Abstract
We study ballisticity conditions for d-dimensional random walks in strong mixing environments, with underlying dimension \(d\ge 2\). Specifically, we introduce an effective polynomial condition similar to that given by Berger et al. (Comm. Pure Appl. Math. 77:1947–1973, 2014). In a mixing setup we prove that this condition implies the corresponding stretched exponential decay, and obtain an annealed functional central limit theorem for the random walk process centered at the limiting velocity. This paper complements previous work of Guerra (Ann. Probab. 47:3003–3054, 2019) and completes the answer about the meaning of condition \((T')|\ell \) in a mixing setting, an open question posed by Comets and Zeitouni (Ann. Probab. 32:880–914, 2004).
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E. Guerra was supported by CAPES PNPD20130824, Nucleus Millennium SMCDS NC130062 and CONICYT FONDECYT Postdoctorado 3180255. G. Valle was partially supported by FAPERJ grants E-26/203.048/2016 and E-26/202.636/2019 and CNPq grants 421383/2016-0 and 308006/2018-6. M. E. Vares was partially supported by CNPq grant 305075/2016-0, FAPERJ grant E-26/203.048/2016 and FAPERJ grant E-26/202.636/2019.
Appendices
Proof of Lemma 2.1
This proof is very similar to that of [8, Lemma 2.2]. However this reference missed a proof for the equivalence of \((T^\gamma )|_{l_0}\) in Definition 1.3 with conditions (i), (ii) and (iii). Thus for the sake of completeness we provide a proof here for the case \(\gamma <1\).
Proof of Lemma 2.1
The proof of \((i)\Rightarrow (ii)\) can be found in [25, pages 516-517]. In sequence we show that \((ii)\Rightarrow (iii)\), \((iii)\Rightarrow (i)\), \((iii)\Rightarrow (T^\gamma )|_{l_0}\) and \((T^\gamma )|_{l_0}\Rightarrow (i)\).
We turn to prove \((ii)\Rightarrow (iii)\). By (ii), there exist \(b, \ {\hat{r}}, \ {\widetilde{c}}>0\) such that for large L there are finite subsets \(\varDelta _L\) with \(0\in \varDelta _L \subset \{x\in \mathbb {Z}^d: x\cdot l_0\ge -bL\}\cap \{x\in {\mathbb {R}}^d: |x|_2\le {\hat{r}}L\}\) and
Furthermore, since we are considering the outer boundary of our discrete sets as \(\varDelta _L\), by a simple geometric argument we can replace \(\varDelta _L\) by its intersection with \(\{x\in {\mathbb {Z}}^d: \,x\cdot l_0<L\}\), so without loss of generality we can assume that \(\varDelta _L\subset \{x\in {\mathbb {Z}}^d: \, x\cdot l_0<L \}\). Consider the box \({{\widetilde{B}}}_{L, {\hat{r}},b, l_0}(x)\) defined by
where \({\widetilde{R}}\) is a rotation on \({\mathbb {R}}^d\) with \({\widetilde{R}}(l_0)=e_1\). Its frontal boundary is defined as in (1.6). We then have that \(\varDelta _L \subset {{\widetilde{B}}}_{L, {\hat{r}}, b, l_0}(0)\), and consequently for large L,
Notice that if \(b\le 1\), we choose c in (iii) as \({{\hat{r}}}\), and we finish the proof. If \(b>1\), we proceed as follows: let \(N=bL\) and consider the box
defined according (2.1). We introduce for each integer \(i\in [1,\lfloor b \rfloor ]\) a sequence \((T_i)_{1 \le i \le \lfloor b \rfloor }\) of \(({\mathcal {F}}_n)_{n\ge 0}\)-stopping times via
We also introduce the stopping time S which codifies the unlikely walk exit from box \({\widetilde{B}}_{L, {\hat{r}}, b, l_0}(X_0)\) and is defined by:
The previous definitions imply
It is convenient at this point to introduce boundary sets \(F_i\), \(i\in [1, \lfloor b \rfloor ]\) as follows
We also define, for \(i\in [1,\lfloor b \rfloor ]\), good environmental events \(G_i\) via
Observe that the left hand of inequality (A.2) is greater than
In turn, writing for simplicity \({{\widetilde{B}}}_{L, {\hat{r}}, b, l_0}(y)\) as B(y) for \(y\in {\mathbb {Z}}^d\), the last expression equals
where we have made use of the Markov property to obtain the last inequality. We iterate this process to see
Notice that using Chebyshev’s inequality, we have for \(i\in [1,\lfloor b \rfloor ]\) and large L
From (A.3), the fact that b is finite and independent of L and the estimate (A.4), there exists \(\eta >0\) such that for large N
and this ends the proof of the required implication.
To prove the implication \((iii)\Rightarrow (i)\), we fix a rotation R on \({\mathbb {R}}^d\), with \(R(e_1)=l_0\) and such that R is the underlying rotation of hypothesis in (iii). For small \(\alpha \) we define \(2(d-1)\)-directions \(l_{+i}\) and \(l_{-i}\), \(i\in [2,d]\)
Following a similar argument as the one in [10, Proposition 4.2], but using \(\gamma \)-stretched exponential decay instead of polynomial one, we conclude that there exists \(\alpha > 0\) sufficiently small such that for each \(2 \le i \le d\) there are some \(r_i>0\), with
Observe that from (A.5), we finish the proof that (iii) implies (i) by taking
and then observing that for \(i\in [0,2(d-1)]\) we have
Finally, it is straightforward by the previous argument to see that (iii) implies \((T^\gamma )|_{l_0}\), indeed for \(\alpha \) as above, the set
is open in \({\mathbb {S}}^{d-1}\) and contains \(l_0\). Furthermore, for any \(b>0\) we have
for each direction \(\ell \in {\mathcal {A}}_{\alpha }\), an thus we have that \((T^\gamma )|_{l_0}\) holds. Conversely, assume \((T^\gamma )|_{l_0}\), then there exits \(\varepsilon > 0\) such that
for each direction \(\ell \in \left( l_0+\{x\in {\mathbb {R}}^d: |x|_2<\varepsilon \}\right) \cap {\mathbb {S}}^{d-1}=: \varTheta (\varepsilon )\). Since the set \(\varTheta (\varepsilon )\) is not flat and in fact has a non-zero curvature, any d-different elements \(l_1\), \(l_2\), \(\ldots ,\) \(l_d\) of \(\varTheta (\varepsilon )\) are linearly independents and thus they span \({\mathbb {R}}^d\). Take data: \(l_0,\) \(l_1,\) \(\ldots ,\) \(l_d\), \(a_0=1,\) \(a_1=1,\) \(\ldots ,\) \(a_d=0\) and arbitrary positive numbers \(b_0,\) \(b_1,\) \(\ldots ,\) \(b_d\) and it is clear that they generate an \(l_0\)-directed system of slabs of order \(\gamma \), which ends the proof. \(\square \)
On the proof of Proposition 2.2
As we mentioned in Remark 2.1, the proof of Proposition 2.2 is analogous to that of [8, Proposition 4.1]. It is just a matter of following the proof using the subadditivity of \(h(u) = u^\gamma \) at each step where the exponent is decomposed. However there are some mistakes in the proof of [8, Proposition 4.1]. We present here the needed correction, showing how one can keep \(\mathbf{c }\) in equation [8, (4.11)] less than or close to 1. This is related to the existence of the constant \(c_3\) in display (2.11).
-
(i)
The problem starts with display [8, (4.6)]. We should not bound
$$\begin{aligned} \sum _{0\le n\le L^2-1}\overline{E}_0\left[ \exp \left( c \kappa ^L X_{S_{k+1}}\cdot l\right) ,\, t^n_k<\infty ,\, B_{n,k},\, A_{n,k}\right] \end{aligned}$$(B.1)from above by
$$\begin{aligned} \sum _{ n\ge L^2}\overline{E}_0\left[ \exp \left( c \kappa ^L X_{S_{k+1}}\cdot l\right) ,\, t^n_k<\infty ,\, B_{n,k},\, A_{n,k}\right] , \end{aligned}$$(B.2)because it is not a sharp bound. In order to get a better upper bound, we work with both terms separately. Specifically, we have that
$$\begin{aligned}&\sum _{0\le n\le L^2-1}\overline{E}_0\left[ \exp \left( c \kappa ^L X_{S_{k+1}}\cdot l\right) ,\, t^n_k<\infty ,\, B_{n,k},\, A_{n,k}\right] \nonumber \\&\quad \le \sum _{0\le n\le L^2-1}\overline{E}_0\left[ \exp \left( c \kappa ^L X_{S_{k+1}}\cdot l\right) ,\, t^n_k<\infty ,\, A_{n,k}\right] \nonumber \\&\quad = \kappa ^L\sum _{0\le n\le L^2-1}\overline{E}_0\left[ \exp \left( c \kappa ^L X_{S_{k+1}}\cdot l\right) ,\, t^n_k<\infty \right] \end{aligned}$$(B.3)for the first term in [8, (4.6)]. Then, we use [8, (4.7)] to get that there exists \(L_0\in |\ell |_1{\mathbb {N}}\) such that for each \(L\ge L_0\) we have
$$\begin{aligned}&\sum _{0\le n\le L^2-1}\overline{E}_0\left[ \exp \left( c \kappa ^L X_{S_{k+1}}\cdot l\right) ,\, t^{(0)}_k<\infty ,\, B_{n,k},\, A_{n,k}\right] \\&\quad \le \left( 2L^2\right) \kappa ^L\overline{E}_0\left[ \exp \left( c \kappa ^L M_k\right) ,\, t^{(0)}_k<\infty ,\,\right] , \end{aligned}$$where we have used that for large L, one has
$$\begin{aligned} \sum _{0\le n\le L^2-1}\exp \left( c\kappa ^L\left( \overline{c}L+n|l|_\infty \right) \right) \le 2L^2. \end{aligned}$$ -
(ii)
We now turn to get an appropriate or corrected upper bound for the expression in (B.2). As it is shown after [8, (4.7)], we have that for \(n\ge L^2\) the term in (B.2) is smaller than
$$\begin{aligned} \kappa ^L{\overline{E}}_0\left[ \exp \left( c\kappa ^L(M_k+ n|l|_\infty +{\overline{c}} L)\right) , \,t^{(n)}_k<\infty ,\, B_{n,k}\right] . \end{aligned}$$(B.4)In order to estimate the last expression, we use the proof of [10, Lemma 6.6], to see that there exists a constant \(c_{43}>1\) such that for all \(n\ge L^2\) and integer \(k\ge 0\) we have
$$\begin{aligned} Q[B_{n,k}]\le \left( 1-c_{43}L^2\kappa ^L\right) ^{\left[ \frac{n}{L^2}\right] }. \end{aligned}$$(B.5)Therefore, using inequality (B.5) in expression (B.4) we get
$$\begin{aligned}&\kappa ^L{\overline{E}}_0\left[ \exp \left( c\kappa ^L(M_k+ n|\ell |_\infty +{\overline{c}} L)\right) , \,t^{(n)}_k,\, B_{n,k}\right] \nonumber \\&\quad \le L^2\kappa ^L\exp \left( c{\overline{c}}L\kappa ^L\right) \sum _{j=1}^\infty \exp \left( jc|l|_\infty \kappa ^L (L+1)^2\right) \left( 1-c_{43}L^2\kappa ^L\right) ^j\nonumber \\&\qquad \times {\overline{E}}_0\left[ \exp \left( c\kappa ^L M_k\right) , t^{(0)}_k<\infty \right] \nonumber \\&\quad \le L^2\kappa ^L\exp \left( c\overline{c}L\kappa ^L\right) \frac{1}{1-\exp \left( c|l|_\infty \kappa ^L (L+1)^2\right) (1-c_{43}L^2\kappa ^L)}. \end{aligned}$$(B.6)By Taylor’s expansion up to order 1, for all \(L\ge L_1\) (we assume \(L_1>L_0\)) there exists a constant \(c_2=c_2(L_1)\) such that
$$\begin{aligned} \exp \left( c|l|_\infty \kappa ^L (L+1)^2\right) (1-c_{43}L^2\kappa ^L)\le 1+(cc_2|l|_\infty -c_{43})L^2\kappa ^L. \end{aligned}$$As a result, we take c small enough so that \(c_{43} - cc_2|l|_\infty> \xi ^{-1} > 1\) and hence, we have (this corrects inequality [8, (4.8)]):
$$\begin{aligned}&\sum _{ n\ge L^2}\overline{E}_0\left[ \exp \left( c \kappa ^L X_{S_{k+1}}\cdot l\right) ,\, t^n_k<\infty ,\, B_{n,k},\, A_{n,k}\right] \end{aligned}$$(B.7)$$\begin{aligned}&\quad \le \xi \exp \left( c{\overline{c}}L\kappa ^L\right) \overline{E}_0\left[ e^{\kappa ^LM_k}, t^{(0)}_k<\infty \right] . \end{aligned}$$(B.8)We summarize the previous estimates as follows,
$$\begin{aligned}&{\overline{E}}_0[e^{c\kappa ^LX_{S_{k+1}}\cdot l}, S_{k+1}<\infty ] \nonumber \\&\quad \le \left( 2L^2\kappa ^L+ \xi \exp \left( c\overline{c}L\kappa ^L\right) \right) {\overline{E}}_0\left[ e^{c\kappa ^LM_k}, t^{(0)}_k<\infty \right] . \end{aligned}$$(B.9)Using inequality [8, (4.9)] and the estimate immediately after [8, (4.10)] into (B.9) we have
$$\begin{aligned} {\overline{E}}_0\left[ e^{c\kappa ^L X_{S_{k+1}\cdot l}}, S_{k+1}<\infty \right] \le {\overline{E}}_0\left[ \mathcal Ce^{c\kappa ^L{\overline{M}}},D'<\infty \right] ^k, \end{aligned}$$(B.10)where the constant \({\mathcal {C}}(L)\) is defined by
$$\begin{aligned} {\mathcal {C}}:=\left( 2L^2\kappa ^L+\xi \exp \left( c\overline{c}L\kappa ^L\right) \right) \exp \left( C\sum _{x\in \partial ^rH, y\in \partial ^r \varLambda }e^{-g|x-y|_1}\right) . \end{aligned}$$We remark that the sets H and \(\varLambda \) were defined after [8, (4.4)]. We finally use the first display in [8, page 3022] to see that
$$\begin{aligned} \lim _{L\rightarrow \infty }{\mathcal {C}}=\xi < 1 \end{aligned}$$and we notice that this is enough to correct the convergence of the series since \(P_0[D'<\infty ]<1\).
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Guerra, E., Valle, G. & Vares, M.E. Polynomial ballisticity conditions and invariance principle for random walks in strong mixing environments. Probab. Theory Relat. Fields 182, 685–750 (2022). https://doi.org/10.1007/s00440-021-01106-9
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DOI: https://doi.org/10.1007/s00440-021-01106-9
Keywords
- Random walk in random environment
- Ballisticity conditions
- Invariance principle
- Mixing environment
Mathematics Subject Classification
- Primary 60K37
- Secondary 82D30