(y)} W\left( X_s, y' \right) \ \textrm{d}s \right) \right| \\&\le {\mathbb {E}}_{{\mathbb {Q}}} \left| \int _0^t \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle }} \xi _y(\omega ) W\left( X_s, y \right) \ \textrm{d}s - \int _0^t \sum _{y \in {\mathcal {V}}_{0}^{\langle M\rangle }}\xi _y(\omega ) \sum _{y' \in \pi _{M}^{-1}(y)} W\left( X_s, y' \right) \ \textrm{d}s \right| . \end{aligned}$$

Now, observe that for every $x \in {\mathcal {K}}^{\langle \infty \rangle }$ we have

$$\begin{aligned} \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle }} \xi _y(\omega ) W\left( x, y \right) = \sum _{y \in {\mathcal {V}}_{0}^{\langle M \rangle }} \xi _y(\omega ) W\left( x, y \right) + \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle } \setminus {\mathcal {V}}_{0}^{\langle M \rangle }} \xi _y(\omega ) W\left( x, y \right) \end{aligned}$$

and

$$\begin{aligned} \sum _{y \in {\mathcal {V}}_{0}^{\langle M\rangle }} \sum _{y' \in \pi _{M}^{-1}(y)} \xi _{y'}(\omega ) W\left( x, y' \right){} & {} = \sum _{y \in {\mathcal {V}}_{0}^{\langle M\rangle }} \xi _{y}(\omega ) W\left( x, y \right) \\{} & {} \quad + \sum _{y \in {\mathcal {V}}_{0}^{\langle M\rangle }} \sum _{y' \in \pi _{M}^{-1}(y) \backslash \{y\}} \xi _{y'}(\omega ) W\left( x, y' \right) . \end{aligned}$$

By this observation, the Fubini theorem and the fact that all the lattice random variables together with the profile function W are nonnegative, we get that the above expectation can be estimated above by

$$\begin{aligned}&{\mathbb {E}}_{{\mathbb {Q}}}\int _0^t \left( \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle } \setminus {\mathcal {V}}_{0}^{\langle M \rangle }} \xi _y(\omega ) W\left( X_s, y \right) \right. \left. + \sum _{y \in {\mathcal {V}}_{0}^{\langle M\rangle }} \sum _{y' \in \pi _{M}^{-1}(y) \backslash \{y\}} \xi _{y'}(\omega ) W\left( X_s, y' \right) \right) \textrm{d}s \\&\qquad \le 2 \mathbb {E_Q}\xi \int _{0}^{t} \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle } \backslash {\mathcal {V}}_{0}^{\langle M \rangle }} W\left( X_s, y \right) \textrm{d}s. \end{aligned}$$

As $X_s \in {\mathcal {D}}_M$ for $0\le s \le t$ and $y \in {\mathcal {V}}_{0}^{\langle \infty \rangle } \backslash {\mathcal {V}}_{0}^{\langle M \rangle }$, we have $d_{\left\lfloor M/2 \right\rfloor }(X_s, y)>2$ for $0\le s\le t$. This gives

$$\begin{aligned} I_1(t,M)&\le \frac{2\kappa }{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {D}}_M} p(t,x,x) {\textbf{E}}^{x,x}_{t} \left[ \int _0^t \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle } \backslash {\mathcal {V}}_{0}^{\langle M \rangle }} W(X_s(\omega ), y) \textrm{d}s ; t<\tau _{{\mathcal {D}}_M }\right] \mu (\textrm{d}x)\\&\le c_5 \sup _{z \in {\mathcal {K}}^{\langle \infty \rangle }} \sum _{y \in {\mathcal {V}}_0^{\langle \infty \rangle } \backslash B_{\lfloor M/2 \rfloor }(z,1)} W(z,y), \end{aligned}$$

with a constant $c_5>0$ independent of M. By (W3) this is a term of a convergent series.

To show that $I_2(t,M)$ is a term of a convergent series, we just follow the steps from the proof of (a) for an appropriate set ${\mathcal {D}}^{'}_M \subset {\mathcal {D}}_M$. The proof of (b) is finished. $\square$

Proof of Lemma 3.3

The proof of this lemma follows the main steps from the proof of [28, Lemma 3.2], but it substantially differs in some details which are critical for the argument. We will focus on explanation of these differences.

First note that due to Lemma 3.2(a), similarly as in the proof of [28, Lemma 3.2], we only need to establish (3.2). We consider a family of measures $(\nu _M)_{M \in \mathbb {Z_+}}$ given by

$$\begin{aligned} \nu _M:= \left( \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {K}}^{\langle M\rangle }} p(t,x,x) {\textbf{P}}_{t}^{x,x} \mu (dx) \right) ^{\otimes 2} \otimes {\mathbb {Q}}^{\otimes 3}, \quad M \in {\mathbb {Z}}_{+}, \end{aligned}$$

(B.1)

defined with the product space ${\widetilde{\Omega }} = D([0,t],{\mathcal {K}}^{\langle \infty \rangle })^2 \times \Omega ^3$, and nondecreasing sequence of positive integers $\left( a_M\right) _{M \in {\mathbb {Z}}_{+}}$ such that $a_M \le M$, $M \in {\mathbb {Z}}_{+}$. Its values will be chosen later in the proof.

For $m \in {\mathbb {Z}}_{+}$ we set

$$\begin{aligned} V^{\omega ,m}(x):= \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle } \cap B_m(x,1)} \xi _{y}(\omega ) W(x,y) \end{aligned}$$

(B.2)

and

$$\begin{aligned} {\widetilde{V}}^{\omega ,m}(x):= \sum _{y \in {\mathcal {V}}_{0}^{\langle \infty \rangle } \setminus B_m(x,1)} \xi _{y}(\omega ) W(x,y), \end{aligned}$$

(B.3)

where the ball $B_m(x,1)$ is taken in the m-graph metric, i.e. for $x \in {\mathcal {K}}^{\langle \infty \rangle } \backslash {\mathcal {V}}^{\langle \infty \rangle }_m$ it is equal to $\Delta _m(x)$ – the only m-complex containing x; for $x \in {\mathcal {V}}^{\langle \infty \rangle }_m$ it is a sum of m-complexes attached to x (there are $\text {rank}(x) \in \{1,2,3\}$ of them). We also denote for $M \in {\mathbb {Z}}_{+}$

$$\begin{aligned} F_M(w,\omega ):= \textrm{e}^{-\int _0^t V^{\omega ,a_M}(X_s(w)) \textrm{d}s} \quad \text {and} \quad {\widetilde{F}}_M(w,\omega ):= \textrm{e}^{-\int _0^t {\widetilde{V}}^{\omega ,a_M}(X_s(w)) \textrm{d}s}. \end{aligned}$$

With this notation we have

$$\begin{aligned} {\mathbb {E}}_{{\mathbb {Q}}} \left[ \Lambda _M^D - {\mathbb {E}}_{{\mathbb {Q}}} \Lambda _M^D \right] ^2&= \int _{{\widetilde{\Omega }}} \prod _{i=1}^{2} \left( F_M(w_i, \omega _0) {\widetilde{F}}_M(w_i, \omega _0) - F_M(w_i, \omega _i) {\widetilde{F}}_M(w_i, \omega _i) \right) \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \times {\textbf{1}}_{\{t<\tau _{{\mathcal {K}}^{\langle M \rangle }}(w_i)\}} \cdot \textrm{d}\nu _M (w_1,w_2,\omega _0,\omega _1,\omega _2) \\&=: \int _{{\widetilde{\Omega }}} {\mathcal {X}}(w_1,w_2,\omega _0,\omega _1,\omega _2) \textrm{d}\nu _M (w_1,w_2,\omega _0,\omega _1,\omega _2). \end{aligned}$$

We now consider the partition of ${\widetilde{\Omega }}$ into three disjoint sets:

$$\begin{aligned} D_0^M&:= \left\{ (w_1,w_2) \in D([0,t],{\mathcal {K}}^{\langle \infty \rangle })^2 : \text {for every } s_1,s_2 \in [0,t] \ d_{a_M} \left( X_{s_1}(w_1),X_{s_2}(w_2)\right)>2 \right\} \times \Omega ^3,\\ D_1^M&:= \bigg \{(w_1,w_2) \in D([0,t],{\mathcal {K}}^{\langle \infty \rangle })^2 : d_{a_M+3}\left( X_{0}(w_1),X_{0}(w_2)\right) >2 \text { and there exist } s_1,s_2 \in (0,t] \\&\ \ \ \ \ \text { such that } d_{a_M}\left( X_{s_1}(w_1),X_{s_2}(w_2)\right) \le 2 \bigg \} \times \Omega ^3,\\ D_2^M&:= {\widetilde{\Omega }} \backslash \left( D_0^M \cup D_1^M\right) . \end{aligned}$$

We will integrate ${\mathcal {X}}$ over each of these sets separately. Let us point out that all these sets are now defined with the m-graph metric $d_m(x,y)$. We also use this opportunity to correct the definition of the sets $D_0^M$ and $D_1^M$ on p. 1272 in [28]: they also should be defined with $s_1,s_2 \in [0,t]$ as above instead of single $s \in [0,t]$. Also, $a_M$ should be used in the definition of $D_0^M$ instead of $c_M$.

By following the argument in the proof of the quoted lemma, we get

$$\begin{aligned}{} & {} \prod _{i=1}^{2} \left( F_M(w_i,\omega _0) {\widetilde{F}}_M(w_i, \omega _0) - F_M(w_i,\omega _i) {\widetilde{F}}_M(w_i, \omega _i) \right) \\{} & {} \quad \le \prod _{i=1}^{2} \left( F_M(w_i,\omega _0) - F_M(w_i,\omega _i) \right) \\{} & {} \qquad + 2 - \left( {\widetilde{F}}_M(w_1, \omega _0) {\widetilde{F}}_M(w_2, \omega _2) + {\widetilde{F}}_M(w_2, \omega _0) {\widetilde{F}}_M(w_1, \omega _1) \right) . \end{aligned}$$

For a given $M \in {\mathbb {Z}}_{+}$ and a path $X_s(w)$, the functional $F_M(w,\cdot )$ depends only on those fractal lattice points from $V_{0}^{\langle \infty \rangle }$ which are in the set $X_{[0,t]}^{a_M}(w):= \bigcup _{s\in [0,t]} B_{a_M}(X_s(w),1)$. From the definition of $D_0^M$ we see that on this set we have $X_{[0,t]}^{a_M}(w_1) \cap X_{[0,t]}^{a_M}(w_2) = \emptyset$ and therefore, the random variables $F_M(w_1,\omega _0) - F_M(w_1,\omega _1)$ and $F_M(w_2,\omega _0) - F_M(w_2,\omega _2)$ are ${\mathbb {Q}}^{\otimes 3}$-independent. In consequence,

$$\begin{aligned} \int _{D_0^M} \prod _{i=1}^{2} \left( F_M(w_i,\omega _0) - F_M(w_i,\omega _i) \right) {\textbf{1}}_{\{t<\tau _{{\mathcal {K}}^{\langle M \rangle }}(w_i)\}} \textrm{d}\nu _M (w_1,w_2,\omega _0,\omega _1,\omega _2) =0 \end{aligned}$$

and, by following the argument in the proof of the cited lemma, including Jensen’s inequality and the assumption that all lattice random variables are nonnegative and integrable, we obtain

$$\begin{aligned} \int _{D_0^M} {\mathcal {X}}\ \textrm{d}\nu _M \le c t \, \mathbb {E_Q}\xi \sup _{x \in {\mathcal {K}}^{\langle \infty \rangle }} \sum _{y \notin B_{a_M}(x,1)} W(x,y), \end{aligned}$$

(B.4)

for a constant $c>0$, independent of M.

On the set $D_1^M$ we have

$$\begin{aligned} \sup _{s \in (0,t]} d_{a_M} \left( X_0(w_i), X_s(w_i) \right) >2, \quad \text {for }i=1 \text { or } i=2. \end{aligned}$$

(B.5)

This can be seen as follows (see Fig. 1 for illustration). From the definition of $D_1^M$ we have that $d_{a_M+3} \left( X_0(w_1), X_0(w_2) \right) >2$, what means that $X_0(w_1)$ and $X_0(w_2)$ are in separate $(a_M+3)$-complexes (light grey complexes in the figure). If, on the contrary to (B.5), for both $i \in \{1,2\}$ and all $s \in (0,t]$ were $d_{a_M} \left( X_0(w_i), X_s(w_i) \right) \le 2$, then for both $i=1,2$ the entire path $X_{s}(w_i)$ would be bounded inside $B_{a_M}(X_0(w_i),2)$ (dark grey complexes in the figure). That would mean that these two paths of the process up to time t are not closer to each other than in two different $a_M$-complexes attached to the vertices of a common $(a_M+3)$-complex (white in the figure). In fact, they would be in separate $(a_M+1)$-complexes inside two different $(a_M+2)$-complexes. This would mean that $d_{a_M} \left( X_{s_1}(w_1), X_{s_2}(w_2) \right) \ge 6$ for all $s_1, s_2 \in (0,t]$, because the path realizing the graph distance must pass through another $a_M$-complex inside $(a_M+1)$-complex containing $X_{s_1}(w_1)$, then through another $(a_M+1)$-complex inside $(a_M+2)$-complex containing $X_{s_1}(w_1)$, so at least two more $a_M$-complexes. By symmetry, it must go next through at least three $a_M$-complexes inside $(a_M+2)$-complex containing $X_{s_2}(w_2)$.

Fig. 1

$X_0(w_i)$ are in separable $(a_M+3)$-complexes (light grey). If (B.5) does not hold, then the entire paths $X_s(w_i)$, $s \in [0,t]$, must be restricted to the respective dark grey $a_M$-complexes, contradicting the definition of $D_1^M$ (colour figure online)

By [27, Lemma A.2] this implies that

$$\begin{aligned} \sup _{s \in (0,t]} |X_0(w_i), X_s(w_i)| \ge c_1 L^{a_M }, \quad \text {for }i=1 \text { or } i=2, \end{aligned}$$

with a constant $c_1$ independent of M, and since $0\le F_M(w_i,\omega _k) \le 1$, $i=1,2$, $k=0,1,2$, the integral over $D_1^M$ can be estimated by

$$\begin{aligned} \frac{1}{\mu \left( {\mathcal {K}}^{\langle M\rangle }\right) } \int _{{\mathcal {K}}^{\langle M\rangle }} p(t,x,x) {\textbf{P}}_{t}^{x,x} \left[ \sup _{s \in (0,t]} |X_0(w_i), X_s(w_i)| \ge c_1 L^{a_M } \right] \mu (\textrm{d}x).\qquad \end{aligned}$$

(B.6)

Using the same argument as in the same step of the proof of the cited lemma, we then get that the above expression is less than or equal to

$$\begin{aligned} c_2 \sup _{x \in {\mathcal {K}}^{\langle \infty \rangle }} {\textbf{P}}^x \left[ \sup _{s \in (0,t/2]} |X_0(w_i), X_s(w_i)| \ge c_1 L^{a_M } \right] , \end{aligned}$$

(B.7)

with some $c_2$, independent of M.

Since the integrand ${\mathcal {X}}$ is not bigger than 1, it is enough to estimate the measure of $D_2^M$. We have

$$\begin{aligned} \nu _M(D_2^M) \le \frac{ c_3 \mu \{(x,y) \in {\mathcal {K}}^{\langle M \rangle } \times {\mathcal {K}}^{\langle M \rangle }: d_{a_M+3}\left( x,y\right) \le 2 \}}{\left( \mu ({\mathcal {K}}^{\langle M \rangle })\right) ^2} \le \frac{ 4 c_3 N^{a_M+3}}{N^{M}},\qquad \nonumber \\ \end{aligned}$$

(B.8)

with some $c_3$ independent of M.

We may choose $a_M=\lfloor M/4 \rfloor$. Then (B.4) is a term of a convergent series by (W3) and (B.7) is a term of convergent series by Lemma A.1. $\square$