# Frequently visited sites of the inner boundary of simple random walk range

Izumi Okada

###### Abstract

This paper considers the question: how many times does a simple random walk
revisit the most frequently visited site among the inner boundary points? It
is known that in ${\mathbb{Z}}^{2}$, the number of visits to the most
frequently visited site among all of the points of the random walk range up to
time $n$ is asymptotic to $\pi^{-1}(\log n)^{2}$, while in ${\mathbb{Z}}^{d}$
$(d\geq 3)$, it is of order $\log n$. We prove that the corresponding number
for the inner boundary is asymptotic to $\beta_{d}\log n$ for any $d\geq 2$,
where $\beta_{d}$ is a certain constant having a simple probabilistic
expression.

##  1 Introduction

Many works have studied properties of the trajectory of a simple random walk.
These properties include the growth rate of the trajectory’s range, location
of the most frequently visited site, the number of the visits to this site,
the number of the sites of frequent visits, and so forth. There remain many
interesting unsolved questions concerning these properties. The most
frequently visited site among all the points of the range (of the walk of
finite length) is called a favorite site and a site which is revisited many
times (in a certain specified sense) is called a frequently visited site.
About fifty years ago, Erdős and Taylor [4] posed a problem concerning a
simple random walk in ${\mathbb{Z}^{d}}$: how many times does the random walk
revisit the favorite site (up to a specific time)? Open problems concerning
the favorite site are raised by Erdős and Révész [5], [6] and Shi and Tóth
[13] but remain unsolved so far. By Lifshits and Shi [10] it is known that the
favorite site of a $1$-dimensional random walk tends to be far from the
origin, but almost nothing is known about its location for multi-dimensional
walks.

In this paper, we focus on the most frequently visited site among the inner
boundary points of the random walk range, rather than among all of the points
of the range, and propose the question: how many times does a random walk
revisit the most frequently visited site among the inner boundary points?
Here, we briefly state our result and compare it with known results for the
favorite site. Let $M_{0}(n)$ be the number of visits to the favorite site by
the walk up to time $n$ and $M(n)$ be that of the most frequently visited site
among the inner boundary points. In Theorem 2.1, we will prove that for $d\geq
2$

$\displaystyle\lim_{n\to\infty}\frac{M(n)}{\log n}=\frac{1}{-\log
P(T_{0}<T_{b})}\quad\text{ a.s.}$

Here, $T_{a}$ is the first time the random walk started at the origin hits $a$
after time $0$, and $b$ is a neighbor site of the origin. To compare, a
classical result of Erdős and Taylor [4] says that for $d\geq 3$,

$\displaystyle\lim_{n\to\infty}\frac{M_{0}(n)}{\log n}=\frac{1}{-\log
P(T_{0}<\infty)}\quad\text{ a.s.},$

and for $d=2$, $M_{0}(n)/(\log n)^{2}$ is bounded away from zero and infinity
a.s. (the limit exists and is identified [2] as mentioned later in Section 2).

These results illuminate the geometric structure of the random walk range as
well as the nature of recurrence or transience of random walks. We are able to
infer that the favorite site is outside the inner boundary from some time
onwards with probability one. This may appear intuitively clear; it seems
improbable for the favorite point to continue to be an inner boundary point
since it must be visited many times, but our result further shows that there
are many inner boundary points that are visited many times, with amounts
comparable to that of the favorite point for $d\geq 3$. In addition, the
growth order of $M_{0}(n)$ is the same for all $d\geq 2$, meaning the phase
transition which occurs between $d=2$ and $d\geq 3$ for $M_{0}(n)$ does not
occur for $M(n)$.

In Theorem 2.2, which is a strong claim in comparison to Theorem 2.1, we will
provide an explicit answer to the question of how many frequently visited
sites among the inner boundary points exist.

The upper bounds for both Theorem 2.1 and Theorem 2.2 are obtained using the
idea in [4]. The Chebyshev inequality and the Borel-Cantelli lemma are also
used in the same way as in [4]. On the other hand, $M(n)$ is not monotone,
while $M_{0}(n)$ is monotone. We work with the walk and its trajectory at the
times $2^{k}$ and find a process that is monotone and a bit larger than
$M(n)$, but with the desired asymptotics.

On the other hand, the idea for the proof of the lower bound is different from
that for the known results. In [2], a Brownian occupation measure was used in
the proof. Rosen [12] provided another proof to the result of [2], in which he
computed a crossing number instead of the number of the frequently visited
site. In this paper, we use the Chebyshev inequality and the Borel-Cantelli
lemma as in [12] but for the number of the frequently visited sites among the
inner boundary points. In addition, as the proof of the upper bound, we
estimate a number slightly smaller than the number of the frequently visited
site among the inner boundary points.

We conclude this introduction by mentioning some known results about the inner
boundary points of the random walk range that are closely related to the
present subject. Let $L_{n}$ be the number of the inner boundary points up to
time $n$. In [1], it is noticed that the entropy of a random walk is
essentially governed by the asymptotic of $L_{n}$. In [11], the law of large
numbers for $L_{n}$ is shown and $\lim L_{n}/n$ is identified for a simple
random walk on ${\mathbb{Z}}^{d}$ for every $d\geq 1$. Let $J_{n}^{(p)}$
denote the number of $p$-multiplicity points in the inner boundary and be
defined as

$\displaystyle J_{n}^{(p)}=\sharp$ $\displaystyle\\{S_{i}\in\partial
R(n):\sharp\\{m:0\leq m\leq n,S_{m}=S_{i}\\}=p\\},$

where $\partial R(n)$ is the set of the inner boundary of
$\\{S_{0},S_{1},...,S_{n}\\}$ and $\sharp A$ is the number of elements in $A$.
In [11], it is also shown that for a simple random walk in $\mathbb{Z}^{2}$,
with $p\geq 1$,

$\displaystyle\frac{{\pi}^{2}}{2}\leq\lim_{n\to\infty}EL_{n}\times\frac{(\log
n)^{2}}{n}$ $\displaystyle\leq 2{\pi}^{2},$
$\displaystyle\frac{\tilde{c}^{p-1}\pi^{2}}{4}\leq\lim_{n\to\infty}EJ_{n}^{(p)}\times\frac{(\log
n)^{2}}{n}$ $\displaystyle\leq\tilde{c}^{p-1}\pi^{2},$ (1)

where $\tilde{c}=P(T_{0}<T_{b})$ for any/some neighbor site $b$ of the origin.
These may be compared with the results for the entire range; according to [7],
$\sharp\\{S_{i}:0\leq i\leq n\\}$ in ${\mathbb{Z}}^{2}$ is asymptotic to $\pi
n/\log n$ and the asymptotic form of the number of $p$-multiplicity points in
it is independent of $p$.

##  2 Framework and Main Results

Let $\\{S_{k}\\}_{k=0}^{\infty}$ be a simple random walk on the
$d$-dimensional square lattice ${\mathbb{Z}}^{d}$. Let $P^{a}$ denote the
probability of the simple random walk starting at $a$; we simply write $P$ for
$P^{0}$. Let ${\mathbb{N}}=\\{1,2,3,...\\}$ and for $n\in{\mathbb{N}}$, set
$R(n)=\\{S_{0},S_{1},\ldots,S_{n}\\}$ as the random walk range up to the time
$n$. We call $z\in\mathbb{Z}^{d}$ a neighbor of $a\in\mathbb{Z}^{d}$ if
$|a-z|=1$. Let ${\cal N}(a)$ be the set of all neighbors of $a$ defined as

${\cal N}(a)=\\{z\in{\mathbb{Z}}^{d}:|a-z|=1\\}.$

The inner boundary of $A\subset\mathbb{Z}^{d}$ is denoted by $\partial A$,
that is

$\partial A=\\{x\in A:{\cal N}(x)\not\subset A\\}.$

We denote the number of visits to $x$ of $S_{m}$, $0\leq m\leq n$ by $K(n,x)$.
That is,

$K(n,x)=\sum_{j=0}^{n}1_{\\{S_{j}=x\\}}.$

Moreover, we set

$M(n):=\max_{x\in\partial R(n)}K(n,x).$

Clearly, this is the maximal number of visits of the random walk of length $n$
to $\partial R(n)$, the inner boundary of $R(n)$. Let $T_{x}$ denote the first
passage time to $x$: $T_{x}=\inf\\{m\geq 1:S_{m}=x\\}$. We are now ready to
state our main theorems. The first theorem provides us with sharp asymptotic
behavior of $M(n)$.

######  Theorem 2.1.

For $d\geq 2$

$\displaystyle\lim_{n\to\infty}\frac{M(n)}{\log n}=\beta_{d}\quad\text{
a.s.},$

where

$\displaystyle\beta_{d}=\frac{1}{-\log P(T_{0}<T_{b})}$

for any $b\in{\cal N}(0)$. Note that $P(T_{0}<T_{b})$ does not depend on the
choice of $b\in{\cal N}(0)$, but rather depends only on the dimension.

Leading to the second main theorem, we first define $\Theta_{n}(\delta)$ for
$n\in\mathbb{N}$ and $0<\delta<1$ as

$\displaystyle\Theta_{n}(\delta):=\sharp\\{x\in\partial
R(n):\frac{K(n,x)}{\log n}\geq\beta_{d}\delta\\}.$

This is the cardinality of points in $\partial R(n)$ whose number of visits is
comparable to the maximal order appearing in Theorem 2.1, with a ratio greater
than or equal to $\delta$. Our second main theorem exhibits the sharp
logarithmic asymptotic behavior of $\Theta_{n}(\delta)$ as $n\to\infty$.

######  Theorem 2.2.

For $d\geq 2$ and $0<\delta<1$,

$\displaystyle\lim_{n\to\infty}\frac{\log\Theta_{n}(\delta)}{\log
n}=1-\delta\quad a.s.$

We compare our main results to the corresponding results for the whole random
walk range $R(n)$. We denote the quantity corresponding to $M(n)$ by
$M_{0}(n)$. That is, $M_{0}(n)=\max_{x\in{\mathbb{Z}^{d}}}K(n,x)$ where
$M_{0}(n)$ represents the maximal number of visits of the random walk to a
single site in the entire random walk range until time $n$. Erdős and Taylor
[4] showed that for $d\geq 3$

$\displaystyle\lim_{n\to\infty}\frac{M_{0}(n)}{\log n}=\frac{1}{-\log
P(T_{0}<\infty)}\quad\text{ a.s.}$

For $d=2$, they obtained

$\displaystyle\frac{1}{4\pi}\leq\liminf_{n\to\infty}\frac{M_{0}(n)}{(\log
n)^{2}}\leq\limsup_{n\to\infty}\frac{M_{0}(n)}{(\log
n)^{2}}\leq\frac{1}{\pi}\quad\text{ a.s.},$

and conjectured that the limit exists and equals $1/\pi$ a.s. Forty years
later, Dembo et al. [2] verified this conjecture and also showed how many
frequently visited sites of order $(\log n)^{2}$ there are in the following
sense. Let $d=2$. Then, for $0<a<1$, define

$\Theta_{0,n}=\sharp\\{x\in{\mathbb{Z}^{2}}:\frac{K(n,x)}{(\log
n)^{2}}\geq\frac{a}{\pi}\\}.$

Then

$\displaystyle\lim_{n\to\infty}\frac{\log\Theta_{0,n}}{\log n}=1-a\quad\text{
a.s.}$

In view of these results, Theorem 2.1 entails the following corollary.

######  Corollary 2.1.

For $d\geq 2$, the favorite site does not appear in the inner boundary from
some time onwards a.s. In other words, $M_{0}(n)>M(n)$ for all but finitely
many $n$ with probability one.

The $p$-th hitting times $T^{p}_{x}$ for $p=0,1\,\ldots$ and the partial
ranges $R[l,n]$ that we are now to define play significant roles. Let
$T_{x}^{0}=\inf\\{j\geq 0:S_{j}=x\\}$ and for $p\geq 1$,

$\displaystyle T_{x}^{p}=\inf\\{j>T_{x}^{p-1}:S_{j}=x\\}$ (2)

with the convention $\inf\emptyset=\infty$. For $l,n\in{\mathbb{N}}$ let

$R[l,n]=\\{S_{l},S_{l+1},...,S_{n}\\}$

if $n\geq l$ and $R[l,n]=\emptyset$ if $l>n$ and $R[0,-1]=\emptyset$. The
inner boundary of the random walk range $R[l,n]$ is denoted simply by
$\partial R[l,n]$ as it is for $R(n)$. It is noted that $T_{x}=T_{x}^{0}$ for
$x\neq S_{0}$ and $T_{x}=T_{x}^{1}$ if $x=S_{0}$. Also, $R(n)=R[0,n]$.

In the proofs given in the remainder of this paper we denote contextual
constants by $C$ or $c$. In addition, $\lceil a\rceil$ denotes the smallest
integer $n$ with $n\geq a$, and $A^{c}$ denotes a complementary set of $A$.

##  3 The upper bound in Theorem 2.1

Here, we prove the following proposition.

######  Proposition 3.1.

For $d\geq 2$

$\limsup_{n\to\infty}\frac{M(n)}{\log n}\leq\beta_{d}\quad\text{ a.s.}$

Unlike the proof of the lower bound below, the proof of Proposition 3.1 will
be performed independently of the dimension $d$. As mentioned above, neither
$M(n)$ nor $\Theta_{n}(\delta)$ is monotone in $n\in\mathbb{N}$. To mitigate
this issue, we introduce the random variables. For $\beta>0$, we set

$\displaystyle\tilde{\Theta}_{n}(\beta)=\sharp\\{x\in\partial
R(T_{x}^{\lceil\beta\log
n/2\rceil}):K(n,x)\geq\lceil\beta\log\frac{n}{2}\rceil\\}$

($T_{x}^{p}$ is defined by (2)). This is a modification of
$\Theta_{n}(\beta/\beta_{d})$ made by relaxing the constraint of being on the
inner boundary. Note that $\tilde{\Theta}_{n}(\beta)$ vanishes for all
sufficiently large $n\in\mathbb{N}$ if $\beta>\beta_{d}$.

######  Lemma 3.1.

For $\beta>0$ there exists $C>0$ such that for any $n\in\mathbb{N}$

$\displaystyle E[\tilde{\Theta}_{n}(\beta)]\leq
Cn^{1-\frac{\beta}{\beta_{d}}}.$

###### Proof.

First we introduce the elementary property. For any intervals $I_{0}$,
$I_{1}$, $I_{2}\subset{\mathbb{N}}\cup\\{0\\}$ with $I_{0}\subset I_{1}\subset
I_{2}$, it holds that

$\displaystyle R(I_{0})\cap\partial R(I_{2})\subset\partial R(I_{1}).$ (3)

Note that we can write

$\displaystyle\tilde{\Theta}_{n}(\beta)=\sum_{l=0}^{n}1_{B_{l,n}},$ (4)

where

$\displaystyle B_{l,n}=\\{S_{l}\in R(l-1)^{c}\cap\partial
R(T_{S_{l}}^{\lceil\beta\log
n/2\rceil}),K(n,S_{l})\geq\lceil\beta\log\frac{n}{2}\rceil\\}.$

Since $K(l-1,S_{l})=0$ on $\\{S_{l}\in R(l-1)^{c}\\}$, for $l\leq n$

$\displaystyle P(B_{l,n})=$ $\displaystyle
P(\sum_{j=0}^{n}1_{\\{S_{j}=S_{l}\\}}\geq\lceil\beta\log\frac{n}{2}\rceil,S_{l}\in
R(l-1)^{c}\cap\partial R(T_{S_{l}}^{\lceil\beta\log n/2\rceil}))$
$\displaystyle=$ $\displaystyle
P(\sum_{j=l}^{n}1_{\\{S_{j}=S_{l}\\}}\geq\lceil\beta\log\frac{n}{2}\rceil,S_{l}\in
R(l-1)^{c}\cap\partial R(T_{S_{l}}^{\lceil\beta\log n/2\rceil}))$
$\displaystyle\leq$ $\displaystyle
P(\sum_{j=l}^{n}1_{\\{S_{j}=S_{l}\\}}\geq\lceil\beta\log\frac{n}{2}\rceil,S_{l}\in\partial
R[l,T_{S_{l}}^{\lceil\beta\log n/2\rceil}])$ $\displaystyle=$ $\displaystyle
P(K(n-l,0)\geq\lceil\beta\log\frac{n}{2}\rceil,0\in\partial
R(T_{0}^{\lceil\beta\log n/2\rceil-1})).$

Here, the inequality comes from (3) with $I_{0}=\\{l\\}$,
$I_{1}=[l,T_{S_{l}}^{\lceil\beta\log n/2\rceil}]$ and
$I_{2}=[0,T_{S_{l}}^{\lceil\beta\log n/2\rceil}]$. The last equality follows
from the Markov property and the translation invariance for $S_{l}$. In
addition, by applying the Markov property repeatedly, we obtain

$\displaystyle P(K(n-l,0)\geq\lceil\beta\log\frac{n}{2}\rceil,0\in\partial
R(T_{0}^{\lceil\beta\log n/2\rceil-1}))$ $\displaystyle\leq$ $\displaystyle
P(T_{0}^{\lceil\beta\log n/2\rceil-1}<\infty,0\in\partial
R(T_{0}^{\lceil\beta\log n/2\rceil-1}))$ $\displaystyle=$ $\displaystyle
P(\cup_{b\in{\cal N}(0)}\\{T_{0}^{\lceil\beta\log n/2\rceil-1}<T_{b}\\})$
$\displaystyle\leq$ $\displaystyle 2dP(T_{0}<T_{b})^{\lceil\beta\log
n/2\rceil-1}\leq Cn^{-\frac{\beta}{\beta_{d}}}.$

Hence, the assertion holds by $E[\tilde{\Theta}_{n}(\beta)]\leq n\max_{1\leq
l\leq n}P(B_{l,n})$ which follows from (4). ∎

###### Proof of Proposition 3.1.

Since $M(n)$ is not monotonically increasing in $n$, we instead first consider
$\tilde{M}(n)=\max_{l\leq n}M(l)$. If $\tilde{M}(n)>m$, there exist
$x_{1}\in\mathbb{Z}^{d}$ and $n_{1}\leq n$ such that
$\tilde{M}(n)=M(n_{1})=K(n_{1},x_{1})$ and $x_{1}\in\partial R(n_{1})$.
Therefore, for such $n_{1}$, $m$ and $x_{1}$, it holds that $T_{x_{1}}^{m}\leq
n_{1}$ and hence $x_{1}\in\partial R(n_{1})\cap\partial R(T_{x_{1}}^{m})$
holds. Further, $K(n_{1},x_{1})\leq K(n,x_{1})$. Accordingly, we have

$\displaystyle P(\tilde{M}(n)\geq\lceil\beta\log\frac{n}{2}\rceil)$
$\displaystyle\leq$ $\displaystyle P(\cup_{x\in\mathbb{Z}^{2}}\\{x\in\partial
R(T_{x}^{\lceil\beta\log
n/2\rceil}),K(n,x)\geq\lceil\beta\log\frac{n}{2}\rceil\\}).$

Thus, by Lemma 3.1 and the Chebyshev inequality, we obtain

$\displaystyle P(\tilde{M}(n)\geq\lceil\beta\log\frac{n}{2}\rceil)\leq
P(\tilde{\Theta}_{n}(\beta)\geq 1)\leq Cn^{1-\frac{\beta}{\beta_{d}}}.$

By using the Borel-Cantelli lemma for any $\beta>\beta_{d}$, we can show that
the events $\\{\tilde{M}(2^{k})>\beta\log 2^{k-1}\\}$ happen only finitely
often with probability one. Therefore, it holds that for any
$\beta>\beta_{d}$,

$\displaystyle\limsup_{k\to\infty}\frac{\tilde{M}(2^{k})}{\beta\log
2^{k-1}}\leq 1\quad\text{ a.s. }$ (5)

Now we consider $M(n)$. For any $k$, $n\in{\mathbb{N}}$ with $2^{k-1}\leq
n<2^{k}$ we have

$\frac{M(n)}{\log n}\leq\frac{\tilde{M}(2^{k})}{\log 2^{k-1}},$

and so with (5), for any $\beta>\beta_{d}$

$\displaystyle\limsup_{n\to\infty}\frac{M(n)}{\beta\log n}\leq 1\quad\text{
a.s. }$

Therefore, the proof is completed. ∎

##  4 The lower bound in Theorem 2.1

###  4.1 Reduction of the lower bound of Theorem 2.1 to key lemmas

Our goal in this section is to prove the following Proposition 4.1.

######  Proposition 4.1.

For $d\geq 2$

$\displaystyle\liminf_{n\to\infty}\frac{M(n)}{\log n}\geq\beta_{d}\quad\text{
a.s.}$ (6)

Unlike Sections $4.3$ and $4.4$, the argument of Sections $4.1$ and $4.2$ will
be performed independently of the dimension $d$. In what follows, we discuss
the proof for each fixed $\beta<\beta_{d}$. Let

$u_{n}=\lceil\exp(n^{2})\rceil.$

In this section, we will reduce the proof of Proposition 4.1 to two key lemmas
given below (Lemmas 4.2 and 4.3). For $b\in{\cal N}(0)$ and
$A\subset\mathbb{Z}^{d}$, we define $\partial_{b}A$ as

$\partial_{b}A:=\\{x\in A:x+b\not\in A\\}.$

We can extend the property (3) in the following way: for any intervals
$I_{0}$, $I_{1}$, $I_{2}\subset{\mathbb{N}}\cup\\{0\\}$ with $I_{0}\subset
I_{1}\subset I_{2}$, it holds that

$\displaystyle
R(I_{0})\cap\partial_{b}R(I_{2})\subset\partial_{b}R(I_{1})\subset\partial
R(I_{1}).$ (7)

Let us define $\tilde{Q}_{n}$ as

$\displaystyle\tilde{Q}_{n}:=\sharp\\{x\in
R(u_{n-1})\cap\partial_{b}R(u_{n}),T_{x}^{\lceil\beta n^{2}\rceil}\leq
u_{n-1}\\}.$

We begin by providing a sufficient condition for the inequality (6) asserted
in Proposition 4.1 to be true by means of $\tilde{Q}_{n}$.

######  Lemma 4.1.

If

$\displaystyle P(\tilde{Q}_{n}\geq 1\quad\quad\text{ for all but finitely many
}n)=1,$ (8)

for any $\beta\in(0,\beta_{d})$ then the inequality (6) holds.

###### Proof.

Set

$L(n):=\max_{x\in R(u_{n-1})\cap\partial_{b}R(u_{n})}K(u_{n-1},x).$

Note that $L(n)\geq\beta n^{2}$ if $\tilde{Q}_{n}\geq 1$. Hence, it holds that

$\displaystyle P(L(n)\geq\beta n^{2}\quad\quad\text{ for all but finitely many
}n)=1.$

Moreover, by (7), for any $m$, $n\in{\mathbb{N}}$ with $u_{m-1}\leq n<u_{m}$
we have

$\displaystyle R(u_{m-1})\cap\partial_{b}R(u_{m})\subset\partial R(n)$ (9)

and hence $L(m)\leq\max_{x\in\partial R(n)}K(u_{m-1},x)\leq\max_{x\in\partial
R(n)}K(n,x)\leq M(n)$. Therefore, we conclude that for any $\beta<\beta_{d}$

$\displaystyle\liminf_{n\to\infty}\frac{M(n)}{\beta\log
n}\geq\liminf_{m\to\infty}\frac{L(m)}{\beta\log u_{m}}\geq 1\quad\text{
a.s.,}$

as per (6) and as desired. ∎

In order to verify the condition (8), we introduce a new quantity $Q_{n}$ by
modifying the definition of $\tilde{Q}_{n}$. To do this, we first introduce
several notions. Set $T_{x,l}^{0}:=\inf\\{j\geq l:S_{j}=x\\}$ and
$T_{x,l}^{p}:=\inf\\{j>T_{x,l}^{p-1}:S_{j}=x\\}$. Note that
$T_{x,0}^{a}=T_{x}^{a}$ for any $a\in{\mathbb{N}}$ and $x\in\mathbb{Z}^{d}$,
and that $T_{S_{l}}^{p}=T_{S_{l},l}^{p}$ holds for each $p$ on the event
$S_{l}\notin R(l-1)$. Note that $T_{S_{l},l}^{p}$ is a stopping time while
$T_{S_{l}}^{p}$ is not. Although we can state the key lemmas without using
this notion, we introduce it for later use. For $k\in{\mathbb{N}}$, let
$h_{k}=\beta\log P(T_{0}<T_{b}\wedge k)+1$. Since
$\lim_{k\to\infty}h_{k}=1-\beta/\beta_{d}>0$, we have $h_{k}>0$ for all
sufficiently large $k$. We fix such a $k$ and simply denote $h_{k}$ by $h$
hereafter. Let

$I_{n}:=[\frac{u_{n-1}}{n^{2}},u_{n-1}-k\lceil\beta_{d}n^{2}\rceil]\cap{\mathbb{N}}.$

For any $l\in I_{n}$, we introduce the events $E_{l,n}$ and $A_{l,n}$ defined
by

$\displaystyle E_{l,n}:=\\{T_{S_{l},l}^{j}-T_{S_{l},l}^{j-1}<k\text{ for any
}1\leq j\leq\lceil\beta n^{2}\rceil\\},$

and

$\displaystyle A_{l,n}:=\\{S_{l}\in R(l-1)^{c}\cap\partial_{b}R(u_{n})\\}\cap
E_{l,n}.$

Then, we set

$\displaystyle Q_{n}:=\sum_{l\in I_{n}}1_{A_{l,n}}.$ (10)

Although $I_{n}$, $A_{l,n}$ and $Q_{n}$ depend on the choice of parameters
$\beta$ and $k$, we do not indicate such dependence explicitly by symbols. By
the definition of $I_{n}$,
$I_{n}\subset[0,u_{n-1}]\cap(\mathbb{N}\cup\\{0\\})$ and $l+k\lceil\beta
n^{2}\rceil\leq u_{n-1}$ hold. These facts imply $Q_{n}\leq\tilde{Q}_{n}$. As
we will see, the verification of condition (8) is reduced to the following two
estimates for $Q_{n}$.

######  Lemma 4.2.

Let $\beta<\beta_{d}$ and take $k\in{\mathbb{N}}$ so that $h=h_{k}>0$ as
above. Then, there exists $c>0$ such that for any $n\in{\mathbb{N}}$, the
following hold:

(i) When $d=2$,

$\displaystyle EQ_{n}\geq\frac{c\exp(hn^{2}-2n)}{n^{4}}.$

(ii) When $d\geq 3$,

$\displaystyle EQ_{n}\geq c\exp(hn^{2}-2n).$

######  Lemma 4.3.

Let $\beta<\beta_{d}$ and take $k\in{\mathbb{N}}$ so that $h=h_{k}>0$ as
above. Then, there exists $C>0$ such that for any $n\in{\mathbb{N}}$, the
following hold:

(i) When $d=2$,

$\displaystyle\mathrm{Var}(Q_{n})\leq
C\bigg{(}\frac{\exp(hn^{2}-2n)}{n^{4}}\bigg{)}^{2}\frac{\log n}{n^{2}}.$

(ii) When $d\geq 3$,

$\displaystyle\mathrm{Var}(Q_{n})\leq
C\exp(2hn^{2}-4n)\times\frac{1}{n^{10}}.$

Now we give a proof of Proposition 4.1 by assuming Lemmas 4.2 and 4.3 are
true.

###### Deduction of Proposition 4.1 from Lemmas 4.2 and 4.3.

If Lemmas 4.2 and 4.3 hold, then we only need to prove the assumption of Lemma
4.1 to obtain Proposition 4.1. Take $k\in{\mathbb{N}}$ and $h>0$ as above. By
the Chebyshev inequality, we have

$\displaystyle
P(|Q_{n}-EQ_{n}|>\frac{EQ_{n}}{2})\leq\frac{4\mathrm{Var}(Q_{n})}{(EQ_{n})^{2}}.$
(11)

By Lemmas 4.2 and 4.3, we can see that there exists $C>0$ such that the
following is true:

$\displaystyle\frac{\mathrm{Var}(Q_{n})}{(EQ_{n})^{2}}\begin{cases}&\leq\frac{C\log
n}{n^{2}}\quad\quad\quad\quad\text{if }d=2,\\\
&\leq\frac{C}{n^{10}}\quad\quad\quad\quad\quad\text{ if }d\geq 3.\end{cases}$

As a result, the right hand side of (11) is summable. Since
$|b-a|\leq\frac{a}{2}$ implies $b\geq\frac{a}{2}$ for $a,b\geq 0$, the Borel-
Cantelli lemma yields

$\displaystyle P(Q_{n}\geq\frac{1}{2}EQ_{n}\quad\quad\text{ for all but
finitely many }n)=1.$ (12)

Since $h>0$, Lemma 4.2 implies $EQ_{n}\geq 2$ for all sufficiently large $n$.
Hence, the assertion of Lemma 4.1 holds by combining this fact with (12). ∎

###  4.2 Preparations for the proof of Lemmas 4.2 and 4.3

In this section, we estimate $P(A_{l,n})$ using the strong Markov property.
For later use, we will consider more general events than $A_{l,n}$. For any
$n^{\prime},l,\tilde{n}\in{\mathbb{N}}\cup\\{0\\}$ with $n^{\prime}\leq
l\leq\tilde{n}$ and $n\in{\mathbb{N}}$, let

$\displaystyle F_{n^{\prime},l,\tilde{n},n}=\\{S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},\tilde{n}]\\}\cap E_{l,n},$

which we will sometimes denote $F(n^{\prime},l,\tilde{n},n)$ for typographical
reasons. Note that $F_{0,l,u_{n},n}=A_{l,n}$ holds.

######  Lemma 4.4.

There are constants $c$, $C>0$ such that for any
$n^{\prime},l,\tilde{n}\in{\mathbb{N}}\cup\\{0\\}$ with $n^{\prime}\leq
l\leq\tilde{n}$ and $n,a\in{\mathbb{N}}$ with
$l+k\lceil\beta_{d}n^{2}\rceil\leq a\leq\tilde{n}$

$\displaystyle P(F_{n^{\prime},l,\tilde{n},n})\leq P(T_{0}<T_{b}\wedge
k)^{\lceil\beta n^{2}\rceil}P(T_{0}\wedge T_{b}>l-n^{\prime})\times
P(T_{b}>\tilde{n}-a)$

and

$\displaystyle P(F_{n^{\prime},l,\tilde{n},n})\geq P(T_{0}<T_{b}\wedge
k)^{\lceil\beta n^{2}\rceil}P(T_{0}\wedge T_{b}>l-n^{\prime})\times
P(T_{b}>\tilde{n}).$

###### Proof.

We first remark that $l<T_{S_{l},l}^{\lceil\beta n^{2}\rceil}$ holds and,
hence, ${\cal F}(l)\subset{\cal F}(T_{S_{l},l}^{\lceil\beta n^{2}\rceil})$. By
taking a conditional expectation with respect to ${\cal
F}(T_{S_{l},l}^{\lceil\beta n^{2}\rceil})$, we obtain

$\displaystyle P(F_{n^{\prime},l,\tilde{n},n})=$ $\displaystyle
E[1{\\{S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l,n}}$ $\displaystyle\times
P(S_{l}\in\partial_{b}R[T_{S_{l},l}^{\lceil\beta n^{2}\rceil},\tilde{n}]|{\cal
F}(T_{S_{l},l}^{\lceil\beta n^{2}\rceil}))].$ (13)

On the event $E_{l,n}$, we have

$\displaystyle 0\leq T_{S_{l},l}^{\lceil\beta n^{2}\rceil}\leq
k\lceil\beta_{d}n^{2}\rceil+T_{S_{l},l}^{0}.$

Since $l=T_{S_{l},l}^{0}$, our choice of $a$ and this inequality imply

$\displaystyle 0\leq T_{S_{l},l}^{\lceil\beta n^{2}\rceil}\leq a.$ (14)

The Markov property and the translation invariance for $S_{l}$ yield

$\displaystyle P(S_{l}\in\partial_{b}R[T_{S_{l},l}^{\lceil\beta
n^{2}\rceil},\tilde{n}]|{\cal F}(T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}))=P(0\in\partial_{b}R(\tilde{n}-t))|_{t=T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}}.$ (15)

Substituting (15) for (13) and keeping (14) in mind, we obtain

$\displaystyle P(F_{n^{\prime},l,\tilde{n},n})\leq$ $\displaystyle
P(\\{S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l,n})$ $\displaystyle\times
P(0\in\partial_{b}R(\tilde{n}-a)),$ (16)

and

$\displaystyle P(F_{n^{\prime},l,\tilde{n},n})\geq$ $\displaystyle
P(\\{S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l,n})$ $\displaystyle\times
P(0\in\partial_{b}R(\tilde{n})).$ (17)

Thus, the proof is reduced to the estimate of the common first factor in the
right hand side of (16) and (17). If we take a conditional expectation with
respect to ${\cal F}(l)$, by the Markov property and the translation
invariance for $S_{l}$, we obtain

$\displaystyle P(\\{S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l,n})$ $\displaystyle=$ $\displaystyle P(S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},l])\times
P(\\{0\in\partial_{b}R(T_{0}^{\lceil\beta n^{2}\rceil})\\}\cap E_{0,n}).$ (18)

By the choice of $h$, it holds that

$\displaystyle P(\\{0\in\partial_{b}R(T_{0}^{\lceil\beta n^{2}\rceil})\\}\cap
E_{0,n})=P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil},$ (19)

where there exist $c$, $C>0$ such that for any $n\in\mathbb{N}$

$\displaystyle c\exp((h-1)n^{2})\leq P(T_{0}<T_{b}\wedge k)^{\lceil\beta
n^{2}\rceil}\leq C\exp((h-1)n^{2}).$ (20)

By considering a time-reversal, we obtain

$\displaystyle P(S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},l])=P(0\in
R[1,l-n^{\prime}]^{c}\cap\partial_{b}R(l-n^{\prime})).$ (21)

In addition, for $m\in\mathbb{N}$, we have $P(0\in
R[1,m]^{c}\cap\partial_{b}R(m))=P(T_{0}\wedge T_{b}>m)$, and
$P(0\in\partial_{b}R(m))=P(T_{b}>m)$. Therefore, by (16), (17), (18), (19) and
(21), the desired formulas hold. ∎

######  Remark 4.1.

We substitute $T_{S_{l},l}^{\lceil\beta n^{2}\rceil}$ for $\tilde{n}$ of
$F_{n^{\prime},l,\tilde{n},n}$. That is, for any
$n^{\prime},l\in{\mathbb{N}}\cup\\{0\\}$ with $n^{\prime}\leq l$, we write

$\displaystyle F(n^{\prime},l,T_{S_{l},l}^{\lceil\beta
n^{2}\rceil},n)=\\{S_{l}\in
R[n^{\prime},l-1]^{c}\cap\partial_{b}R[n^{\prime},T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l,n}.$

By the same argument, we obtain the following: for any
$n^{\prime},l\in{\mathbb{N}}\cup\\{0\\}$ with $n^{\prime}\leq l$

$\displaystyle P(F(n^{\prime},l,T_{S_{l},l}^{\lceil\beta n^{2}\rceil},n))\leq
P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil}P(T_{0}\wedge
T_{b}>l-n^{\prime}).$ (22)

(See the argument after (18).)

######  Corollary 4.1.

For any $l\in I_{n}$ and all sufficiently large $n\in\mathbb{N}$ with
$u_{n}/n^{11}\leq u_{n}-u_{n-1}$,

$\displaystyle P(A_{l,n})\leq$ $\displaystyle P(T_{0}\wedge T_{b}>l)\times
P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil}$ $\displaystyle\times
P(T_{b}>\frac{u_{n}}{n^{11}})$ (23)

and

$\displaystyle P(A_{l,n})\geq$ $\displaystyle P(T_{0}\wedge T_{b}>l)\times
P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil}$ $\displaystyle\times
P(T_{b}>u_{n}).$ (24)

###### Proof.

For $l\in I_{n}$

$\displaystyle l=T_{S_{l},l}^{0}\leq u_{n-1}-k\lceil\beta_{d}n^{2}\rceil$

holds. Therefore, by using Lemma 4.4 with $F_{0,l,u_{n},n}=A_{l,n}$ and
substituting $u_{n-1}$ for $a$ in the assumption of Lemma 4.4 we obtain (24)
and

$\displaystyle P(A_{l,n})\leq$ $\displaystyle P(T_{0}\wedge T_{b}>l)\times
P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil}$ $\displaystyle\times
P(T_{b}>u_{n}-u_{n-1}).$

Therefore, we obtain (23) for all sufficiently large $n\in\mathbb{N}$. ∎

###  4.3 Proof of Lemmas 4.2 and 4.3 for $d\geq 3$

By Corollary 4.1, we obtain the following estimate of $P(A_{l,n})$.

######  Lemma 4.5.

There exist constants $C$, $c>0$ such that for any $n\in\mathbb{N}$ and $l\in
I_{n}$

$\displaystyle P(A_{l,n})\leq$ $\displaystyle C\exp((h-1)n^{2})$ (25)
$\displaystyle P(A_{l,n})\geq$ $\displaystyle c\exp((h-1)n^{2}).$ (26)

Moreover, for any $n\in\mathbb{N}$ and $l\in I_{n}$

$\displaystyle P(A_{l,n})=$ $\displaystyle P(T_{0}\wedge T_{b}=\infty)\times
P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil}\times P(T_{b}=\infty)$
$\displaystyle+O(\exp((h-1)n^{2}-cn^{2})).$ (27)

###### Proof.

Since (20) and (27) yield (25) and (26), we only need to prove (27). First, we
introduce some estimates of hitting times. Since
$P(S_{n}=0)=O(n^{-\frac{d}{2}})$, we obtain $P(T_{0}=n)=O(n^{-\frac{d}{2}})$
and, hence, for $d\geq 3$ and $b\in{\cal N}(0)$,

$\displaystyle P(n\leq T_{0}<\infty)$
$\displaystyle\leq\sum_{m=n}^{\infty}O(n^{-\frac{d}{2}})=O(n^{-\frac{d}{2}+1}).$
(28)

In addition, by the Markov property and the translation invariance for
$b^{\prime}$ we have

$\displaystyle P(T_{0}\geq a+1)=\frac{1}{2d}\sum_{b^{\prime}\in{\cal
N}(0)}P^{b^{\prime}}(T_{0}\geq a)=\frac{1}{2d}\sum_{b^{\prime}\in{\cal
N}(0)}P(T_{-b^{\prime}}\geq a)=P(T_{b}\geq a),$ (29)

for $a\in\mathbb{N}$. Hence, (28) yields

$\displaystyle P(n\leq T_{b}<\infty)$ $\displaystyle=O(n^{-\frac{d}{2}+1}).$
(30)

Moreover, it holds that

$\displaystyle P(n\leq T_{0}\wedge T_{b}<\infty)\leq P(\\{n\leq
T_{0}<\infty\\}\cup\\{n\leq T_{b}<\infty\\})$ $\displaystyle=$ $\displaystyle
P(n\leq T_{0}<\infty)+P(n\leq T_{b}<\infty),$

and so with (28) and (30),

$\displaystyle P(n\leq T_{0}\wedge T_{b}<\infty)$
$\displaystyle=O(n^{-\frac{d}{2}+1}).$ (31)

Therefore, by (28), (30) and (31) there exists $c>0$ for any $n\in\mathbb{N}$
and $l\in I_{n}$

$\displaystyle P(T_{0}\wedge T_{b}>l)=P(T_{0}\wedge
T_{b}=\infty)+O(\exp(-cn^{2})),$ (32) $\displaystyle
P(T_{b}>u_{n})=P(T_{b}=\infty)+O(\exp(-cn^{2})),$ (33) $\displaystyle
P(T_{b}>\frac{u_{n}}{n^{11}})=P(T_{b}=\infty)+O(\exp(-cn^{2})).$ (34)

Substituting (32), (33) and (34) for the right hand sides of (23) and (24), we
obtain the desired formula. ∎

###### Proof of Lemma 4.2 for $d\geq 3$.

Since for all sufficiently large $n\in\mathbb{N}$

$\displaystyle\sharp I_{n}\geq
u_{n-1}-k\lceil\beta_{d}n^{2}\rceil-\frac{u_{n-1}}{n^{2}}\geq cu_{n-1}$ (35)

holds, by (26) and the definition of $Q_{n}$ in (10), we obtain

$\displaystyle EQ_{n}\geq\sharp I_{n}\times\min_{l\in I_{n}}P(A_{l,n})\geq
c\exp(hn^{2}-2n),$

as required. ∎

To prove Lemma 4.3, we decompose $I_{n}\times I_{n}$ into three parts
$J_{n,j}$, $j=1,2,3$ defined by

$\displaystyle J_{n,1}:=\\{(l,l^{\prime})\in I_{n}\times I_{n}:0\leq
l^{\prime}-l\leq k\lceil\beta_{d}n^{2}\rceil\\},$ $\displaystyle
J_{n,2}:=\\{(l,l^{\prime})\in I_{n}\times
I_{n}:k\lceil\beta_{d}n^{2}\rceil<l^{\prime}-l\leq
2\lceil\frac{u_{n-1}}{n^{10}}\rceil\\},$ $\displaystyle
J_{n,3}:=\\{(l,l^{\prime})\in I_{n}\times
I_{n}:2\lceil\frac{u_{n-1}}{n^{10}}\rceil<l^{\prime}-l\\}.$

For all sufficiently large $n\in\mathbb{N}$,
$k\lceil\beta_{d}n^{2}\rceil<2\lceil u_{n-1}/n^{10}\rceil$ holds and hence
$J_{n,2}$ is non-empty. By a simple computation,

$\displaystyle\mathrm{Var}(Q_{n})=$ $\displaystyle EQ_{n}^{2}-(EQ_{n})^{2}$
$\displaystyle\leq$ $\displaystyle 2\sum_{l,l^{\prime}\in I_{n},l\leq
l^{\prime}}(E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]-E[1_{A_{l,n}}]E[1_{A_{l^{\prime},n}}])$
$\displaystyle\leq$ $\displaystyle 2(\sum_{(l,l^{\prime})\in
J_{n,1}}E[1_{A_{l,n}}]+\sum_{(l,l^{\prime})\in
J_{n,2}}E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]$ $\displaystyle+$
$\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}(E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]-E[1_{A_{l,n}}]E[1_{A_{l^{\prime},n}}])).$
(36)

######  Lemma 4.6.

There exists $C>0$ such that for any $n\in\mathbb{N}$

$\displaystyle\sum_{(l,l^{\prime})\in J_{n,1}}E[1_{A_{l,n}}]\leq
Cn^{2}\exp(hn^{2}-2n)$ (37)

and

$\displaystyle\sum_{(l,l^{\prime})\in
J_{n,2}}E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]\leq
C\exp(2hn^{2}-4n)\times\frac{1}{n^{10}}.$ (38)

######  Remark 4.2.

This Lemma also holds for $d=2$ by the same proof.

###### Proof.

First, we show (37). By the definition, we have $\sharp J_{n,1}\leq
k\lceil\beta_{d}n^{2}\rceil u_{n-1}$. Thus, (25) yields

$\displaystyle\sum_{(l,l^{\prime})\in J_{n,1}}E[1_{A_{l,n}}]\leq\sharp
J_{n,1}\times\max_{l\in I_{n}}P(A_{l,n})\leq Cn^{2}\exp(hn^{2}-2n).$

Hence, we obtain (37). To show (38), let us introduce additional notations. We
define

$\displaystyle\tilde{A}_{l,n}:=\\{S_{l}\in\partial_{b}R[l+1,T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l,n}.$ (39)

Note that

$\displaystyle\tilde{A}_{l,n}=F(l,l,T_{S_{l},l}^{\lceil\beta n^{2}\rceil},n),$
$\displaystyle\tilde{A}_{l,n}\cap\\{S_{l}\in
R(l-1)^{c}\cap\partial_{b}R(l)\\}\cap\\{S_{l}\in\partial_{b}R[T_{S_{l},l}^{\lceil\beta
n^{2}\rceil},u_{n}]\\}=A_{l,n}$

and

$\displaystyle\tilde{A}_{l,n}\in\sigma\\{S_{j}-S_{l}:j\in[l,T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}]\\}.$

By Remark 4.1, we obtain

$\displaystyle P(\tilde{A}_{l,n})=P(F(l,l,T_{S_{l},l}^{\lceil\beta
n^{2}\rceil},n))\leq C\exp((h-1)n^{2}).$ (40)

We obtain (38) as follows: by the definition of $A_{l,n}$ and
$\tilde{A}_{l,n}$, we have ${A}_{l,n}\subset\tilde{A}_{l,n}$ and
${A}_{l^{\prime},n}\subset\tilde{A}_{l^{\prime},n}$. In addition,
$\tilde{A}_{l^{\prime},n}$ is independent of $\tilde{A}_{l,n}$ for
$(l,l^{\prime})\in J_{n,2}$. Thus,

$\displaystyle E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]\leq
E[1_{\tilde{A}_{l,n}}1_{\tilde{A}_{l^{\prime},n}}]=E[1_{\tilde{A}_{l,n}}]E[1_{\tilde{A}_{l^{\prime},n}}],$

and so by (40),

$\displaystyle\sum_{(l,l^{\prime})\in
J_{n,2}}E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]=$ $\displaystyle\sharp
J_{n,2}\times C(\exp((h-1)n^{2}))^{2}$ $\displaystyle\leq$ $\displaystyle
C\exp(2hn^{2}-4n)\times\frac{1}{n^{10}}.$

Therefore, we obtain (38). ∎

###### Proof of Lemma 4.3 for $d\geq 3$.

We estimate the last sum appearing in (36). To this end, set

$\displaystyle A^{\prime}_{l,n}:=$ $\displaystyle\\{S_{l}\in
R(l-1)^{c}\cap\partial_{b}R(l+\lceil\frac{u_{n-1}}{n^{10}}\rceil)\\}\cap
E_{l,n}$ (41) $\displaystyle A^{\prime\prime}_{l^{\prime},n}:=$
$\displaystyle\\{S_{l^{\prime}}\in
R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime}-1]^{c}\cap\partial_{b}R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,u_{n}]\\}\cap
E_{l^{\prime},n}.$ (42)

Note that

$\displaystyle
A^{\prime}_{l,n}=F_{0,l,l+\lceil\frac{u_{n-1}}{n^{10}}\rceil,n},\quad
A^{\prime\prime}_{l^{\prime},n}=F_{l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime},u_{n},n}.$

By Lemma 4.4, (32) and (34), we can estimate $P(A^{\prime}_{l,n})$ and
$P(A^{\prime\prime}_{l^{\prime},n})$ as follows: for any $l\in I_{n}$ and all
sufficiently large $n\in\mathbb{N}$ with $u_{n-1}/n^{11}\leq(\lceil
u_{n-1}/n^{10}\rceil-k\lceil\beta n^{2}\rceil)\wedge l$

$\displaystyle
P(A^{\prime}_{l,n})=P(F_{0,l,l+\lceil\frac{u_{n-1}}{n^{10}}\rceil,n})$
$\displaystyle\leq$ $\displaystyle P(T_{0}\wedge T_{b}>l)\times
P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil}\times
P(T_{b}>\frac{u_{n-1}}{(n-1)^{11}})$ $\displaystyle\leq$ $\displaystyle
P(T_{b}=\infty)P(T_{0}\wedge T_{b}=\infty)P(T_{0}<T_{b}\wedge k)^{\lceil\beta
n^{2}\rceil}+O(\exp((h-1)n^{2}-cn^{2}))$ (43)

and for any $l^{\prime}\in I_{n}$ and all sufficiently large $n\in\mathbb{N}$
with $u_{n}/n^{11}\leq u_{n}-(l^{\prime}+k\lceil\beta n^{2}\rceil)$ and
$u_{n-1}/n^{11}\leq\lceil u_{n-1}/n^{10}\rceil\leq l^{\prime}$

$\displaystyle
P(A^{\prime\prime}_{l^{\prime},n})=P(F_{l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime},u_{n},n})$
$\displaystyle\leq$ $\displaystyle P(T_{0}\wedge
T_{b}>\lceil\frac{u_{n-1}}{n^{10}}\rceil)\times P(T_{0}<T_{b}\wedge
k)^{\lceil\beta n^{2}\rceil}\times P(T_{b}>\frac{u_{n}}{n^{11}})$
$\displaystyle\leq$ $\displaystyle P(T_{b}=\infty)P(T_{0}\wedge
T_{b}=\infty)P(T_{0}<T_{b}\wedge k)^{\lceil\beta
n^{2}\rceil}+O(\exp((h-1)n^{2}-cn^{2})).$ (44)

Therefore, by (27), (43) and (44), we obtain

$\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}(E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]-E[1_{A_{l,n}}]E[1_{A_{l^{\prime},n}}])$
$\displaystyle\leq$ $\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}(E[1_{A^{\prime}_{l,n}}]E[1_{A^{\prime\prime}_{l^{\prime},n}}]-E[1_{A_{l,n}}]E[1_{A_{l^{\prime},n}}])$
$\displaystyle\leq$ $\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}(P(T_{b}=\infty)^{2}P(T_{0}\wedge T_{b}=\infty)^{2}P(T_{0}<T_{b}\wedge
k)^{2\lceil\beta n^{2}\rceil}$ $\displaystyle-P(T_{b}=\infty)^{2}P(T_{0}\wedge
T_{b}=\infty)^{2}P(T_{0}<T_{b}\wedge k)^{2\lceil\beta n^{2}\rceil}$
$\displaystyle+O(\exp(2(h-1)n^{2}-cn^{2})))$ $\displaystyle\leq$
$\displaystyle C(\exp(hn^{2}-2n))^{2}\times\exp(-cn^{2}).$ (45)

By (37), (38) and (45), the right hand side of (36) is bounded by
$\exp(2hn^{2}-4n)\times 1/n^{10}$. This completes the proof of Lemma 4.3 for
$d\geq 3$. ∎

###  4.4 Proof of Lemmas 4.2 and 4.3 for $d=2$

First, we state a lemma that is important for our proof of Lemma 4.3.

######  Lemma 4.7.

There exists $C>0$ such that for any $n\in\mathbb{N}$ and
$x\in{\mathbb{Z}^{2}}$ with $0<|x|<n\sqrt{u_{n-1}}$

$\displaystyle P(T_{0}\wedge
T_{x}>\lceil\frac{u_{n}}{n}\rceil)\leq\frac{\pi}{(n+1)^{2}}+C\frac{\log
n}{n^{4}}.$ (46)

Moreover, it holds that for $b\in{\cal N}(0)$

$\displaystyle P(T_{b}\wedge
T_{x+b}>\lceil\frac{u_{n}}{n}\rceil)\leq\frac{\pi}{(n+1)^{2}}+C\frac{\log
n}{n^{4}}.$ (47)

###### Proof.

We prove only the first claim since the second one follows from (46) by a
similar observation as in (29). Decomposing the whole event by means of the
last exit time from $\\{0,x\\}$ by time $\lceil u_{n}/n\rceil$, we obtain

$\displaystyle 1=$ $\displaystyle\sum_{k=0}^{\lceil
u_{n}/n\rceil}P(S_{k}=0)P^{0}(0,x\notin R[1,\lceil\frac{u_{n}}{n}\rceil-k])$
$\displaystyle+$ $\displaystyle\sum_{k=0}^{\lceil
u_{n}/n\rceil}P(S_{k}=x)P^{x}(0,x\notin R[1,\lceil\frac{u_{n}}{n}\rceil-k]).$
(48)

By the local central limit theorem (see Theorem $1.2.1$ in [8]), there exists
$c>0$ such that for any $k\in\mathbb{N}\cup\\{0\\}$, $x\in{\mathbb{Z}^{2}}$

$\displaystyle\begin{cases}&\displaystyle{P(S_{k}=x)\geq\frac{2}{\pi
k}\exp(-\frac{|x|^{2}}{k})-\frac{c}{k^{2}}}\quad\quad\text{ if
}k\rightleftharpoons x\\\
&\displaystyle{P(S_{k}=x)=0}\quad\quad\quad\quad\quad\quad\quad\quad\text{ if
}k+1\rightleftharpoons x,\end{cases}$

where $k\rightleftharpoons x=(x_{1},x_{2})(\in{\mathbb{Z}^{2}})$ means that
$k+x_{1}+x_{2}$ is even. Let

$\displaystyle\gamma(n)=P(T_{0}\wedge
T_{x}>\lceil\frac{u_{n}}{n}\rceil)=P(0,x\notin
R[1,\lceil\frac{u_{n}}{n}\rceil]).$

By the invariance property of $S_{n}$ under an isomorphism of
${\mathbb{Z}^{2}}$, for $a\leq\lceil u_{n}/n\rceil$

$\gamma(n)\leq P^{0}(0,x\notin R[1,a])=P^{0}(-x,0\notin
R[1,a])=P^{x}(0,x\notin R[1,a]).$

Note that $\sum_{k=1}^{\lceil u_{n}/n\rceil}\frac{2}{\pi
k}1_{\\{k\rightleftharpoons x\\}}=\sum_{m=1}^{\lceil
u_{n}/n\rceil/2}\frac{1}{\pi m}>\frac{1}{\pi}\log(1+\lceil u_{n}/n\rceil/2)$
holds. Then, by (48) we obtain

$\displaystyle 1\geq$ $\displaystyle\bigg{(}\sum_{k=1}^{\lceil
u_{n}/n\rceil}(\frac{2}{\pi k}-\frac{c}{k^{2}})1_{\\{k\rightleftharpoons
0\\}}+\sum_{k=1}^{\lceil u_{n}/n\rceil}\bigg{(}\frac{2}{\pi
k}\exp(-\frac{|x|^{2}}{k})-\frac{c}{k^{2}}\bigg{)}1_{\\{k\rightleftharpoons
x\\}}\bigg{)}\gamma(n)$ $\displaystyle\geq$
$\displaystyle\bigg{(}\frac{n^{2}}{\pi}-\frac{\log n}{\pi}-c+\sum_{k=\lceil
n|x|^{2}\rceil}^{\lceil u_{n}/n\rceil}\frac{2}{\pi
k}\exp(-\frac{|x|^{2}}{k})1_{\\{k\rightleftharpoons x\\}}\bigg{)}\gamma(n)$
$\displaystyle\geq$ $\displaystyle\bigg{(}\frac{n^{2}}{\pi}-\frac{\log
n}{\pi}-c+\sum_{k=\lceil n|x|^{2}\rceil}^{\lceil u_{n}/n\rceil}\frac{2}{\pi
k}\exp(-\frac{1}{n})1_{\\{k\rightleftharpoons x\\}}\bigg{)}\gamma(n)$
$\displaystyle\geq$ $\displaystyle\bigg{(}\frac{n^{2}}{\pi}-\frac{\log
n}{\pi}-c+\sum_{k=n^{3}u_{n-1}}^{\lceil u_{n}/n\rceil}\frac{2}{\pi
k}\exp(-\frac{1}{n})1_{\\{k\rightleftharpoons x\\}}\bigg{)}\gamma(n)$
$\displaystyle\geq$ $\displaystyle\frac{1}{\pi}\bigg{(}n^{2}-\log
n-c+n^{2}-(n-1)^{2}-4\log n-(1-\exp(-\frac{1}{n}))2n\bigg{)}\gamma(n)$
$\displaystyle\geq$ $\displaystyle\frac{1}{\pi}\bigg{(}(n+1)^{2}-5\log
n-c\bigg{)}\gamma(n).$

Thus the assertion follows from an easy rearrangement. ∎

To prove Lemma 4.2, we first introduce the following lemma.

######  Lemma 4.8.

There exist constants $C$, $c>0$ such that for any $n\in\mathbb{N}$ and $l\in
I_{n}$

$\displaystyle P(A_{l,n})$ $\displaystyle\leq\frac{C\exp((h-1)n^{2})}{n^{4}},$
(49) $\displaystyle P(A_{l,n})$
$\displaystyle\geq\frac{c\exp((h-1)n^{2})}{n^{4}}.$ (50)

In addition, for any $n\in\mathbb{N}\cap\\{1\\}^{c}$ and $l\in I_{n}$

$\displaystyle P(A_{l,n})=\frac{\pi^{2}P(T_{0}<T_{b}\wedge k)^{\lceil\beta
n^{2}\rceil}}{2n^{2}(n-1)^{2}}+O(\frac{\exp((h-1)n^{2})\log n}{n^{6}}).$ (51)

###### Proof.

Since (20) and (51) yield (49) and (50), we only prove (51). First we
introduce the following estimates: for any $M>0$ there exists $C>0$ such that
for any $n\in\mathbb{N}$

$\displaystyle\frac{\pi}{n^{2}}-C\frac{\log n}{n^{4}}\leq
P(T_{b}>\frac{u_{n}}{n^{M}})\leq\frac{\pi}{n^{2}}+C\frac{\log n}{n^{4}},$ (52)
$\displaystyle\frac{\pi}{2n^{2}}-C\frac{\log n}{n^{4}}\leq P(T_{0}\wedge
T_{b}>\frac{u_{n}}{n^{M}})\leq\frac{\pi}{2n^{2}}+C\frac{\log n}{n^{4}}.$ (53)

Since we know

$P(T_{0}>n)=\frac{\pi}{\log n}+O\bigg{(}\frac{1}{(\log n)^{2}}\bigg{)}$

(see $(2.5)$ in [4]), the assertion (52) follows by a simple calculation of
(29). For the latter assertion, we already know a weaker estimate of (53)
involving only the leading term. (See Lemma $3.3$ in [11].) We can obtain the
error term of (53) by modifying the proof in [11] along the argument in [4] in
a straightforward way. Thus, we omit the proofs of (52) and (53). From (52)
and (53), we already have estimates of each term in (23) and (24). Indeed, for
any $l\in I_{n}$ and all sufficiently large $n\in\mathbb{N}$, (52) yields

$\displaystyle P(T_{b}>\frac{u_{n}}{n^{11}})=\frac{\pi}{n^{2}}+O(\frac{\log
n}{n^{4}}),$ (54) $\displaystyle P(T_{b}>u_{n})=\frac{\pi}{n^{2}}+O(\frac{\log
n}{n^{4}}).$

Since $l\in I_{n}$, (53) implies

$\displaystyle P(T_{0}\wedge T_{b}>l)=\frac{\pi}{2(n-1)^{2}}+O(\frac{\log
n}{n^{4}}).$ (55)

Therefore, by substituting these estimates for the right hand sides of (23)
and (24) we obtain the desired formula. ∎

###### Proof of Lemma 4.2 for $d=2$.

Recall (35). Then, (10) and (50) yield

$\displaystyle EQ_{n}\geq\sum_{l\in
I_{n}}\frac{c\exp((h-1)n^{2})}{n^{4}}\geq\frac{c\exp(hn^{2}-2n)}{n^{4}}$

for any $n\in\mathbb{N}$, as desired. ∎

###### Proof of Lemma 4.3 for $d=2$.

By the same argument for $d\geq 3$, we obtain (36) for $d=2$. We consider the
estimate of the right hand side of (36). The first term and the second term of
the right hand side of (36) are already estimated by Lemma 4.6. (Note Remark
4.2.) To estimate the third term, we will give a uniform upper bound of
$P(A_{l,n}\cap A_{l^{\prime},n})$ for $(l,l^{\prime})\in J_{n,3}$. Here,
uniform means that the bound is independent of the choice of
$(l,l^{\prime})\in J_{n,3}$. Instead of using
$A^{\prime\prime}_{l^{\prime},n}$ in (42) as we did when $d\geq 3$, we use
more complicated events. Let

$\displaystyle
A^{\prime\prime\prime}_{l,l^{\prime},n}:=A^{\prime\prime}_{l^{\prime},n}\cap\\{S_{l}\in\partial_{b}R[T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil},u_{n}]\\}.$

Recall the definition of $A^{\prime}_{l,n}$ in (42). By the definition of
$A_{l,n}$ and ${A^{\prime}}_{l,n}$, we have ${A}_{l,n}\subset
A^{\prime}_{l,n}$ and ${A}_{l,n}\cap A_{l^{\prime},n}\subset
A^{\prime\prime\prime}_{l,l^{\prime},n}$. Note that $A^{\prime}_{l,n}$ is not
independent of $A^{\prime\prime\prime}_{l,l^{\prime},n}$ for
$(l,l^{\prime})\in J_{n,3}$. Denote the event
$0<|S_{l^{\prime}}-S_{l}|<n\sqrt{u_{n-1}}$ by $D_{1}$, and the event
$|S_{l^{\prime}}-S_{l}|\geq n\sqrt{u_{n-1}}$ by $D_{2}$. Since
$S_{l^{\prime}}\notin R(l^{\prime}-1)$ on $A_{l^{\prime},n}$, we have
$\\{S_{l}=S_{l^{\prime}}\\}\cap A_{l^{\prime},n}=\emptyset$ for
$l<l^{\prime}$. Thus, $A_{l^{\prime},n}=(A_{l^{\prime},n}\cap
D_{1})\cup(A_{l^{\prime},n}\cap D_{2})$ and therefore $A_{l,n}\cap
A_{l^{\prime},n}\subset(A^{\prime}_{l,n}\cap
A^{\prime\prime\prime}_{l,l^{\prime},n}\cap
D_{1})\cup(\tilde{A}_{l,n}\cap\tilde{A}_{l^{\prime},n}\cap D_{2})$ holds.
Then, the following holds:

$\displaystyle
E[1_{A_{l,n}}1_{A_{l^{\prime},n}}]\leq(E[1_{A^{\prime}_{l,n}}1_{A^{\prime\prime\prime}_{l,l^{\prime},n}}1_{D_{1}}]+E[1_{\tilde{A}_{l,n}}1_{\tilde{A}_{l^{\prime},n}}1_{D_{2}}]).$
(56)

Hence, by putting (54) and (55) into the right hand side of the inequalities
given in Lemma 4.4 we can see that there exists $C>0$ such that for any $l\in
I_{n}$ and all sufficiently large $n\in\mathbb{N}$ with
$u_{n-1}/(n-1)^{11}\leq(\lceil u_{n-1}/n^{10}\rceil-k\lceil\beta
n^{2}\rceil)\wedge l$

$\displaystyle
P(A^{\prime}_{l,n})=P(F_{0,l,l+\lceil\frac{u_{n-1}}{n^{10}}\rceil,n})$
$\displaystyle\leq$ $\displaystyle P(T_{0}\wedge T_{b}>l)\times
P(T_{0}<T_{b}\wedge k)^{\lceil\beta n^{2}\rceil}\times
P(T_{b}>\frac{u_{n-1}}{(n-1)^{11}})$ $\displaystyle\leq$
$\displaystyle\frac{\pi}{2(n-1)^{2}}\times P(T_{0}<T_{b}\wedge k)^{\lceil\beta
n^{2}\rceil}\times\frac{\pi}{(n-1)^{2}}+C\frac{\exp((h-1)n^{2})\log
n}{n^{6}}.$ (57)

Taking tha conditional probability of the event $A^{\prime}_{l,n}\cap
A^{\prime\prime\prime}_{l,l^{\prime},n}\cap D_{1}$ on ${\cal
F}(T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta n^{2}\rceil})$ and using (57),
we see that for any $l,l^{\prime}\in I_{n}$,

$\displaystyle
E[1_{A^{\prime}_{l,n}}1_{A^{\prime\prime\prime}_{l,l^{\prime},n}}1_{D_{1}}]$
$\displaystyle=$ $\displaystyle
E[1_{A^{\prime}_{l,n}}1_{D_{1}}1\\{\\{S_{l^{\prime}}\notin
R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime}-1],$ $\displaystyle
S_{l^{\prime}}\in\partial_{b}R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l^{\prime}n}\\}$ $\displaystyle
P(S_{l^{\prime}},S_{l}\in\partial_{b}R[T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil},u_{n}]|{\cal F}(T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil}))].$ (58)

Note that $T_{x^{\prime},l}^{\lceil\beta n^{2}\rceil}<u_{n-1}$ if
$(l,l^{\prime})\in J_{n,3}$. Hence, by (47), we see that for all sufficiently
large $n\in\mathbb{N}$ with $u_{n}-u_{n-1}\geq u_{n}/n$ and
$x,x^{\prime}\in{\mathbb{Z}^{2}}$ with $0<|x-x^{\prime}|<n\sqrt{u_{n-1}}$, it
holds that

$\displaystyle P(x,x^{\prime}\in\partial_{b}R[T_{x^{\prime},l}^{\lceil\beta
n^{2}\rceil},u_{n}]|{\cal F}(T_{x^{\prime},l}^{\lceil\beta n^{2}\rceil}))$
$\displaystyle=$ $\displaystyle
P(0,x-x^{\prime}\in\partial_{b}R(u_{n}-t))|_{t=T_{x^{\prime},l}^{\lceil\beta
n^{2}\rceil}}$
$\displaystyle\leq\max_{0<|x-x^{\prime}|<n\sqrt{u_{n-1}}}P(T_{b}\wedge
T_{x-x^{\prime}+b}>\lceil\frac{u_{n}}{n}\rceil)\leq\frac{\pi}{(n+1)^{2}}+\frac{C\log
n}{n^{4}}.$

By the inequalities in the last line restricted to $x=S_{l}$ and
$x^{\prime}=S_{l^{\prime}}$, the right hand side of (58) is bounded by

$\displaystyle E[1_{A^{\prime}_{l,n}}1\\{\\{S_{l^{\prime}}\in
R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime}-1]^{c}\cap\partial_{b}R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil}\\}\cap E_{l^{\prime},n}\\}]$
$\displaystyle\times\bigg{(}\frac{\pi}{(n+1)^{2}}+\frac{C\log
n}{n^{4}}\bigg{)}.$ (59)

Moreover, it holds that

$\displaystyle E[1_{A^{\prime}_{l,n}}1\\{\\{S_{l^{\prime}}\in
R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime}-1]^{c}\cap\partial_{b}R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l^{\prime},n}\\}]$ $\displaystyle=$ $\displaystyle
E[1_{A^{\prime}_{l,n}}]E[1\\{\\{S_{l^{\prime}}\in
R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime}-1]^{c}\cap\partial_{b}R[l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil}]\\}\cap E_{l^{\prime},n}\\}]$ $\displaystyle=$ $\displaystyle
E[1_{A^{\prime}_{l,n}}]E[1F(l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime},T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil},n)].$ (60)

By substituting (22) for Remark 4.1, we obtain

$\displaystyle
P(F(l^{\prime}-\lceil\frac{u_{n-1}}{n^{10}}\rceil,l^{\prime},T_{S_{l^{\prime}},l^{\prime}}^{\lceil\beta
n^{2}\rceil},n))$ $\displaystyle\leq$ $\displaystyle\frac{\pi
P(T_{0}<T_{b}\wedge k)^{\lceil\beta
n^{2}\rceil}}{2(n-1)^{2}}+C\frac{\exp((h-1)n^{2})\log n}{n^{4}}.$

Therefore, (57), (59) and (60) yield

$\displaystyle
E[1_{A^{\prime}_{l,n}}1_{A^{\prime\prime\prime}_{l,l^{\prime},n}}1_{D_{1}}]$
$\displaystyle\leq$ $\displaystyle\frac{\pi^{4}P(T_{0}<T_{b}\wedge
k)^{2\lceil\beta
n^{2}\rceil}}{4(n-1)^{6}(n+1)^{2}}+C\frac{\exp(2(h-1)n^{2})\log n}{n^{10}}.$
(61)

Now, we turn to the estimate of
$E[1_{\tilde{A}_{l,n}}1_{\tilde{A}_{l^{\prime},n}}1_{D_{2}}]$. From the large
deviation result (see $(11)$ in [9]), there exist $C$, $c>0$ such that for any
$n$, $m\in\mathbb{N}\cap\\{1\\}^{c}$ with $m\leq u_{n-1}$

$\displaystyle P(|S_{m}|\geq n\sqrt{u_{n-1}})\leq Ce^{-cn}.$ (62)

Thus, by the strong Markov property, we can estimate
$E[1_{\tilde{A}_{l,n}}1_{\tilde{A}_{l^{\prime},n}}1_{D_{2}}]$ for any
$(l,l^{\prime})\in J_{n,3}$ as

$\displaystyle E[1_{\tilde{A}_{l,n}}1_{\tilde{A}_{l^{\prime},n}}1_{D_{2}}]=$
$\displaystyle E[1_{\tilde{A}_{l,n}}1_{D_{2}}]E[1_{\tilde{A}_{l^{\prime},n}}]$
$\displaystyle=$ $\displaystyle E[1_{\tilde{A}_{l,n}}E[1_{D_{2}}|{\cal
F}(T_{S_{l},l}^{\lceil\beta n^{2}\rceil})]]E[1_{\tilde{A}_{l^{\prime},n}}]$
$\displaystyle=$ $\displaystyle
E[1_{\tilde{A}_{l,n}}P(|S_{l^{\prime}-l-t}|\geq
n\sqrt{u_{n-1}})_{t=T_{S_{l},l}^{\lceil\beta
n^{2}\rceil}}]E[1_{\tilde{A}_{l^{\prime},n}}]$ $\displaystyle\leq$
$\displaystyle E[1_{\tilde{A}_{l,n}}\max_{|l-l^{\prime}|\leq
u_{n-1}}P(|S_{l^{\prime}-l}|\geq
n\sqrt{u_{n-1}})]E[1_{\tilde{A}_{l^{\prime},n}}]$ $\displaystyle=$
$\displaystyle E[1_{\tilde{A}_{l,n}}]\max_{m\leq u_{n-1}}P(|S_{m}|\geq
n\sqrt{u_{n-1}})E[1_{\tilde{A}_{l^{\prime},n}}]$ $\displaystyle\leq$
$\displaystyle C\frac{\exp(2(h-1)n^{2})}{e^{cn}}.$ (63)

The last inequality comes from (40) and (62). Finally, by (51), (61) and (63),
we obtain the following estimate. Since $\sharp J_{n,3}\leq(u_{n-1})^{2}$, for
any $n\in{\mathbb{N}}\cap\\{0\\}^{c}$,

$\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}(E[1_{A^{\prime}_{l,n}}1_{A^{\prime\prime\prime}_{l,l^{\prime},n}}1_{D_{1}}]+E[1_{\tilde{A}_{l,n}}1_{\tilde{A}_{l^{\prime},n}}1_{D_{2}}]-E[1_{A_{l,n}}]E[1_{A_{l^{\prime},n}}])$
$\displaystyle\leq$ $\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}\bigg{(}\frac{\pi^{4}P(T_{0}<T_{b}\wedge k)^{2\lceil\beta
n^{2}\rceil}}{4(n-1)^{6}(n+1)^{2}}-\frac{\pi^{4}P(T_{0}<T_{b}\wedge
k)^{2\lceil\beta n^{2}\rceil}}{4(n-1)^{4}n^{4}}$
$\displaystyle+C\frac{\exp(2(h-1)n^{2})}{e^{cn}}+C\frac{\exp(2(h-1)n^{2})\log
n}{n^{10}}\bigg{)}$ $\displaystyle\leq$ $\displaystyle
C\sum_{(l,l^{\prime})\in J_{n,3}}\frac{\exp(2(h-1)n^{2})\log n}{n^{10}}$
$\displaystyle\leq$ $\displaystyle
C\bigg{(}\frac{\exp(hn^{2}-2n)}{n^{4}}\bigg{)}^{2}\times\frac{\log n}{n^{2}}.$
(64)

The second inequality comes from the fact that there exists $C>0$ such that
for any $n\in\mathbb{N}\cap\\{1\\}^{c}$

$\displaystyle\frac{1}{(n-1)^{6}(n+1)^{2}}-\frac{1}{(n-1)^{4}n^{4}}\leq\frac{C}{n^{10}}.$
(65)

By (37), (38) and (64), the right hand side of (36) is bounded by
$(\exp(hn^{2}-2n)/n^{4})^{2}\times\log n/n^{2}$. Therefore, we obtain the
proof of Lemma 4.3 for $d=2$. ∎

######  Remark 4.3.

We observe what happens if we try to estimate the third term of the right hand
side of (36) in the case $d=2$ by the same argument as in the case $d\geq 3$.
Recall the definition of $A^{\prime\prime}_{l^{\prime},n}$ in (42). Then, by
substituting (22) for Lemma 4.4, we can see that for any $l^{\prime}\in
I_{n}$,

$P(A^{\prime\prime}_{l^{\prime},n})\leq\frac{\pi^{2}P(T_{0}<T_{b}\wedge
k)^{\lceil\beta n^{2}\rceil}}{2(n-1)^{2}n^{2}}+O(\frac{\exp((h-1)n^{2})\log
n}{n^{6}}).$

Hence, if we choose $A^{\prime\prime}_{l^{\prime},n}$ instead of
$A^{\prime\prime\prime}_{l,l^{\prime},n}$ in (56),

$\displaystyle E[1_{A^{\prime}_{l,n}}1_{A^{\prime\prime}_{l^{\prime},n}}]=$
$\displaystyle E[1_{A^{\prime}_{l,n}}]E[1_{A^{\prime\prime}_{l^{\prime},n}}]$
$\displaystyle\leq$ $\displaystyle\frac{\pi^{4}P(T_{0}<T_{b}\wedge
k)^{2\lceil\beta n^{2}\rceil}}{4(n-1)^{6}n^{2}}+O(\frac{\exp(2(h-1)n^{2})\log
n}{n^{10}}).$

Based on this estimate, we obtain

$\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}(E[1_{A^{\prime}_{l,n}}1_{A^{\prime\prime}_{l^{\prime},n}}]-E[1_{A_{l,n}}]E[1_{A_{l^{\prime},n}}])$
$\displaystyle\leq$ $\displaystyle\sum_{(l,l^{\prime})\in
J_{n,3}}\bigg{(}\frac{\pi^{4}P(T_{0}<T_{b}\wedge k)^{2\lceil\beta
n^{2}\rceil}}{4(n-1)^{6}n^{2}}-\frac{\pi^{4}P(T_{0}<T_{b}\wedge
k)^{2\lceil\beta n^{2}\rceil}}{4(n-1)^{4}n^{4}}$
$\displaystyle+O(\frac{\exp(2(h-1)n^{2})\log n}{n^{10}})\bigg{)}$
$\displaystyle\leq$ $\displaystyle C\sum_{(l,l^{\prime})\in
J_{n,3}}\frac{\exp(2(h-1)n^{2})}{n^{9}}$ $\displaystyle\leq$ $\displaystyle
C(\frac{\exp(hn^{2}-2n)}{n^{4}})^{2}\times\frac{1}{n}.$

The second inequality comes from the fact that there exists $C>0$ such that
for any $n\in\mathbb{N}\cap\\{1\\}^{c}$

$\displaystyle\frac{1}{(n-1)^{6}n^{2}}-\frac{1}{(n-1)^{4}n^{4}}\leq\frac{C}{n^{9}}.$

(Compare with (65).) Consequently, with the aid of Lemma 4.2, we obtain

$\frac{\mathrm{Var}(Q_{n})}{c^{2}(EQ_{n})^{2}}\leq\frac{C}{n}.$

This estimate is not sufficient to apply the Borel-Cantelli lemma as we did in
the proof of Theorem 2.1 in Section $4.1$. Thus, we cannot use the Borel-
Cantelli lemma here.

##  5 Proof of Theorem 2.2

###  5.1 Proof of the upper bound of Theorem 2.2 for $d\geq 2$

###### Proof.

Note that if we substitute $\beta_{d}\delta$ for $\beta$ in Lemma 3.1, we
obtain $E[\tilde{\Theta}_{n}(\beta_{d}\delta)]=O(n^{1-\delta})$. By the
Chebyshev inequality, we find that for any $\epsilon>0$, there exists $C>0$

$\displaystyle
P\bigg{(}\tilde{\Theta}_{n}(\beta_{d}\delta)\geq\bigg{(}\frac{n}{2}\bigg{)}^{1-\delta+\epsilon}\bigg{)}<Cn^{-\epsilon}.$

Using the Borel-Cantelli lemma, we see that the events
$\\{\tilde{\Theta}_{2^{k}}(\beta_{d}\delta)\geq
2^{(k-1)(1-\delta+\epsilon)}\\}$ happen only finitely often with probability
one. Hence, it holds that for any $\epsilon>0$

$\displaystyle\limsup_{k\to\infty}\frac{\log\tilde{\Theta}_{2^{k}}(\beta_{d}\delta)}{\log
2^{k-1}}\leq 1-\delta+\epsilon\quad{a.s.}$ (66)

Note that if $K(n,x)\geq\lceil\beta_{d}\delta\log n\rceil$, for all $k$,
$n\in{\mathbb{N}}$ with $2^{k-1}\leq n<2^{k}$
$T_{x}^{\lceil\beta_{d}\delta\log 2^{k-1}\rceil}\leq n$ and
$K(2^{k},x)\geq\lceil\beta_{d}\delta\log 2^{k-1}\rceil$, and hence
${\Theta}_{n}(\delta)\leq\tilde{\Theta}_{2^{k}}(\beta_{d}\delta)$ holds. Thus,
for all $k$, $n\in{\mathbb{N}}$ with $2^{k-1}\leq n<2^{k}$, we have

$\frac{\log{\Theta}_{n}(\delta)}{\log
n}\leq\frac{\log\tilde{\Theta}_{2^{k}}(\beta_{d}\delta)}{\log 2^{k-1}}.$

Therefore, with (66) we obtain for any $\epsilon>0$,

$\displaystyle\limsup_{n\to\infty}\frac{\log{\Theta}_{n}(\delta)}{\log n}\leq
1-\delta+\epsilon\quad{a.s.}$

The desired upper bound holds by combining these bounds. ∎

###  5.2 Proof of the lower bound of Theorem 2.2 for $d\geq 2$

###### Proof.

We closely follow the argument in the proof of Lemma $4.2$ with
$\beta=\beta_{d}\delta$. Take $k$ and $h_{k}$ as in section $4.1$. Set

$\displaystyle W_{n}(\beta_{d}\delta)=\sharp\\{x\in\partial R(u_{n})\cap
R(u_{n-1}):K(u_{n-1},x)\geq\lceil\beta_{d}\delta n^{2}\rceil\\}.$

Note that $W_{n}(\beta_{d}\delta)\geq Q_{n}$ holds for any $n\in{\mathbb{N}}$.
Indeed, if $x\in\partial_{b}R(u_{n})$, then $x\in\partial R(u_{n})$. Moreover,
if $l\in I_{n}$, then $l+k\lceil\beta_{d}\delta n^{2}\rceil\leq u_{n-1}$ holds
for all sufficiently large $n\in{\mathbb{N}}$. Therefore, since we know (12)
and Lemma 4.2, we have

$\displaystyle
P(W_{n}(\beta_{d}\delta)\geq\frac{1}{2}EQ_{n}\geq\frac{c\exp(h_{k}n^{2}-2n)}{n^{4}}\quad\quad\text{
for all but finitely many }n)=1.$

Let $u_{m-1}\leq n<u_{m}$. Note that $K(u_{m-1,x})\leq K(n,x)$ holds for all
$x\in\mathbb{Z}^{d}$ and by virtue of (3), $\partial R(u_{m})\cap
R(u_{m-1})\in\partial R(n)$. Hence,
$W_{m}(\beta_{d}\delta)\leq\Theta_{n}(\delta)$ holds. Therefore, it holds that

$\displaystyle\liminf_{n\to\infty}\frac{\log\Theta_{n}(\delta)}{\log n}\geq
h_{k}\quad\text{ a.s.}$

Since $h_{k}\to 1-(\beta_{d}\delta)/\beta_{d}$ as $k\to\infty$, the desired
result holds, completing the proof. ∎

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# Control of vortex chirality in a symmetric ferromagnetic ring using
ferromagnetic nanoelement

Uladzislau Makartsou<EMAIL_ADDRESS>Institute of Spintronics and
Quantum Information, Faculty of Physics, Adam Mickiewicz University,
Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Mathieu Moalic  Institute
of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz
University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Mateusz Zelent
Institute of Spintronics and Quantum Information, Faculty of Physics, Adam
Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland
Michal Mruczkiewicz  Institute of Electrical Engineering, Slovak Academy of
Sciences, Dubravska cesta 9, SK-841-04 Bratislava, Slovakia and Centre For
Advanced Materials Application CEMEA, Slovak Academy of Sciences, Dubravska
cesta 9, 845 11 Bratislava, Slovakia  Maciej Krawczyk<EMAIL_ADDRESS>Institute of Spintronics and Quantum Information, Faculty of Physics, Adam
Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland

###### Abstract

Controlling the vortex chirality in ferromagnetic nanodots and nanorings has
been a topic of investigation for the last few years. Many control methods
have been proposed and it has been found that the control is related to the
breaking of the circular symmetry. In this paper, we present a theoretical
study demonstrating the control of chirality in ferromagnetic nanoring without
directly breaking its symmetry, but instead by placing elongated ferromagnetic
nanoelement inside the ring, Here, the stray magnetostatic field exerted by
the asymmetrically placed nanoelement determines the movement of the domain
walls upon remagnetization of the nanoring and the resulting chirality in the
remanence. This approach allows the chirality of the vortex state to be
controlled and also promises its control in a dense array of nanorings, thus
suitable for spintronic and magnonic applications.

††preprint: APS/123-QED

##  I Introduction

An advantage of the soft ferromagnetic disks and rings over the monodomain
nanoparticles for the development of magnetic memory and reprogrammable logic
devices is vortex chirality.[1, 2, 3, 4, 5] The key factor determining the
chirality of ferromagnetic nanorings (NRs) is the control of domain wall (DW)
motion [6, 7]. Understanding and control of the transition process from an
onion state (OS) to a vortex state (VS) opens opportunities not only in the
study and development of applications based on the static magnetization
configuration and dynamic properties related to the chirality of NRs [8, 9,
10] but may also contribute to devices based on DW dynamics in arrays of NRs
[11, 12]. The transition from the OS to the VS can be induced by reducing the
in-plane magnetic field. This results in moving head-to-head (HTH) or tail-to-
tail (TTT) DWs perpendicular to the field direction towards either the left or
right side of the ring. The direction of DWs movement is spontaneous and
random, leading to the NR obtaining a flux-closure state with either clockwise
(CW) or counterclockwise (CCW) chirality. This indicates that the DW motion
and vortex chirality control presented in our paper can be applied to a wide
variety of spintronic and magnonic applications.

Chirality control usually involves breaking the circular symmetry of the NR,
as a result of which the direction of movement of the DWs when interacting
with an external magnetic field is determined by the energy difference between
the asymmetric sides of the ring [13, 14]. For instance, as the asymmetry is
introduced as a de-centered ring, i.e., one side of the ring has a larger
width than the other, the direction of the external field applied in the plane
parallel to these sides, determines the VS chirality [15, 16, 17]. Similarly
to VS in full disks, deformation of edges, like notches, also allows to
control chirality by causing energy splitting under in-plane applied magnetic
fields [18, 19, 5].

Our approach proposes abandoning the direct change in a ring symmetry in
favour of its altering by the magnetization of an asymmetrically located
single-domain ferromagnetic nanoelement (NE) with high shape anisotropy,
placed inside the NR. Its interaction with HTH-TTT DWs can provide the desired
control of vortex chirality. The NE must have the following properties in
order to prevent its magnetization from changing by external magnetic field
and stray fields produced by NR: sufficiently strong anisotropy and
sufficiently high magnetic moment. In numerical simulations, we found that an
NE having a shape of ring segment and located asymmetrically within the NR
would be most suitable.

Chirality control by magnetostatic coupling with nanomagnets in ferromagnetic
nanodisks has recently been implemented by placing rhombus elements near the
disk edges [20]. Our approach adds additional flexibility to the process of
controlling chirality and motion of DWs. Placing NE inside the ring can allow
to create a dense array of NRs [21], and even an array of overlapping rings
[12], with controlled chirality. This is important in the context of
interconnected rings, which have recently been demonstrated to perform well in
terms of reservoir computing system [12, 22].

##  II Geometry and simulation method

We study an isolated soft ferromagnetic Fe nanoring with an inner diameter
$d_{\mathrm{in}}=500$ nm, an outer diameter $d_{\mathrm{out}}=800$ nm and
thickness $t=80$ nm. Such dimensions allow for VS stabilization at remanence.
To control the magnetization chirality we placed the NE, made also from Fe,
inside the ring at distance of 25 nm from the inner wall of the NR. The shape
of the NE can be regarded as a part of an NR with a width of 25 nm and sharp
ends. The structure under investigation is shown in Fig. 1(a). The shape of
the NE gives shape anisotropy, with switching field around 126 mT, which is
higher than the coercive field of the NR at 100 mT. In this paper, we compare
3 variants of the system: (1) non-NE configuration, i.e., the reference
system, NR without the NE, (2) the parallel configuration with the external
magnetic field parallel to the magnetization of the NE, and lastly (3) the
anti-parallel configuration with the external field opposite to the
magnetization of the NE.

Figure 1: (a) Schematic representation of the system under consideration: the
ferromagnetic nanoelement (NE) inside the ferromagnetic ring (NR). (b) The
grains structure representation used in simulations.

All simulations were carried out using MuMax3, a GPU accelerated micromagnetic
simulation program [23]. To implement the system in the simulations, we
discretized it with $512\times 512\times 7$ cells with cell size $\approx
1.57\times 1.57\times 11.42$ nm3 for a total size of $805\times 805\times 80$
nm3 along the x, y and z axes, respectively. Because the magnetization is not
uniform throughout the thickness and it would not be practical to show all 7
layers used in simulations, we will present in figure only the average
magnetization across the thickness rather than a selected $z$-layer.

We used magnetic parameters from the experimental paper of Miyawaki, et al.
[24]. These are: the saturation magnetization $M_{\mathrm{S}}$= 1600 kA/m, the
uni-axial magnetocrystalline anisotropy along the $z$-axis (out-of-plane
direction) with constant $K=47$ kJ/m3, and the exchange stiffness constant
$A_{\mathrm{ex}}$ = 21 pJ/m.[24] Using a Voronoi tessellation (see, Fig.
1(b)), we have added magnetic grains in the NR to show that our results are
robust even for imperfect materials. The grains are 20 nm on average and each
of them has a random value of $M_{\mathrm{S}}$ obtained from a normal
distribution of mean 1600 kA/m, and with a standard deviation
$\sigma_{M_{\mathrm{S}}}$ of 2% (32 kA/m). We also reduced the exchange
coupling between grains uniformly by 5% as compared to $A_{\mathrm{ex}}$
value. The introduction of this inhomogeneity results in a fully deterministic
simulations for non-NE configuration, always leading to a single VS with CW or
CCW chirality, for a given pseudo-random number defining grains and
distribution of $M_{\mathrm{S}}$ among the grains.

The simulations for the statistical analysis (shown in Fig. 2) for the
parallel [Fig. 2(b)] and non-NE configuration [Fig. 2(a)] were run as follows.
First, we apply a global external magnetic field $B_{\mathrm{ext}}$ of 1 T
along the $y$-axis, which saturates the NR and the NE if present, then we
decrease $B_{\mathrm{ext}}$ by 1 mT steps until we reach 0 mT. For every step,
we use the conjugate gradient method to find the ground state of the
magnetization. For the antiparallel configuration [Fig. 2(c)], we could not
apply such a strong external field as it would remagnetize the NE in the
parallel configuration. So here, we start from $B_{\mathrm{ext}}$ = 100 mT,
which is sufficient to maintain the OS for the NR, but not enough to switch
the NE. In this case, we also had to set the initial NE magnetization in the
opposite direction to the external field to achieve the desired configuration.
Then, the system was relaxed and finally we demagnetized the rings to 0 mT in
1 mT steps.

##  III Results and discussion

###  III.0.1 Chirality control demonstration

To show that the NE determines the final magnetization state of the NR, we
conducted a statistical analysis of the remagnetization of the NR with
decreasing the external magnetic field according to the procedure described in
Sec. II. We ran 100 simulations for 3 configurations: non-NE configuration
[Fig. 2(a)], parallel [Fig. 2(b)] and anti-parallel [Fig. 2(c)]. For each
simulation, we used a different random seed which resulted in a different
organisation of the grain structure.

In Fig. 2(a) we observe that the CW to CCW states are obtained in 60$\%$ and
40$\%$ of cases, respectively. The statistics change significantly when we
introduce NE. From Fig. 2(b) and (c) we see that we have full control of the
VS chirality at remanence by the magnetization orientation of NE. 100% of the
simulations show a CCW configuration with a NE magnetized parallel to
$B_{\mathrm{ext}}$, and 100% of the simulations show a CW configuration for
the opposite case.

Figures 2(d) and (e) show the effect of the magnetization saturation of the NE
($M_{\mathrm{S,NE}}$) on the chirality control for the parallel and anti-
parallel configurations, respectively. We decreased the value of
$M_{\mathrm{S,NE}}$ from 1600 kA/m to 1200, 800 and 400 kA/m while keeping the
value of $M_{\mathrm{S}}$ in the NR unchanged, i.e., at $M_{\mathrm{S}}$ =
1600 $\pm 5\%$ kA/m. We run 100 simulations in each case recording the final
states. For both configurations we observe that lowering $M_{\mathrm{S,NE}}$
reduces the degree of chirality control. This indicates that to control VS
chirality the NE needs to have sufficiently strong magnetic moment, which can
be guaranteed by sufficiently large $M_{\mathrm{S}}$ or volume of the NE.

Interestingly, for the anti-parallel configuration shown in Fig. 2(e), the
dependence is nonmonotonic. When the magnetization of NE is set to 800 kA/m,
it causes the NE to switch its magnetization to be parallel to the nearest
part of NR during the remagnetization process, resulting in the loss of
chirality control. We observe that 40% and 60% of the VSs are CW and CCW,
respectively (similarly to the non-NE configuration). However, when the
magnetization of the NE is set to 400 kA/m, the in-plane magnetization changes
to the out-of-plane magnetization. This is due to a perpendicular magnetic
anisotropy which influence is enhanced with reduced $M_{\mathrm{S}}$. This
process needs a separate investigations which are beyond the scope of this
paper.

Figures 2(f) and (g) show the statistics of the chirality in dependence on the
$\sigma_{M_{\mathrm{S}}}$ distribution in the grains, which was determined
using Voronoi tessellation. For both NE configurations, the effect of the
degree of chirality control decrease as $\sigma_{M_{\mathrm{S}}}$increases.
However, for antiparallel configuration this process is much slower in respect
to the parallel configuration. Figs. 2(h) and (i) show the statistics in
dependence on the separation between NE and NR. With increasing distance, the
effect of chirality control decreases for both configurations but up to 150 nm
we still have over 80% of control. This indicates that by selecting the
position of the NE relative to the edge of the NR, we can tune the occurrence
of chirality of a given type to a given probability.

The results demonstrate that we have a stable systematic control of the VS
chirality using the NE during the remagnetization process. To elucidate the
chirality control mechanisms, we performed a detailed hysteresis loop
analysis.

Figure 2: (a) The statistic of the 100 micromagnetic simulation results for
the NR without the NE (non-NE configuration) with different random
distribution of parameters among the grains. (b) and (c) The statistics of the
micromagnetic simulations with the same grain realizations but for the NR with
NE in parallel and anti-parallel configurations, respectively. At the bottom
the static magnetization configurations in remanence are shown. Here, the
color map indicates the vector orientation, where hue indicates the
orientation of the magnetization according to the diagram in the right-bottom
corner. (d), (e) Statistic of the chirality control in dependence on the
decrease of the magnetization saturation of NE. (f), (g) Chirality control
depending on the increase in the distribution of magnetization saturation in
grains. (h), (i) Chirality control in dependence on the separation between the
NE and the inner edge of NR.

###  III.0.2 Remagnetization procedure

NR and NE, made of the same material, exhibit different values of the
switching field due to differences in shape anisotropy. This means that the
magnetization reversal process of the NE will occur at different magnetic
fields compared to the NR.[25] Furthermore, it is important to note that the
coupled NR-NE system significantly alters the value of the NE switching field,
it changes from $H_{\mathrm{S}}$ = 94 mT to $H_{\mathrm{S}}$ = 126 mT.

Figure 3:  (a) The simulations of the hysteresis loop, where: for non-NE and
parallel configuration, the simulations start at full saturation at -2000 mT
to 2000 mT and opposite from 2000 mT to -2000 mT, for antiparallel
configuration, the simulations start at full saturation at -2000 mT and
interrupted at point 110 mT, then from 110 mT to -2000 mT. Magnetization
configuration at selected magnetic fields: (b),(c),(d) – in non-NE
configuration, (e),(f),(g) – parallel configuration, (h),(i),(j) –
antiparallel configuration. The colormap for magnetization orientation is the
same as that shown on Fig. 2.

We will analyse the hysteresis loop in the three scenarios related to the
three configurations considered above. In the first scenario, we consider NR
in the absence of NE (non-NE configuration). The hysteresis loop is shown in
Fig. 3(a) with a blue solid line. We start simulations by applying a large
field of -2000 mT along the -$y$ direction, decreasing its magnitude to 0 mT,
and then increasing it to the ring saturation in the opposite direction, i.e.,
2000 mT. This process is then repeated by reversing the direction of the
field. The HTH and TTT DWs appear when the magnitude of the field is less than
400 mT. As the field is decreased further, the DWs start to deform. The
magnetization structures at 110 and 25 mT are shown in Fig. 3(b) and (c),
respectively. The remagnetization of the right part of the ring occurs at -24
mT, and a CCW state stabilises at remanence, as shown in Fig. 3(d). However,
the chirality of the ring depends on the random distribution of the grains, as
shown in Fig. 2(a). Thus, for other grain distributions, a CW state also may
stabilize.

The second scenario is for the NR-NE system in the parallel configuration, we
also start with full saturation at -2000 mT. The hysteresis loop is shown in
Fig. 3(a) with a yellow dashed line, and it develops similarly to the previous
scenario. Fig. 3(e) shows the magnetization of the NR and NE at 110 mT. As
shown in Fig. 3(f), the magnetization of the system is just before switching
from an OS to a VS at -30 mT. This switching occurs at a higher field than for
the non-NE configuration. In this scenario, the NE is located on the right
side of the NR, leading to a CCW magnetization chirality at remanence, as
shown in Fig. 3(g). Importantly, the final state in this scenario does not
depend on the random distribution of the grains, as demonstrated in Fig. 2(b).
Starting from a positive value of the external magnetic field, i.e.,
$B_{\mathrm{ext}}$ = 2000 mT, we always reach a CCW chirality at remanence.

In the third scenario, we simulate a truncated hysteresis loop for the NR-NE
system that allows for an antiparallel configuration, where the magnetization
of the NE and the nearest side of the NR are aligned in opposite directions.
We begin at an external magnetic field of -2000 mT along the $y$-direction and
follow the main loop, decreasing the magnitude of the field as in the previous
scenario. However, in this simulation, we interrupt the process at 110 mT
[Fig. 3(h)], when the magnetization of the NE and the nearest side of the NR
are antiparallel. Then, we reverse the direction of the changes in the
external magnetic field. Fig. 3(i) shows the magnetization at 33 mT, just
before the demagnetization of the NR. The magnetization switches in the part
of the ring closest to the NE, establishing a CW chirality in the ring, as
shown in Fig. 3(j) and corresponding to Fig. 2(c).

The results show that regardless of the direction of the external magnetic
field at the starting point of the field change, parallel or antiparallel to
the direction of magnetization in NE, the direction of magnetization in NE is
the same as the direction of magnetization of the nearest side of NR at
remanence, which determines the VS chirality. Further conclusion is that the
chirality control takes place at fields close to the switching field when one
of the vertical parts of the ring changes the magnetization orientation as a
result of DW motion in defined direction. This is clearly visible in the
movies showing remagnetisation in NR provided in Supplementary Material. Thus,
chirality is determined by the direction of movement of the DWs to the left or
right part of the ring from the vertical symmetry axis, which is initiated by
the magnetostatic interactions between NR and NE.

###  III.0.3 Discussion

The stray field produced by NE interacts with the DWs and changes the internal
field in the NR, so introduces the additional element to the system that
control the direction of DWs propagation. In Fig. 4 we present schematically
DW changes with decreasing magnetic field in the 3 scenarios of
remagnetization discussed in the previous sections.

Figure 4:  Schematic representation of the remagnetization process in a ring.
Starting from the onion state (State 1), with decreasing magnetic field
(applied vertically) the DWs move to the right or left (State 2) determining
VS chirality at remanence (State 4) via annihilation at switching field (State
3). The chirality of the VS (State 4) is not controlled in the non-NE case
(a), (b). With the NE magnetization oriented parallel (c) or antiparllel (d)
to the external magnetic field in a remanence the ring has determined the
chirality, CCW and CW, respectively. (e) Schematic representation of the
effect of the magnetostatic stray field from the NE on the DWs in NR.

In the non-NE configuration, Figs. 4(a) and (b), we can have two equivalent
final configurations, CCW and CW, respectively. In the state 1 the DWs are in
the HTH and TTT configuration. State 2 shows the DWs position at decreased
field, where the DWs are placed with small tilts to the external magnetic
field. The move of the DWs to the left or to the right are fully equivalent
for perfectly symmetric NR, which leads to an uncontrolled VS chirality.
States 3 and 4 show the process of annihilation of DWs and the final state,
CCW or CW, respectively.

Fig. 4(c-d) schematically illustrates the stages of DW evolution during the
remagnetization process for both parallel (c) and antiparallel (d)
configurations. The process starts from State 1, where the ring has an onion
state with HTH-TTT DWs in line with the external magnetic field. In State 2,
the DWs begin to move, influenced by the magnetization of the NR. The NR
induces a stray field, resulting in an asymmetrical distribution of the
effective field between the left and right parts of the NR, as shown in Fig.
4(e) and Fig. S2(a-b) in Supplementary Material. For the parallel
configuration of the NE and NR, the stray field from NE (left part of Fig.
4(e)) pushes the DWs to the left side of the NR (State 3), ending with a CCW
chirality in remanence (State 4). For the antiparallel configuration (right
part of Fig. 4(e)), the stray field from NE pushes DWs to the right arm of NR,
causing the DWs to annihilate on the right side (State 3) and resulting in a
CW chirality in remanence (State 4).

Figure 5:  Numerical simulations of azimuthal component of the averaged torque
in the ring in dependence on the magnetic field. In (a), we showed the results
of two simulations for anti-parallel [curve (1)] and parallel [curve (2)]
magnetization of the NE with respect to the magnetic field, which ends with CW
and CCW VSs, respectively. In both simulations the same random seed was used.
(b) The results for non-NE configuration with two different random seed
numbers, which determine the chirality of the VS, CW (3) and CCW (4) in
remanence. For (4) we used in simulations the same random seed as in (1) and
(2). On the right part of the plots, the magnetization configuration and local
distribution of the azimuthal component of the torque in NR are shown for four
considered cases, at the fields just before the switching (marked by dots on
the left plots). The colormap for magnetization orientation is the same as
that shown in Fig. 2.

As described above, we brake the circular symmetry of the magnetic system by
introducing NE, which creates a difference in the effective field
distributions between the left and right parts of the NR and determines the
direction of the DWs motion (see also Supplementary Materials) [2, 26]. Thus,
the stray field produced by NE determines the direction of the torque exerted
on the DWs in NR. To measure the effect of the NE on the NR, we extract from
micromagnetic simulations magnetic torque for all the cases presented above.
We use the function, which is defined in Mumax3, and returns Cartesian
components of the magnetic torque in (T) units. The torque is saved
immediately after the change in the external magnetic field, before starting
the relaxation procedure. We transform the torque for each discretization cell
from the Cartesian to the 2D polar coordinate system. To obtain a single
measure, we averaged the azimuthal component of the torque field across all
spatial dimensions of the NR, resulting in a scalar,
$\overline{\tau}_{\varphi}$, indicating whether the torque generated by the NE
causes the NR’s magnetization to rotate CW ($\overline{\tau}_{\varphi}$ $<$ 0)
or CCW ($\overline{\tau}_{\varphi}$ $>$ 0).

In Fig. 5 (left parts) we present $\overline{\tau}_{\varphi}$ in dependence on
the magnetic field for configurations considered above. For simulations of the
curves (1), (2), and (4) we import magnetization texture for some selected
seed from one of our previous analysis for non-NE configuration at the field
100 mT [Fig. 2(a)], which ends with CCW VS. Then, we set manually the NE
inside the ring with anti-parallel [curve (1)] and parallel [curve (2)]
magnetization, and perform simulations with decreasing magnetic field,
extracting $\tau_{\varphi}$ at each field step. The simulation (3) in non-NE
configuration, was performed for other random seed to show the remagnetization
of the NR to the CW state. We see that the sign of $\overline{\tau}_{\varphi}$
at fields just before the switching field points out the VS chirality in
remanence in all presented configurations. Moreover, for the case (1), where
the magnetization orientation of NE was artificially reversed at the starting
field 100 mT, the $\overline{\tau}_{\varphi}$ from the positive value changes
the sign at smaller fields, and ends with negative values expected for CW
chirality at this configuration.

##  IV Conclusions

In summary, we have demonstrated with micromagnetic simulations the systematic
control of the vortex chirality in a symmetric ferromagnetic ring by a
ferromagnetic nanoelement placed inside the ring. NE, by exerting stray
magnetostatic field, changes symmetry of HTH-TTT DWs in the onion state, and
during the remagnetization process determines the direction of the DWs
movement and finally VS chirality in remanence. To control chirality NE
requires sufficiently strong shape anisotropy to keep monodomain state and to
have a switching field higher than the switching field of NR (without NE). In
addition, the NE should have a sufficiently large magnetic moment (through
large saturation magnetization or volume) to create a stray magnetostatic
field that will determine the direction of DW motion. We show that this can be
achieved by making NE with material in the shape similar to the part of the
inner side of the ring, which simplify eventual fabrication process. In
addition, we demonstrated the resistance of this method to the variability of
geometric and material parameters of the system and random NR disturbances.
All this make the experimental implementation of the proposed system using
existing technologies possible and make it useful for spintronic and magnonic
applications.

###### Acknowledgements.

The research that led to these results has received funding from the National
Science Center of Poland, project no. 2020/37/B/ST3/03936.MM acknowledges
funding from the Slovak Grant Agency APVV (grant number APVV-19-0311(RSWFA)),
and supported by Research & Innovation Operational Programme funded by the
ERDF-ITMS project code 313021T081. The simulations were partially performed at
the Poznan Supercomputing and Networking Center (Grant No. 398).

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