i.e. the set of “reversed” increments (recall the definition of the set
$I_{sb}$ in Eq. 111). We further denote by

$\stackrel{{\scriptstyle\leftarrow}}{{\mu_{sb}}}=\sum_{i=0}^{\infty}\alpha\gamma^{i}\cdot\delta_{(i,-1)}+\sum_{i=1}^{\infty}\alpha\gamma^{i}\cdot\delta_{(-i,i-1)},$
(120)

the “reversed” distribution on $\stackrel{{\scriptstyle\leftarrow}}{{I_{sb}}}$
induced by $\mu_{sb}$ (see also the right-hand side of Fig. 13).

Figure 13: Left: Some of the increments in the set $I_{sb}$ are plotted
together with the corresponding probability weights given by $\mu_{sb}$.
Right: Some increments in the set
$\stackrel{{\scriptstyle\leftarrow}}{{I_{sb}}}$ are plotted together with the
corresponding probability weights given by
$\stackrel{{\scriptstyle\leftarrow}}{{\mu_{sb}}}$.

For all $n\in\mathbb{Z}_{>0}$, we define the following additional probability
measure

$\nu_{sb}^{n}=\frac{1}{Z_{n}}\sum_{(h,\ell)\in\mathcal{L}^{n}_{sb}}\gamma^{2\ell+h}\delta_{(h,\ell)},$
(121)

where $\mathcal{L}_{sb}^{n}\coloneqq\\{\text{Labels at level $n$ in the
generating tree for semi-Baxter permutations}\\}$ and the normalizing constant
satisfies $Z_{n}=\sum_{(h,\ell)\in\mathcal{L}^{n}_{sb}}\gamma^{2\ell+h}.$

Let $(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\geq 1}$ be a
two-dimensional random walk with increments distributed as
$\stackrel{{\scriptstyle\leftarrow}}{{\mu_{sb}}}$ and starting probability
$\nu_{sb}^{n}(h,\ell)=\mathbb{P}(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(1)=(h,\ell))$.
Denote by $\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{W}_{sb}^{n}}}$ the
set of two-dimensional walks $(x_{i})_{i\in[n]}$ in the non-negative quadrant
with increments in $\stackrel{{\scriptstyle\leftarrow}}{{I_{sb}}}$ and such
that $x_{n}=(0,0)$.

######  Proposition 5.8.

Conditioning on the event
$\left\\{(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\in[n]}\in\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{W}_{sb}^{n}}}\right\\}$,
the walk $(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\in[n]}$ is
a uniform walk in
$\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{W}_{sb}^{n}}}$.

###### Proof.

Fix
$(x_{i})_{i\in[n]}\in\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{W}_{sb}^{n}}}$.
It is enough to show that
$\mathbb{P}\left((\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\in[n]}=(x_{i})_{i\in[n]}\right)$
is independent of the choice of $(x_{i})_{i\in[n]}$. To do that, assume that
the walk $(x_{i})_{i\in[n]}$ is formed by $k$ increments of the form
$(-i,i-1)$ and $n-k-1$ increments of the form $(i,-1)$. Moreover, assume that
the sum of the $k$ increments of the form $(-i,i-1)$ is equal to $(-s,s-k)$
and the sum of the $n-k-1$ increments of the form $(i,-1)$ is equal to
$(q,-n+k+1)$. Under these assumptions we have that if $x_{1}=(h,\ell)$ then
$x_{n}=(h-s+q,\ell+s-n+1)$. Since
$(x_{i})_{i\in[n]}\in\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{W}_{sb}^{n}}}$,
it must hold that $(h-s+q,\ell+s-n+1)=(0,0)$ and so $s=n-1-\ell$ and
$q=n-1-\ell-h$, in particular $s+q=2n-2\ell-h-2$.

Under these assumptions,

$\mathbb{P}\left((\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\in[n]}=(x_{i})_{i\in[n]}\right)=\nu_{sb}^{n}(h,\ell)\cdot\alpha^{n-1}\cdot\gamma^{s+q}=\frac{\gamma^{2\ell+h}}{Z_{n}}\cdot\alpha^{n-1}\cdot\gamma^{2n-2\ell-h-2}=\frac{\alpha^{n-1}\cdot\gamma^{2n-2}}{Z_{n}},$
(122)

and this concludes the proof. ∎

Let $(\bm{W}_{n}(i))_{i\geq 1}$ be the reversed walk obtained from
$(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\geq 1}$. An easy
consequence of 5.8 is the following.

######  Corollary 5.9.

Conditioning on the event
$\left\\{(\bm{W}_{n}(i))_{i\in[n]}\in\mathcal{W}_{sb}^{n}\right\\}$, the walk
$(\bm{W}_{n}(i))_{i\in[n]}$ is a uniform walk in $\mathcal{W}_{sb}^{n}$.

####  5.2.2 Scaling limit of the conditioned two-dimensional walks for semi-
Baxter permutations

We define a rescaled version of the walk $(\bm{W}_{n}(i))_{i\geq
1}=(\bm{X}_{n}(i),\bm{Y}_{n}(i))_{i\geq 1}$: for all $n\geq 1,$ let
${\bm{\mathcal{W}}}_{n}:[0,1]\to\mathbb{R}^{2}$ be the continuous process that
linearly interpolates the following points

${\bm{\mathcal{W}}}_{n}\left(\frac{k}{n}\right)=\left(\frac{\bm{X}_{n}(k)}{\sqrt{2(2+\sqrt{5})n}},\frac{\bm{Y}_{n}(k)}{\sqrt{(1+\sqrt{5})n}}\right),\quad\text{
for all }k\in[n].$

######  Proposition 5.10.

Conditioning on the event
$\left\\{(\bm{W}_{n}(i))_{i\in[n]}\in\mathcal{W}_{sb}^{n}\right\\}$, the
following convergence in the space $\mathcal{C}([0,1],\mathbb{R}^{2})$ holds

${\bm{\mathcal{W}}}_{n}\xrightarrow[n\to\infty]{d}{\bm{\mathcal{E}}}_{\rho},$
(123)

where $\rho=-\frac{1+\sqrt{5}}{4}$.

The proof of this proposition is identical to the proof of 4.3 using the
expression for $\rho$ given below Eq. 118.

####  5.2.3 Scaling limit of coalescent-walk processes and semi-Baxter
permutations

Let
$\overline{\bm{W}}=(\overline{\bm{X}},\overline{\bm{Y}})=(\overline{\bm{X}}(k),\overline{\bm{Y}}(k))_{k\in\mathbb{Z}}$
be a random bi-infinite two-dimensional walk with step distribution $\mu_{sb}$
defined in Eq. 116, and let
$\overline{\bm{Z}}=\operatorname{WC}(\overline{\bm{W}})$ be the corresponding
discrete coalescent-walk process. For convenience, we set
$\overline{\bm{Z}}^{(j)}_{i}=0$ for $i,j\in\mathbb{Z}$, $i<j$.

We further define the following rescaled processes: for all $n\geq
1,u\in\mathbb{R}$, let
$\overline{{\bm{\mathcal{W}}}}_{n}:\mathbb{R}\to\mathbb{R}^{2}$, and
$\overline{{\bm{\mathcal{Z}}}}^{(u)}_{n}:\mathbb{R}\to\mathbb{R}$ be the
continuous processes that interpolate the following points:

$\overline{{\bm{\mathcal{W}}}}_{n}\left(\frac{k}{n}\right)=\left(\frac{\overline{\bm{X}}(k)}{\sqrt{2(2+\sqrt{5})n}},\frac{\overline{\bm{Y}}(k)}{\sqrt{(1+\sqrt{5})n}}\right),\quad\text{for
all}\quad k\in\mathbb{Z},$ (124)

and

$\overline{{\bm{\mathcal{Z}}}}^{(u)}_{n}\left(\frac{k}{n}\right)=\begin{cases}\frac{\overline{\bm{Z}}^{(\lceil
nu\rceil)}_{k}}{\sqrt{(1+\sqrt{5})n}},\quad&\text{when}\quad\overline{\bm{Z}}^{(\lceil
nu\rceil)}_{k}\geq 0,\\\ \frac{\overline{\bm{Z}}^{(\lceil
nu\rceil)}_{k}}{\sqrt{2(2+\sqrt{5})n}},\quad&\text{when}\quad\overline{\bm{Z}}^{(\lceil
nu\rceil)}_{k}\leq 0,\end{cases}\quad\text{for all}\quad k\in\mathbb{Z}.$
(125)

Recall that
$\overline{{\bm{\mathcal{W}}}}_{\rho}=(\overline{{\bm{\mathcal{X}}}}_{\rho},\overline{{\bm{\mathcal{Y}}}}_{\rho})$
denotes a two-dimensional Brownian motion of correlation $\rho$.

######  Theorem 5.11.

Fix $u\in\mathbb{R}$. The following joint convergence in the space
$\mathcal{C}(\mathbb{R},\mathbb{R})^{3}$ holds:

$\left(\overline{{\bm{\mathcal{W}}}}_{n},\overline{{\bm{\mathcal{Z}}}}^{(u)}_{n}\right)\xrightarrow[n\to\infty]{d}\left(\overline{{\bm{\mathcal{W}}}}_{\rho},\overline{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}\right),$
(126)

where

$\rho=-\frac{1+\sqrt{5}}{4}\quad\text{and}\quad q=\frac{1}{2},$ (127)

and $\overline{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}$ is the solution of the SDE
in Eq. 48 driven by $\overline{{\bm{\mathcal{W}}}}_{\rho}$.

5.11 follows from the following lemma, as 4.4 follows from 4.5. Therefore we
just give the proof of the following result.

######  Proposition 5.12.

Fix $u\in\mathbb{R}$. We have the following convergence in
$\mathcal{C}(\mathbb{R},\mathbb{R})$:

$\overline{{\bm{\mathcal{Z}}}}^{(u)}_{n}\xrightarrow[n\to\infty]{d}\overline{{\bm{\mathcal{B}}}}^{(u)},$
(128)

where $\overline{{\bm{\mathcal{B}}}}^{(u)}(t)=0$ for $t<u$ and
$\overline{{\bm{\mathcal{B}}}}_{q}^{(u)}(t)$ is a Brownian motion for $t\geq
u$.

######  Remark 5.13.

We highlight a remarkable difference with the case of strong-Baxter
permutations: the walks of the continuous coalescent-walk process for semi-
Baxter permutations are classical Brownian motions. This was also the case of
Baxter permutations (see [BM20, Theorem 4.5 and Remark 4.3]), but there is a
substantial difference between the walks of the discrete coalescent-walk
process associated with Baxter permutations and the ones associated with semi-
Baxter permutations: in [BM20, Proposition 3.3.] we proved that the walks of
the discrete coalescent-walk process associated with Baxter permutations are
simple random walks with a specific step distribution that is centered. On the
contrary the walks of the discrete coalescent-walk process associated with
semi-Baxter permutations are not even martingales (this can be checked with a
simple computation using Eq. 116 and 5.1). This will force us to first show
that $\overline{{\bm{\mathcal{Z}}}}^{(u)}_{n}$ converges in distribution to a
skew Brownian motion (using some arguments similar to the ones already used
for strong-Baxter permutations) and then to deduce that its parameter is
$1/2$, concluding that it is actually a classical Brownian motion.

In simple words, we can summarize the discussion above as follows:

  * •

walks of the coalescent-walk process for Baxter permutations are symmetric
both in the discrete and in the continuum limit;

  * •

walks of the coalescent-walk process for semi-Baxter permutations are not
symmetric in the discrete case but they become symmetric in the continuum
limit;

  * •

walks of the coalescent-walk process for strong-Baxter permutations are not
symmetric both in the discrete and in the continuum limit.

###### Proof of 5.12.

As explained in 5.13, we prove that
$\overline{{\bm{\mathcal{Z}}}}^{(u)}_{n}\xrightarrow[n\to\infty]{d}\overline{{\bm{\mathcal{B}}}}_{1/2}^{(u)}$,
where $\overline{{\bm{\mathcal{B}}}}_{1/2}^{(u)}(t)=0$ for $t<u$ and
$\overline{{\bm{\mathcal{B}}}}_{1/2}^{(u)}(t)$ is a skew Brownian motion for
$t\geq u$ of parameter $1/2$.

We only consider the case $u=0$, the general proof being similar. Again we
just prove convergence of one-dimensional marginal distributions. We recall
that
$\overline{{\bm{\mathcal{Z}}}}_{n}\coloneqq\overline{{\bm{\mathcal{Z}}}}^{(0)}_{n}:\mathbb{R}\to\mathbb{R}$
is the continuous process defined by linearly interpolating the following
points:

$\overline{{\bm{\mathcal{Z}}}}_{n}\left(\frac{k}{n}\right)=\begin{cases}\frac{\overline{\bm{Z}}_{k}}{\sigma^{\prime}\sqrt{n}},\quad&\text{when}\quad\overline{\bm{Z}}_{k}\geq
0,\\\
\frac{\overline{\bm{Z}}_{k}}{\sigma\sqrt{n}},\quad&\text{when}\quad\overline{\bm{Z}}_{k}\leq
0,\end{cases}\quad\text{for all}\quad k\in\mathbb{Z}.$ (129)

where ${\overline{\bm{Z}}_{k}}\coloneqq{\overline{\bm{Z}}^{(0)}_{k}}$, and
$\sigma^{\prime}=\sqrt{1+\sqrt{5}}$ and $\sigma=\sqrt{2(2+\sqrt{5})}$. For the
rest of the proof we fix a non-negative and Lipschitz continuous function
$\varphi$ with compact support. We want to show that for all $t\geq 0$,

$\mathbb{E}\left[\varphi\left(\overline{{\bm{\mathcal{Z}}}}_{n}(t)\right)\right]\to
1/2\int_{0}^{+\infty}\varphi(y)\frac{2e^{-y^{2}/2t}}{\sqrt{2\pi
t}}dy+1/2\int_{-\infty}^{0}\varphi(y)\frac{2e^{-y^{2}/2t}}{\sqrt{2\pi t}}dy.$
(130)

Since $\varphi$ is Lipschitz continuous it is enough to study

$\mathbb{E}\left[\varphi\left(\frac{\overline{\bm{Z}}_{\lfloor
nt\rfloor}}{\sigma^{\prime}\sqrt{n}}\right);\overline{\bm{Z}}_{\lfloor
nt\rfloor}>0\right]\quad\text{and}\quad\mathbb{E}\left[\varphi\left(\frac{\overline{\bm{Z}}_{\lfloor
nt\rfloor}}{\sigma\sqrt{n}}\right);\overline{\bm{Z}}_{\lfloor
nt\rfloor}<0\right].$ (131)

As before, we just explicitly detail the computations for the first
expectation. It can be decomposed as (recall 5.4)

$\mathbb{E}\left[\varphi\left(\frac{\overline{\bm{Z}}_{\lfloor
nt\rfloor}}{\sigma^{\prime}\sqrt{n}}\right);\overline{\bm{Z}}_{\lfloor
nt\rfloor}>0\right]=\sum_{k=0}^{\lfloor
nt\rfloor}\sum_{\ell=0}^{+\infty}\mathbb{E}\left[\varphi\left(\frac{\overline{\bm{Z}}_{\lfloor
nt\rfloor}}{\sigma^{\prime}\sqrt{n}}\right);\bm{\tau}^{+}_{\ell}=k,(\overline{\bm{Z}}_{i})_{i\in[k+1,\lfloor
nt\rfloor]}>0\right],$ (132)

where $\bm{\tau}^{+}_{0}=0$ and
$\bm{\tau}^{+}_{\ell+1}=\inf\\{i\geq\bm{\tau}^{+}_{\ell}:\overline{\bm{Z}}_{i-1}<0,\overline{\bm{Z}}_{i}=1\\}$
for all $\ell\in\mathbb{Z}_{\geq 0}$. Note that for all
$\ell\in\mathbb{Z}_{\geq 0}$, using the definition in Eq. 114 we have that

$(\overline{\bm{Z}}_{\lfloor
nt\rfloor};\bm{\tau}^{+}_{\ell}=k,(\overline{\bm{Z}}_{i})_{i\in[k+1,\lfloor
nt\rfloor]}>0)\stackrel{{\scriptstyle d}}{{=}}(1+\bm{S}_{\lfloor
nt\rfloor-k};\tau^{+}_{\ell}=k,(1+\bm{S}_{i})_{i\in[\lfloor nt\rfloor-k]}>0),$

where $(\bm{S}_{i})_{i\geq 0}$ denotes a random walk started at zero at time
zero, with step distribution equal to the distribution of
$\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)$, and independent of
$\bm{\tau}^{+}_{\ell}$.Therefore we can write

$\mathbb{E}\left[\varphi\left(\frac{\overline{\bm{Z}}_{\lfloor
nt\rfloor}}{\sigma^{\prime}\sqrt{n}}\right);\overline{\bm{Z}}_{\lfloor
nt\rfloor}>0\right]=\sum_{k=0}^{\lfloor
nt\rfloor}\left(\sum_{\ell=0}^{+\infty}\mathbb{P}(\bm{\tau}^{+}_{\ell}=k)\right)\mathbb{E}\left[\varphi\left(\frac{1+\bm{S}_{\lfloor
nt\rfloor-k}}{\sigma^{\prime}\sqrt{n}}\right);(1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor-k]}>0\right]\\\ =\sum_{k=0}^{\lfloor
nt\rfloor}\left(\sum_{\ell=0}^{+\infty}\mathbb{P}(\bm{\tau}^{+}_{\ell}=k)\right)\mathbb{E}\left[\varphi\left(\frac{1+\bm{S}_{\lfloor
nt\rfloor-k}}{\sigma^{\prime}\sqrt{n}}\right)\middle|(1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor-k]}>0\right]\mathbb{P}\left((1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor-k]}>0\right).$ (133)

We now focus on studying $\mathbb{P}(\bm{\tau}^{+}_{\ell}=k)$. Note that
$(\bm{\tau}^{+}_{\ell})_{\ell\in\mathbb{Z}_{\geq 2}}$ are all equidistributed,
but $\bm{\tau}^{+}_{1}$ has a slightly different distribution (because
$\overline{\bm{Z}}_{\bm{\tau}^{+}_{0}}=\overline{\bm{Z}}_{0}=0$ but
$\overline{\bm{Z}}_{\bm{\tau}^{+}_{\ell}}=1$ for all $\ell\in\mathbb{Z}_{\geq
1}$). We have the following result.

######  Lemma 5.14.

As $k\to\infty$,

$\mathbb{P}(\bm{\tau}^{+}_{2}=k)\sim\frac{\beta}{k^{3/2}},$ (134)

where

$\beta=\frac{1}{\sqrt{2\pi}}\frac{1}{1-\gamma}\left(\frac{1}{\sigma^{\prime}}+\frac{1+\gamma}{\sigma}\right).$
(135)

###### Proof.

Recall that $\overline{\bm{Z}}_{\bm{\tau}^{+}_{1}}=1$. Using Observations 5.3
and 5.4, we can write (see also Fig. 14)

$\mathbb{P}(\bm{\tau}^{+}_{2}=k)=\sum_{s=0}^{k-2}\sum_{\begin{subarray}{c}y\in\mathbb{Z}_{>0}\\\
y^{\prime},y^{\prime\prime}\in\mathbb{Z}_{<0}\end{subarray}}\mathbb{P}(\overline{\bm{Z}}_{s}=y,(\overline{\bm{Z}}_{i})_{i\in[s]}>0)\mathbb{P}(\overline{\bm{Z}}_{s+1}-\overline{\bm{Z}}_{s}=y^{\prime}-y)\\\
\mathbb{P}(\overline{\bm{Z}}_{k-1}=y^{\prime\prime},(\overline{\bm{Z}}_{i})_{i\in[s+1,k-1]}<0|\overline{\bm{Z}}_{s+1}=y^{\prime})\mathbb{P}(\overline{\bm{Z}}_{k}-\overline{\bm{Z}}_{k-1}=1-y^{\prime\prime}).$
(136)

Figure 14: A schema for the event $\left\\{\bm{\tau}^{+}_{2}=k\right\\}$.

Using the definition in Eq. 114 and letting:

  * •

$(\bm{S}_{i})_{i\geq 0}$ be a random walk started at zero at time zero, with
step distribution equal to the distribution of
$\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)$,

  * •

$(\bm{S}^{\prime}_{i})_{i\geq 0}$ be a random walk started at zero at time
zero, with step distribution equal to the distribution of
$-(\overline{\bm{X}}(1)-\overline{\bm{X}}(0))$,

  * •

$(\bm{S}_{i})_{i\geq 0}$ is independent of $(\bm{S}^{\prime}_{i})_{i\geq 0}$,

we can rewrite the previous expression as

$\mathbb{P}(\bm{\tau}^{+}_{2}=k)=\sum_{s=0}^{k-2}\sum_{\begin{subarray}{c}y\in\mathbb{Z}_{>0}\\\
y^{\prime},y^{\prime\prime}\in\mathbb{Z}_{<0}\end{subarray}}\mathbb{P}(1+\bm{S}_{h}=y,(1+\bm{S}_{i})_{i\in[s]}>0)\mathbb{P}(\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)=y^{\prime}-y+1)\\\
\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{k-s-2}=y^{\prime\prime},(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[k-s-2]}<0)\mathbb{P}(-(\overline{\bm{X}}(1)-\overline{\bm{X}}(0))\geq-y^{\prime\prime}).$
(137)

We now focus on
$\sum_{s=0}^{k-2}\mathbb{P}(1+\bm{S}_{s}=y,(1+\bm{S}_{i})_{i\in[s]}>0)\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{k-s-2}=y^{\prime\prime},(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[k-s-2]}<0)$.
From [DW15, Theorem 6] we know that as $m\to\infty$,

$\displaystyle\mathbb{P}(1+\bm{S}_{m}=y,(1+\bm{S}_{i})_{i\in[m]}>0)\sim\frac{c_{1}}{\sigma^{\prime}\sqrt{2\pi}}\frac{\tilde{h}(y)}{m^{3/2}},$
(138)
$\displaystyle\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{m}=y^{\prime\prime},(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[m]}<0)\sim\frac{c_{2}}{\sigma\sqrt{2\pi}}\frac{h^{\prime}(y^{\prime})\tilde{h^{\prime}}(y^{\prime\prime})}{m^{3/2}},$
(139) $\displaystyle\mathbb{P}(1+\bm{S}_{m}=y,(1+\bm{S}_{i})_{i\in[m]}>0)\leq
C\frac{\tilde{h}(y)}{m^{3/2}},\quad\text{for all}\quad y\in\mathbb{Z}_{>0},$
(140)
$\displaystyle\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{m}=y^{\prime\prime},(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[m]}<0)\leq
C\frac{h^{\prime}(y^{\prime})\tilde{h^{\prime}}(y^{\prime\prime})}{m^{3/2}},\quad\text{for
all}\quad y^{\prime},y^{\prime\prime}\in\mathbb{Z}_{<0},$ (141)

where $\tilde{h}$, $h^{\prime}$ and $\tilde{h^{\prime}}$ are the functions
defined in Eq. 195 for the walks $-\bm{S}_{m}$, $\bm{S}^{\prime}_{m}$ and
$-\bm{S}^{\prime}_{m}$ respectively, and

$c_{1}=\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sum_{z\in\mathbb{Z}_{>0}}\tilde{h}(z)\mathbb{P}(\bm{S}_{1}\leq-z)}\quad\text{
and }\quad
c_{2}=\frac{\mathbb{E}[\bm{S}^{\prime}_{\bm{N}^{\prime}}]}{\sum_{z\in\mathbb{Z}_{<0}}\tilde{h}^{\prime}(z)\mathbb{P}(\bm{S}^{\prime}_{1}\geq-z)},$
(142)

with $\bm{N}\coloneqq\inf\\{n>0|\bm{S}_{n}<0\\}$ and
$\bm{N}^{\prime}\coloneqq\inf\\{n>0|\bm{S^{\prime}}_{n}>0\\}$.

Using C.1 and the estimates above, we have, as $k\to\infty$,

$\sum_{s=0}^{k-2}\mathbb{P}(1+\bm{S}_{s}=y,(1+\bm{S}_{i})_{i\in[s]}>0)\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{k-s-2}=y^{\prime\prime},(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[k-s-2]}<0)\\\
\sim\frac{1}{\sqrt{2\pi}k^{3/2}}C(y,y^{\prime},y^{\prime\prime}),$ (143)

where $C(y,y^{\prime},y^{\prime\prime})$ is equal to

$\frac{c_{1}\cdot\tilde{h}(y)}{\sigma^{\prime}}\left(\sum_{s=0}^{\infty}\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{s}=y^{\prime\prime},(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[s]}<0)\right)\\\
+\frac{c_{2}\cdot
h^{\prime}(y^{\prime})\tilde{h^{\prime}}(y^{\prime\prime})}{\sigma}\left(\sum_{s=0}^{\infty}\mathbb{P}(1+\bm{S}_{s}=y,(1+\bm{S}_{i})_{i\in[s]}>0)\right).$
(144)

Therefore, using the last two uniform bounds in Eq. 138, we can conclude that

$\mathbb{P}(\bm{\tau}^{+}_{2}=k)\sim\frac{1}{k^{3/2}}\frac{1}{\sqrt{2\pi}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}_{>0}\\\
y^{\prime},y^{\prime\prime}\in\mathbb{Z}_{<0}\end{subarray}}\mathbb{P}(\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)=y^{\prime}-y+1)\mathbb{P}(-(\overline{\bm{X}}(1)-\overline{\bm{X}}(0))\geq-y^{\prime\prime})C(y,y^{\prime},y^{\prime\prime})=\frac{\beta}{k^{3/2}}.$
(145)

We finally simplify the expression for $\beta$ in order to obtain the
expression in Eq. 135. Note that

$\sum_{s=0}^{\infty}\sum_{y^{\prime\prime}\in\mathbb{Z}_{<0}}\mathbb{P}(-(\overline{\bm{X}}(1)-\overline{\bm{X}}(0))\geq-y^{\prime\prime})\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{s}=y^{\prime\prime},(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[s]}<0)\\\
=\sum_{s=0}^{\infty}\mathbb{P}(y^{\prime}+\bm{S}^{\prime}_{s+1}\geq
0,(y^{\prime}+\bm{S}^{\prime}_{i})_{i\in[s]}<0)=1,$ (146)

because a symmetric random walk started at $y^{\prime}$ becomes eventually
non-negative almost surely. Using the latter identity and substituting the
expressions of $c_{1}$ and $c_{2}$ given in Eq. 142, we obtain that

$\beta=\frac{1}{\sqrt{2\pi}}\Bigg{(}\frac{\mathbb{E}\left[\bm{S}_{\bm{N}}\right]}{\sigma^{\prime}}+\frac{\mathbb{E}\left[\bm{S}^{\prime}_{\bm{N}^{\prime}}\right]}{\sigma}\sum_{\begin{subarray}{c}y\in\mathbb{Z}_{>0}\\\
y^{\prime}\in\mathbb{Z}_{<0}\end{subarray}}h^{\prime}(y^{\prime})\mathbb{P}(\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)=y^{\prime}-y+1)\sum_{s=0}^{\infty}\mathbb{P}(1+\bm{S}_{s}=y,(1+\bm{S}_{i})_{i\in[s]}>0)\Bigg{)}.$
(147)

Noting that

$\sum_{y\in\mathbb{Z}_{>0}}\mathbb{P}(\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)=y^{\prime}-y+1)\mathbb{P}(1+\bm{S}_{s}=y,(1+\bm{S}_{i})_{i\in[s]}>0)=\mathbb{P}(\bm{S}_{s+1}=y^{\prime},(\bm{S}_{i})_{i\in[s]}\geq
0),$ (148)

we obtain the new simplified expression

$\beta=\frac{1}{\sqrt{2\pi}}\left(\frac{\mathbb{E}\left[\bm{S}_{\bm{N}}\right]}{\sigma^{\prime}}+\frac{\mathbb{E}\left[\bm{S}^{\prime}_{\bm{N}^{\prime}}\right]}{\sigma}\sum_{s=0}^{\infty}\sum_{y^{\prime}\in\mathbb{Z}_{<0}}h^{\prime}(y^{\prime})\mathbb{P}(\bm{S}_{s+1}=y^{\prime},(\bm{S}_{i})_{i\in[s]}\geq
0)\right).$ (149)

We now compute an expression for the function $h^{\prime}(x)$ defined in Eq.
195 for the walk $\bm{S}^{\prime}_{m}$,

$h^{\prime}(x)=\begin{cases}1+\sum_{j=1}^{+\infty}\mathbb{P}(\bm{S^{\prime}}^{1}_{\bm{N}^{\prime}}+\dots+\bm{S^{\prime}}^{j}_{\bm{N}^{\prime}}<-x)\quad&\text{if}\quad
x<0,\\\ 0\quad&\text{otherwise},\end{cases}$ (150)

where the $(\bm{S^{\prime}}^{j}_{\bm{N}^{\prime}})_{j\in\mathbb{Z}_{\geq 1}}$
are i.i.d. copies of $\bm{S^{\prime}}_{\bm{N}^{\prime}}$. Recall that

$\mu_{sb}=\sum_{i=0}^{\infty}\alpha\gamma^{i}\cdot\delta_{(-i,1)}+\sum_{i=1}^{\infty}\alpha\gamma^{i}\cdot\delta_{(i,-i+1)},$
(151)

where $\alpha=\sqrt{5}-2$ and $\gamma=\frac{\sqrt{5}-1}{2}$. Therefore

$\mathbb{P}(\overline{\bm{X}}(1)-\overline{\bm{X}}_{0}=z)=\alpha\gamma^{|z|}\quad\text{for
all}\quad z\in\mathbb{Z},$ (152)

and

$\displaystyle\mathbb{P}(\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)=z)=\alpha\gamma^{|z|+1}\quad\text{for
all}\quad z\in\mathbb{Z}_{\leq 0},$ (153)
$\displaystyle\mathbb{P}(\overline{\bm{Y}}(1)-\overline{\bm{Y}}(0)=1)=\frac{\alpha}{1-\gamma}.$
(154)

From D.2 we have that
$\mathbb{P}(\bm{S^{\prime}}_{\bm{N}^{\prime}}=x)=(1-\gamma)\gamma^{x-1}$, for
all $x\in\mathbb{Z}_{>0}$. Therefore, for all $j,z\in\mathbb{Z}_{>0}$,

$\mathbb{P}(\bm{S^{\prime}}^{1}_{\bm{N}^{\prime}}+\dots+\bm{S^{\prime}}^{j}_{\bm{N}^{\prime}}=z)=\sum_{\begin{subarray}{c}z_{1},\dots,z_{j}>0\\\
z_{1}+\dots+z_{j}=z\end{subarray}}\mathbb{P}(\bm{S^{\prime}}^{1}_{\bm{N}^{\prime}}=z_{1},\dots,\bm{S^{\prime}}^{j}_{\bm{N}^{\prime}}=z_{j})\\\
=\sum_{\begin{subarray}{c}z_{1},\dots,z_{j}>0\\\
z_{1}+\dots+z_{j}=z\end{subarray}}(1-\gamma)^{j}\gamma^{z-j}=\mathds{1}_{\\{j\leq
z\\}}(1-\gamma)^{j}\gamma^{z-j}\binom{z-1}{j-1}.$ (155)

Therefore for all $x\in\mathbb{Z}_{<0}$,

$\sum_{j=1}^{+\infty}\mathbb{P}(\bm{S^{\prime}}^{1}_{\bm{N}^{\prime}}+\dots+\bm{S^{\prime}}^{j}_{\bm{N}^{\prime}}<-x)=\sum_{j=1}^{+\infty}\sum_{z=1}^{-x-1}\mathbb{P}(\bm{S^{\prime}}^{1}_{\bm{N}^{\prime}}+\dots+\bm{S^{\prime}}^{j}_{\bm{N}^{\prime}}=z)\\\
=\sum_{z=1}^{-x-1}\gamma^{z}\sum_{j=1}^{z}(1-\gamma)^{j}\gamma^{-j}\binom{z-1}{j-1}=(1+x)(\gamma-1),$
(156)

and we can conclude that

$h^{\prime}(x)=\begin{cases}\gamma+(\gamma-1)x\quad&\text{if}\quad x<0,\\\
0\quad&\text{otherwise}.\end{cases}$ (157)

Therefore

$\beta=\frac{1}{\sqrt{2\pi}}\left(\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sigma^{\prime}}+\frac{\mathbb{E}[\bm{S}^{\prime}_{\bm{N}^{\prime}}]}{\sigma}\sum_{y^{\prime}\in\mathbb{Z}_{<0}}(\gamma+(\gamma-1)y^{\prime})\sum_{s=1}^{\infty}\mathbb{P}(\bm{S}_{s}=y^{\prime},(\bm{S}_{i})_{i\in[s-1]}\geq
0)\right)\\\
=\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sqrt{2\pi}}\left(\frac{1}{\sigma^{\prime}}+\frac{(1-\gamma)}{\sigma}\sum_{y^{\prime}\in\mathbb{Z}_{<0}}(\gamma+(\gamma-1)y^{\prime})\gamma^{-y^{\prime}-1}\right)\\\
=\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sqrt{2\pi}}\left(\frac{1}{\sigma^{\prime}}+\frac{1+\gamma}{\sigma}\right)=\frac{1}{\sqrt{2\pi}}\frac{1}{1-\gamma}\left(\frac{1}{\sigma^{\prime}}+\frac{1+\gamma}{\sigma}\right),$
(158)

where we also used that
$\mathbb{P}(\bm{S}_{\bm{N}}=-x)=(1-\gamma)\gamma^{x-1}=\mathbb{P}(\bm{S}^{\prime}_{\bm{N}^{\prime}}=x)$
and that $\mathbb{E}[-\bm{S}_{\bm{N}}]=\frac{1}{1-\gamma}$ as a consequence of
D.2 togheter with the expressions in Eqs. 152 and 153. ∎

######  Corollary 5.15.

As $k\to+\infty$,

$\sum_{\ell=0}^{+\infty}\mathbb{P}(\bm{\tau}^{+}_{\ell}=k)\sim\frac{1}{\beta}\frac{1}{2\pi\sqrt{k}}\;.$
(159)

###### Proof.

With some simple modifications of the computations in the proof of 5.14 it is
easy to show that $\sqrt{k}\cdot\mathbb{P}(\bm{\tau}^{+}_{1}=k)\to 0$. Then,
it is enough to use the same proof used for 4.7. ∎

With this result in our hands we can now continue to estimate Eq. 133. With
the same argument used for Eq. 93, we have that

$\mathbb{E}\left[\varphi\left(\frac{\overline{\bm{Z}}_{\lfloor
nt\rfloor}}{\sigma^{\prime}\sqrt{n}}\right);\overline{\bm{Z}}_{\lfloor
nt\rfloor}>0\right]\\\ =\sum_{k=0}^{\lfloor
nt\rfloor}\left(\sum_{\ell=0}^{+\infty}\mathbb{P}(\bm{\tau}^{+}_{\ell}=k)\right)\mathbb{E}\left[\varphi\left(\frac{1+\bm{S}_{\lfloor
nt\rfloor-k}}{\sigma^{\prime}\sqrt{n}}\right)\middle|(1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor-k]}>0\right]\mathbb{P}\left((1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor-k]}>0\right)\\\ =\int_{0}^{t}f_{n}^{+}(s)ds+O(1/\sqrt{n}),$ (160)

where for $1\leq k\leq\lfloor nt\rfloor-1$ and
$s\in[\frac{k}{n},\frac{k+1}{n})$,

$f^{+}_{n}(s)\coloneqq
n\left(\sum_{\ell=0}^{+\infty}\mathbb{P}(\bm{\tau}^{+}_{\ell}=k)\right)\mathbb{E}\left[\varphi\left(\frac{1+\bm{S}_{\lfloor
nt\rfloor-k}}{\sigma^{\prime}\sqrt{n}}\right)\middle|(1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor-k]}>0\right]\mathbb{P}\left((1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor-k]}>0\right).$ (161)

and $f^{+}_{n}(s)\coloneqq 0$ on $(0,\frac{1}{n})\cup[\frac{\lfloor
nt\rfloor-1}{n},t)$. Using that (as in Eq. 94)

$\mathbb{E}\left[\varphi\left(\frac{1+\bm{S}_{\lfloor
nt\rfloor}}{\sigma^{\prime}\sqrt{n}}\right)\middle|(1+\bm{S}_{i})_{i\in[\lfloor
nt\rfloor]}>0\right]\sim\frac{1}{t}\int_{0}^{+\infty}\varphi(u)ue^{-u^{2}/2t}du,$
(162)

and that from D.1,

$\mathbb{P}\left((1+\bm{S}_{i})_{i\in[n]}>0\right)\sim
2\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sigma^{\prime}\sqrt{2\pi}}\frac{1}{\sqrt{n}},$
(163)

together with 5.15, we obtain that for all $s\in(0,t)$,

$f^{+}_{n}(s)\sim
n\frac{1}{\beta}\frac{1}{2\pi\sqrt{n}\sqrt{s}}\frac{1}{t-s}\int_{0}^{+\infty}\varphi(u)ue^{-u^{2}/2(t-s)}du\cdot
2\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sigma^{\prime}\sqrt{2\pi}}\frac{h(1)}{\sqrt{n}\sqrt{t-s}}\\\
=\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sigma^{\prime}\sqrt{2\pi}\beta}\frac{1}{\pi\sqrt{s(t-s)}(t-s)}\int_{0}^{+\infty}\varphi(u)ue^{-u^{2}/2(t-s)}du.$
(164)

Therefore, using the same arguments used for Eq. 98, we can conclude that

$\mathbb{E}\left[\varphi\left(\frac{\overline{\bm{Z}}_{\lfloor
nt\rfloor}}{\sigma^{\prime}\sqrt{n}}\right);\overline{\bm{Z}}_{\lfloor
nt\rfloor}>0\right]\xrightarrow{n\to\infty}\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sigma^{\prime}\sqrt{2\pi}\beta}\int_{0}^{+\infty}\varphi(y)\frac{2e^{-y^{2}/2t}}{\sqrt{2\pi
t}}dy.$ (165)

Using the expression
$\beta=\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sqrt{2\pi}}\left(\frac{1}{\sigma^{\prime}}+\frac{1+\gamma}{\sigma}\right)$
in Eq. 158, we can conclude that

$q=\frac{\mathbb{E}[-\bm{S}_{\bm{N}}]}{\sigma^{\prime}\sqrt{2\pi}\beta}=\frac{\frac{1}{\sigma^{\prime}}}{\frac{1}{\sigma^{\prime}}+\frac{1+\gamma}{\sigma}}=\frac{1}{2},$
(166)

where in the last equality we used that $\sigma^{\prime}=\sqrt{1+\sqrt{5}}$,
$\sigma=\sqrt{2(2+\sqrt{5})}$, and $\gamma=\frac{\sqrt{5}-1}{2}$. ∎

The proof of 1.7 follows from 5.11 using exactly the same strategy explained
in Sections 4.3.2 and 4.4.

##  Appendix A An upper-bound for some local probabilities for random walks in
cones

We fix some notation being consistent with the notation used in [DW15] in
order to help the reader to follow the proof of A.1. Let $\bm{S}(n)$ be a
random walk on $\mathbb{R}^{d}$, $d\geq 1$, where

$\bm{S}(n)\coloneqq\bm{X}_{1}+\dots+\bm{X}_{n},$

and $(\bm{X}_{i})_{i}$ is a family of i.i.d. copies of a random variable
$\bm{X}=(\bm{X}^{(1)},\dots,\bm{X}^{(d)})$ with values in $\mathbb{R}^{d}$.
Denote by $\mathbb{S}^{d-1}$ the unit sphere of $\mathbb{R}^{d}$ and $\Sigma$
an open and connected subset of $\mathbb{S}^{d-1}$. Let $K$ be the cone
generated by the rays emanating from the origin and passing through $\Sigma$,
that is, $\Sigma=K\cap\mathbb{S}^{d-1}$. Let $\bm{\tau}_{x}$ be the exit time
from $K$ of the random walk with starting point $x\in K$, that is,

$\bm{\tau}_{x}\coloneqq\inf\\{n\geq 1:x+\bm{S}(n)\notin K\\}.$

We make the same assumptions on $K$ as in [DW15] (see in particular bottom of
page 994). We also recall that the choice of the cone $K$ determines a
corresponding parameter $p$ (see the equations from (2) to (3) in [DW15] for a
precise definition) that plays an important role in many formulas in [DW15].

We also impose, as in [DW15], the following assumptions on the increments of
the random walk:

  * •

$\mathbb{E}[\bm{X}^{(j)}]=0$, $\mathbb{E}[(\bm{X}^{(j)})^{2}]=1$,
$j=1,\dots,d$. In addition, we assume that151515See the bottom part of page
996 in [DW15] for a discussion on the case
$\operatorname{cov}(\bm{X}_{i},\bm{X}_{j})\neq 0$ and the fact that assuming
$\operatorname{cov}(\bm{X}_{i},\bm{X}_{j})=0$ does not restrict the generality
of the results. $\operatorname{cov}(\bm{X}_{i},\bm{X}_{j})=0$.

  * •

We assume that $\mathbb{E}[\bm{X}^{\alpha}]=0$ with $\alpha=p$ if $p>2$ and
some $\alpha>2$ if $p\leq 2$.

In [DW15, Lemma 28] it was shown that

$\mathbb{P}(x+\bm{S}(n)=y,\bm{\tau}_{x}>n)\leq C(x,y)n^{-p-d/2},\quad\text{for
all}\quad n\in\mathbb{Z}_{>0}\text{ and }x,y\in K.$

In this section, we want to make explicit the dependence in $x$ and $y$ of the
constant $C(x,y)$.

######  Proposition A.1.

There exists a constant $C>0$ independent of $x,y,n$ such that

$\mathbb{P}(x+\bm{S}(n)=y,\bm{\tau}_{x}>n)\leq
C(1+|x|^{p})(1+|y|^{p})n^{-p-d/2},\quad\text{for all}\quad
n\in\mathbb{Z}_{>0}\text{ and }x,y\in K.$

In what follows $C$ denotes any constant (possibly different from place to
place) independent of $x,y,n$.

###### Proof.

From [DW15, Lemma 27],

$\mathbb{P}(x+\bm{S}(n)=y,\bm{\tau}_{x}>n)\leq C\cdot
n^{-d/2}\cdot\mathbb{P}(\bm{\tau}_{x}>n/2).$

If we show that

$\mathbb{P}(\bm{\tau}_{x}>n)\leq C(1+|x|^{p})n^{-p/2},$ (167)

then using exactly the same proof as in [DW15, Lemma 28] we can conclude that

$\mathbb{P}(x+\bm{S}(n)=y,\bm{\tau}_{x}>n)\leq
C(1+|x|^{p})(1+|y|^{p})n^{-p-d/2},\quad\text{for all}\quad
n\in\mathbb{Z}_{>0}\text{ and }x,y\in K.$

Therefore it is enough to prove Eq. 167. Note that for all $\varepsilon>0$,

$\mathbb{P}(\bm{\tau}_{x}>n)=\mathbb{P}(\bm{\tau}_{x}>n,\bm{\nu}_{n}\leq
n^{1-\varepsilon})+\mathbb{P}(\bm{\tau}_{x}>n,\bm{\nu}_{n}>n^{1-\varepsilon}),$
(168)

where

$\bm{\nu}_{n}\coloneqq\min\\{k\geq 1:x+\bm{S}(k)\in K_{n,\varepsilon}\\},$

and

$K_{n,\varepsilon}\coloneqq\\{x\in K:\operatorname{dist}(x,\partial K)\geq
n^{1/2-\varepsilon}\\}.$

By [DW15, Lemma 14]

$\mathbb{P}(\bm{\tau}_{x}>n,\bm{\nu}_{n}>n^{1-\varepsilon})\leq\exp\\{-Cn^{\varepsilon}\\}.$
(169)

Therefore, in order to conclude the proof, we have to show that

$\mathbb{P}(\bm{\tau}_{x}>n,\bm{\nu}_{n}\leq n^{1-\varepsilon})\leq
C(1+|x|^{p})n^{-p/2}.$ (170)

By [DW15, Equation (50)],

$\mathbb{P}(\bm{\tau}_{x}>n,\bm{\nu}_{n}\leq n^{1-\varepsilon})\leq C\cdot
n^{-p/2}\Big{(}\mathbb{E}\left[u(x+\bm{S}(\bm{\nu}_{n}));\bm{\tau}_{x}>\bm{\nu}_{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]\\\
+\mathbb{E}\left[|x+\bm{S}(\bm{\nu}_{n})|^{p};\bm{\tau}_{x}>\bm{\nu}_{n},|x+\bm{S}(\bm{\nu}_{n})|>n^{-\varepsilon/8}\sqrt{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]\Big{)},$ (171)

where $u(x)$ is a function defined in [DW15, Equation (3)] that satisfies
$0\leq u(x)\leq C|x|^{p}$. We have the following results (whose proof is
postponed after the end of the proof of this proposition).

######  Claim A.2.

It holds that

$\mathbb{E}\left[u(x+\bm{S}(\bm{\nu}_{n}));\bm{\tau}_{x}>\bm{\nu}_{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]\leq C(1+|x|^{p}).$

######  Claim A.3.

It holds that

$\mathbb{E}\left[|x+\bm{S}(\bm{\nu}_{n})|^{p};\bm{\tau}_{x}>\bm{\nu}_{n},|x+\bm{S}(\bm{\nu}_{n})|>n^{-\varepsilon/8}\sqrt{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]\leq C(1+|x|^{p}).$

Note that Eq. 170 then follows from Eq. 171 and the two claims above. This
concludes the proof of A.1. ∎

It remains to prove Claims A.2 and A.3.

###### Proof of A.2.

From the proof of [DW15, Lemma 21] we have that

$\mathbb{E}\left[u(x+\bm{S}(\bm{\nu}_{n}));\bm{\tau}_{x}>\bm{\nu}_{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]=\mathbb{E}\left[\bm{Y}_{\bm{\nu}_{n}};\bm{\tau}_{x}>\bm{\nu}_{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]\\\
+\mathbb{E}\left[\sum_{k=0}^{\bm{\nu}_{n}-1}f(x+\bm{S}(k));\bm{\tau}_{x}>\bm{\nu}_{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right],$ (172)

where $\bm{Y}_{n}$ is a martingale defined by (see [DW15, Equation (20)])

$\displaystyle\bm{Y}_{0}=v(x),$ (173)
$\displaystyle\bm{Y}_{n}=v(x+\bm{S}(n))-\sum_{k=0}^{n-1}f(x+\bm{S}(k)),\quad
n\geq 0,$ (174)

with $v(\cdot)$ and $f(\cdot)$ two functions defined in [DW15, Section 1.3].
We have that the second expectation in the right-hand side of Eq. 172 is
bounded by

$\mathbb{E}\left[\sum_{k=0}^{\bm{\nu}_{n}-1}f(x+\bm{S}(k));\bm{\tau}_{x}>\bm{\nu}_{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]\leq\mathbb{E}\left[\sum_{k=0}^{\bm{\tau}_{x}-1}|f(x+\bm{S}(k))|;\bm{\tau}_{x}>\bm{\nu}_{n}\right]\leq
C(1+|x|^{p}),$ (175)

where the last bound follows from [DW15, Equation (24)]. For the first
expectation in the right-hand side of Eq. 172, we can write using the
computations after [DW15, Equation (52)],

$\mathbb{E}\left[\bm{Y}_{\bm{\nu}_{n}};\bm{\tau}_{x}>\bm{\nu}_{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\right]=u(x)-\mathbb{E}\left[\bm{Y}_{\bm{\tau}_{x}};\bm{\tau}_{x}\leq\bm{\nu}_{n}\wedge
n^{1-\varepsilon}\right]\\\
-\mathbb{E}\left[u(x+\bm{S}(n^{1-\varepsilon}));\bm{\tau}_{x}>n^{1-\varepsilon},\bm{\nu}_{n}>n^{1-\varepsilon}\right]+\mathbb{E}\left[\sum_{k=0}^{n^{1-\varepsilon}-1}f(x+\bm{S}(k));\bm{\tau}_{x}>n^{1-\varepsilon},\bm{\nu}_{n}>n^{1-\varepsilon}\right],$
(176)

where $x\wedge y=\min\\{x,y\\}$. It remains to bound the four terms above. As
we already mentioned $0\leq u(x)\leq C|x|^{p}$, so it is enough to find an
appropriate bound for the second and fourth expectations . Recalling the
definition of $\bm{Y}_{n}$ in Eq. 173 and using the bounds in [DW15, Equations
(22) and (24)] we have that

$\left|\mathbb{E}\left[\bm{Y}_{\bm{\tau}_{x}};\bm{\tau}_{x}\leq\bm{\nu}_{n}\wedge
n^{1-\varepsilon}\right]\right|\leq C(1+|x|^{p}).$ (177)

Finally, for the fourth expectation in the left-hand side of Eq. 176 we can
use the same arguments already used for Eq. 175 to obtain that

$\mathbb{E}\left[\sum_{k=0}^{n^{1-\varepsilon}-1}f(x+\bm{S}(k));\bm{\tau}_{x}>n^{1-\varepsilon},\bm{\nu}_{n}>n^{1-\varepsilon}\right]\leq
C(1+|x|^{p}).$ (178)

This concludes the proof. ∎

###### Proof of A.3.

As in the proof of [DW15, Lemma 24] we set

$\bm{\mu}_{n}\coloneqq\min\\{j\geq 1:|\bm{X}_{j}|>n^{1/2-\varepsilon/4}\\},$
(179)

and we write

$\displaystyle\mathbb{E}\Big{[}|x+\bm{S}(\bm{\nu}_{n})|^{p};$
$\displaystyle\bm{\tau}_{x}>\bm{\nu}_{n},|x+\bm{S}(\bm{\nu}_{n})|>n^{-\varepsilon/8}\sqrt{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon}\Big{]}$ (180) $\displaystyle=$
$\displaystyle\mathbb{E}\left[|x+\bm{S}(\bm{\nu}_{n})|^{p};\bm{\tau}_{x}>\bm{\nu}_{n},|x+\bm{S}(\bm{\nu}_{n})|>n^{-\varepsilon/8}\sqrt{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon},\bm{\mu}_{n}\leq\bm{\nu}_{n}\right]$ (181) $\displaystyle+$
$\displaystyle\mathbb{E}\left[|x+\bm{S}(\bm{\nu}_{n})|^{p};\bm{\tau}_{x}>\bm{\nu}_{n},|x+\bm{S}(\bm{\nu}_{n})|>n^{-\varepsilon/8}\sqrt{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon},\bm{\mu}_{n}>\bm{\nu}_{n}\right].$ (182)

From the beginning of the proof of [DW15, Lemma 24] we have that the second
expectation in the right-hand side of the equation above is bounded by
$\exp\\{-Cn^{\varepsilon/8}\\}$, therefore we focus on the first expectation
above. Using the first displayed equation after [DW15, Equation (59)] we have
that

$\mathbb{E}\left[|x+\bm{S}(\bm{\nu}_{n})|^{p};\bm{\tau}_{x}>\bm{\nu}_{n},|x+\bm{S}(\bm{\nu}_{n})|>n^{-\varepsilon/8}\sqrt{n},\bm{\nu}_{n}\leq
n^{1-\varepsilon},\bm{\mu}_{n}\leq\bm{\nu}_{n}\right]\\\
\leq\sum_{j=1}^{n^{1-\varepsilon}}\mathbb{E}\left[|x+\bm{S}(\bm{\nu}_{n})|^{p};\bm{\tau}_{x}>j,\bm{\nu}_{n}\leq
n^{1-\varepsilon},j\leq\bm{\nu}_{n},\bm{\mu}_{n}=j\right].$ (183)

We split the sum above in three parts using the bound

$|x+\bm{S}(\bm{\nu}_{n})|^{p}\leq
C\left(|x+\bm{S}(j-1)|^{p}+|\bm{X}_{j}|^{p}+|\bm{S}(\bm{\nu}_{n})-\bm{S}(j)|^{p}\right).$
(184)

For the third term we have that

$\sum_{j=1}^{n^{1-\varepsilon}}\mathbb{E}\left[|\bm{S}(\bm{\nu}_{n})-\bm{S}(j)|^{p};\bm{\tau}_{x}>j,\bm{\nu}_{n}\leq
n^{1-\varepsilon},j\leq\bm{\nu}_{n},\bm{\mu}_{n}=j\right]\\\
\leq\begin{cases}Cn^{-p\varepsilon/4}\mathbb{E}[|\bm{X}|^{p}]\mathbb{E}[\bm{\tau}_{x}]\leq
Cn^{-p\varepsilon/4}(1+|x|^{2}),\quad&\text{if}\quad p>2,\\\
C(1+|x|^{p})n^{-\delta},\text{ with }\delta>0,\quad&\text{if}\quad p\leq
2,\end{cases}$ (185)

where the bound for $p>2$ is obtained using the first and third equations at
page 1026 of [DW15] and [DW15, Lemma 10], while the bound for $p\leq 2$ is
obtained using the first, fourth and fifth equation at page 1026 of [DW15] and
[DW15, Lemma 10].

For the second term we have that

$\sum_{j=1}^{n^{1-\varepsilon}}\mathbb{E}\left[|\bm{X}_{j}|^{p};\bm{\tau}_{x}>j,\bm{\nu}_{n}\leq
n^{1-\varepsilon},j\leq\bm{\nu}_{n},\bm{\mu}_{n}=j\right]\leq\begin{cases}C(1+|x|^{2})\cdot
o(1),\quad&\text{if}\quad p>2,\\\ C(1+|x|^{p})n^{-\delta},\text{ with
}\delta>0,\quad&\text{if}\quad p\leq 2,\end{cases}$ (186)

where the bounds are obtained from the equations on top of page 1027 in
[DW15].

For the first term we have that

$\sum_{j=1}^{n^{1-\varepsilon}}\mathbb{E}\left[|x+\bm{S}(j-1)|^{p};\bm{\tau}_{x}>j,\bm{\nu}_{n}\leq
n^{1-\varepsilon},j\leq\bm{\nu}_{n},\bm{\mu}_{n}=j\right]\\\
\leq\begin{cases}2^{p}(|x|^{p}dn^{-\varepsilon/2}+Cn^{3p/2+1}\exp\\{-Cn^{\varepsilon/8}\\}+Cn^{-p\varepsilon/4}(1+|x|^{2})),\quad&\text{if}\quad
p>2,\\\
2^{p}(|x|^{p}dn^{-\varepsilon/2}+Cn^{3p/2+1}\exp\\{-Cn^{\varepsilon/8}\\}+C(1+|x|^{p})n^{-\delta}),\quad&\text{if}\quad
p\leq 2,\end{cases}$ (187)

where the bounds are obtained from Equation (63) and the three consecutive
equations in [DW15] and the same bounds used for Eq. 185. These three bounds
conclude the proof. ∎

##  Appendix B A toolbox for strong-Baxter walks

In this section we rewrite A.1 using the notation and the specific random
walks considered in Section 4.1.

Recall that $(\bm{W}_{n}(i))_{i\geq 1}$ is a random walk with increments in

$I_{Sb}\coloneqq\\{(-i,0):i\geq 1\\}\cup\\{(0,1)\\}\cup\\{(1,-i):i\geq 0\\}.$
(188)

and with step distribution

$\mu_{Sb}=\sum_{i=1}^{\infty}\alpha\gamma^{i}\cdot\delta_{(-i,0)}+\alpha\theta^{-1}\cdot\delta_{(0,1)}+\sum_{i=0}^{\infty}\alpha\gamma^{-1}\theta^{i}\cdot\delta_{(1,-i)},$
(189)

where the various parameters are given below Eq. 26. Recall also that
$\mathcal{W}^{n}_{Sb}$ denotes the set of two-dimensional walks in the non-
negative quadrant, starting at $(0,0)$, with increments in $I_{Sb}$.
Therefore, setting $K=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{1}\geq 0,x_{2}\geq
0\\}$, we have that

$\mathbb{P}\left((\bm{W}_{n}(i))_{i\in[n]}\in\mathcal{W}_{Sb}^{n}\middle|\bm{W}_{n}(n)=(h,\ell)\right)=\mathbb{P}\left(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(n)=(0,0),(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\in[n]}\in
K\middle|\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(1)=(h,\ell)\right),$
(190)

where we recall that
$(\stackrel{{\scriptstyle\leftarrow}}{{\bm{W}_{n}}}(i))_{i\in[n]}$ is the
time-reversed walk obtained from $(\bm{W}_{n}(i))_{i\in[n]}$. Therefore we
have the following reinterpretation of A.1.

Let $p>0$ be the parameter previously described in Appendix A associated with
the walk $(\bm{W}_{n}(i))_{i\geq 1}$ and the cone $K$ (we do not need its
explicit expression).

######  Corollary B.1.

There exists a constant $C>0$ independent of $h,\ell,n$ such that

$\mathbb{P}\left((\bm{W}_{n}(i))_{i\in[n]}\in\mathcal{W}_{Sb}^{n}\middle|\bm{W}_{n}(n)=(h,\ell)\right)\leq
C(1+|(h,\ell)|^{p})n^{-p-1},\quad\text{for all}\quad
n,h,\ell\in\mathbb{Z}_{\geq 0},$ (191)

where $|(h,\ell)|$ denotes the Euclidean norm of the vector $(h,\ell)$.

We also state the following consequence of [DW15, Theorem 6].

######  Corollary B.2.

Fix $h,\ell\in\mathbb{Z}_{\geq 0}$. There exists a constant $C>0$ independent
of $n$ such that

$\mathbb{P}\left((\bm{W}_{n}(i))_{i\in[n]}\in\mathcal{W}_{Sb}^{n}\middle|\bm{W}_{n}(n)=(h,\ell)\right)\geq
Cn^{-p-1},\quad\text{for all}\quad n\in\mathbb{Z}_{\geq 0}.$ (192)

##  Appendix C Asymptotic estimates

######  Proposition C.1.

Let $(a_{n})_{n\in\mathbb{Z}_{\geq 0}}$ and $(b_{n})_{n\in\mathbb{Z}_{\geq
0}}$ be two sequences of positive numbers such that for some $\alpha>1$:

  * •

$a_{n}\sim a\cdot n^{-\alpha}$, for some $a>0$;

  * •

$b_{n}\sim b\cdot n^{-\alpha}$, for some $b>0$;

Then as $n\to\infty$,

$\sum_{k=0}^{n}a_{k}\cdot b_{n-k}\sim
n^{-\alpha}\left(a\cdot\sum_{k=0}^{\infty}b_{k}+b\cdot\sum_{k=0}^{\infty}a_{k}\right).$

###### Proof.

We split the sum in four parts as follows:
$\sum_{k=0}^{n}a_{k}b_{n-k}=t_{n}+u_{n}+v_{n}+w_{n}\quad$, where

$t_{n}=\sum_{k=0}^{\lfloor n^{\alpha}\rfloor}a_{k}b_{n-k},\quad
u_{n}=\sum_{k=\lfloor n^{\alpha}\rfloor+1}^{\lfloor
n/2\rfloor}a_{k}b_{n-k},\quad v_{n}=\sum_{k=\lfloor n/2\rfloor+1}^{n-\lfloor
n^{\alpha}\rfloor}a_{k}b_{n-k},\quad w_{n}=\sum_{k=n-\lfloor
n^{\alpha}\rfloor+1}^{n}a_{k}b_{n-k}.$ (193)

By assumption, there exists $B>0$ such that for all $n\geq 1$, $b_{n}\leq
B\cdot n^{-\alpha}$. Then

$u_{n}\leq B\sum_{k=\lfloor n^{\alpha}\rfloor+1}^{\lfloor
n/2\rfloor}a_{k}\cdot(n-\lfloor n/2\rfloor)^{-\alpha}=o(n^{-\alpha}).$

Similarly, $v_{n}=o\left(n^{-\alpha}\right)$. It remains to show that

$t_{n}\sim n^{-\alpha}\cdot b\sum_{k=0}^{+\infty}a_{k},\qquad w_{n}\sim
n^{-\alpha}\cdot a\sum_{k=0}^{+\infty}b_{k}.$ (194)

We can write $t_{n}-n^{-\alpha}\cdot
b\sum_{k=0}^{+\infty}a_{k}=x_{n}+y_{n}+z_{n}$, where

$x_{n}=b\sum_{k=0}^{\lfloor
n^{\alpha}\rfloor}a_{k}\left((n-k)^{-\alpha}-n^{-\alpha}\right),\quad
y_{n}=\sum_{k=0}^{\lfloor
n^{\alpha}\rfloor}a_{k}\left(b_{n-k}-b(n-k)^{-\alpha}\right),\quad
z_{n}=-n^{-\alpha}\cdot b\sum_{k=\lfloor n^{\alpha}\rfloor+1}^{+\infty}a_{k}.$

Obviously $\ z_{n}=o\left(n^{-\alpha}\right)$. For $x_{n}$ note that

$0\leq x_{n}\leq b\sum_{k=0}^{\lfloor n^{\alpha}\rfloor}a_{k}\left((n-\lfloor
n^{\alpha}\rfloor)^{-\alpha}-n^{-\alpha}\right)\leq
n^{-\alpha}\left(\left(1-\frac{\lfloor
n^{\alpha}\rfloor}{n}\right)^{-\alpha}-1\right)b\sum_{k=0}^{+\infty}a_{k}.$

Hence $\ x_{n}=o\left(n^{-\alpha}\right)$. Finally for $y_{n}$ we have that

$|y_{n}|\leq\sum_{k=0}^{\lfloor
n^{\alpha}\rfloor}a_{k}(n-k)^{-\alpha}\left|(n-k)^{\alpha}b_{n-k}-b\right|\leq(n-n^{\alpha})^{-\alpha}\left(\sup_{\ell\geq
n-\lfloor
n^{\alpha}\rfloor}|\ell^{\alpha}b_{\ell}-b|\right)\sum_{k=0}^{+\infty}a_{k}.$

Therefore $\ y_{n}=o\left(n^{-\alpha}\right)$. This proves that $t_{n}\sim
n^{-\alpha}\cdot b\sum_{k=0}^{+\infty}a_{k}.$ Similarly, we have that
$w_{n}\sim n^{-\alpha}\cdot a\sum_{k=0}^{+\infty}b_{k}$. ∎

##  Appendix D Local estimates for one-dimensional random walks conditioned to
be positive

Let $(\bm{S}_{n})_{n\in\mathbb{Z}_{\geq 0}}$ be a one-dimensional random walk
started at zero at time zero. Assume that the increments of
$(\bm{S}_{n})_{n\in\mathbb{Z}_{\geq 0}}$ are i.i.d., centered and with finite
variance $\sigma^{2}$. Let

$\bm{\tau}=\inf\\{n>0|\bm{S}_{n}<0\\},$

and $h$ be the function defined by

$h(x)=\begin{cases}1+\sum_{j=1}^{+\infty}\mathbb{P}(\bm{S}^{1}_{\bm{\tau}}+\dots+\bm{S}^{j}_{\bm{\tau}}>-x)\quad&\text{if}\quad
x>0,\\\ 0\quad&\text{otherwise},\end{cases}$ (195)

where the random variables $\bm{S}^{i}_{\bm{\tau}}$ are independent copies of
$\bm{S}_{\bm{\tau}}$. We have the following local estimates (see for
instance161616We remark that the expression in Eq.(5) in [NP19] is wrong. For
a correct expression see [VW09, Eq. (7)]. [NP19, Lemma 2.2] or [VW09, Theorem
4]):

######  Lemma D.1.

The following estimates hold

$\displaystyle\mathbb{P}\left((x+\bm{S}_{i})_{i\in[n]}>0\right)\sim
2\frac{\mathbb{E}[-\bm{S}_{\bm{\tau}}]}{\sigma\sqrt{2\pi}}\frac{h(x)}{\sqrt{n}},\quad\text{for
all}\quad x\in\mathbb{Z}_{>0},$ (196)
$\displaystyle\mathbb{P}(x+\bm{S}_{n}=y,(x+\bm{S}_{i})_{i\in[n]}>0)\sim\frac{c}{\sigma\sqrt{2\pi}}\frac{h(x)\tilde{h}(y)}{n^{3/2}},\quad\text{for
all}\quad x,y\in\mathbb{Z}_{>0},$ (197)
$\displaystyle\mathbb{P}(x+\bm{S}_{n}=y,(x+\bm{S}_{i})_{i\in[n]}>0)\leq
C\frac{h(x)\tilde{h}(y)}{n^{3/2}},\quad\text{for all}\quad
x,y\in\mathbb{Z}_{>0}.$ (198)

where
$c=\frac{\mathbb{E}[-\bm{S}_{\bm{\tau}}]}{\sum_{z\in\mathbb{Z}_{>0}}\tilde{h}(z)\mathbb{P}(\bm{S}_{1}\leq-z)}$,
$C$ is another constant independent of $n,x,y$, and $\tilde{h}$ is the
function defined in Eq. 195 for the walk $(-\bm{S}_{n})_{n\in\mathbb{Z}_{\geq
0}}$.

We also explicitly compute $\mathbb{E}[-\bm{S}_{\bm{\tau}}]$ in a specific
case.

######  Lemma D.2.

Assume that171717Note that we are not assuming any restriction on the
expression for $\mathbb{P}(\bm{S}_{1}=y)$ when $y\in\mathbb{Z}_{\geq 0}$.
there exists two constants $\alpha>0$ and $\gamma>0$ such that
$\mathbb{P}(\bm{S}_{1}=y)=\alpha\gamma^{-y}$ for all $y\in\mathbb{Z}_{<0}$.
Then

$\mathbb{P}(\bm{S}_{\bm{\tau}}=x)=(1-\gamma)\gamma^{-x-1},\quad\text{for
all}\quad
x\in\mathbb{Z}_{<0},\qquad\text{and}\qquad\mathbb{E}[-\bm{S}_{\bm{\tau}}]=\frac{1}{1-\gamma}.$
(199)

###### Proof.

Since $\bm{\tau}$ is a.s. finite, for every $x\in\mathbb{Z}_{<0}$,

$\mathbb{P}(\bm{S}_{\bm{\tau}}=x)=\sum_{h=1}^{\infty}\mathbb{P}(\bm{S}_{h}=x,(\bm{S}_{i})_{i\in[h-1]}\geq
0)\\\ =\sum_{h=1}^{\infty}\sum_{x^{\prime}\in\mathbb{Z}_{\geq
0}}\mathbb{P}(\bm{S}_{h-1}=x^{\prime},(\bm{S}_{i})_{i\in[h-1]}\geq
0)\mathbb{P}(\bm{S}_{h}=x|\bm{S}_{h-1}=x^{\prime}).$ (200)

Using that by assumption
$\mathbb{P}(\bm{S}_{h}=x|\bm{S}_{h-1}=x^{\prime})=\alpha\gamma^{x^{\prime}-x}$
and setting

$C=\sum_{h=1}^{\infty}\sum_{x^{\prime}\in\mathbb{Z}_{\geq
0}}\mathbb{P}(\bm{S}_{h-1}=x^{\prime},(\bm{S}_{i})_{i\in[h-1]}\geq
0)\alpha\gamma^{x^{\prime}},$

we get that $\mathbb{P}(\bm{S}_{\bm{\tau}}=x)=C\cdot\gamma^{-x}$. Using that
$1=\sum_{x\in\mathbb{Z}_{<0}}\mathbb{P}(\bm{S}_{\bm{\tau}}=x)=\sum_{x\in\mathbb{Z}_{<0}}C\cdot\gamma^{-x}=\frac{C\gamma}{1-\gamma}$.
We conclude that $\mathbb{P}(\bm{S}_{\bm{\tau}}=x)=(1-\gamma)\gamma^{-x-1}$.
The statement for the expectation follows with a basic computation. ∎

##  Appendix E Technical proofs

We give here the details of the proofs of Propositions 4.8 and 4.9 that we
skipped in Section 4.3.2.

###### Proof of 4.8.

The convergence in distribution
${{\bm{\mathcal{W}}}}_{n}\xrightarrow[n\to\infty]{d}{\bm{\mathcal{E}}}_{\rho}$
follows from 4.3 together with 4.2.

Now let $0<\varepsilon<\min\\{u,(1-u)\\}$. Note that
$\left(({{\bm{\mathcal{W}}}}_{n}-{{\bm{\mathcal{W}}}}_{n}(u))|_{[u,1-\varepsilon]},{\bm{\mathcal{Z}}}^{(u)}_{n}|_{[u,1-\varepsilon]}\right)$
is a measurable functional of $(\bm{W}_{n}(k)-\bm{W}_{n}(\lfloor\varepsilon
n\rfloor))_{\lfloor\varepsilon n\rfloor\leq
k\leq\lfloor(1-\varepsilon)n\rfloor}$. Using 4.4 together with standard
absolute continuity arguments (see for instance [Bor21c, Proposition A.1]), we
obtain that

$\left(({{\bm{\mathcal{W}}}}_{n}-{\bm{\mathcal{W}}}_{n}(u))|_{[u,1-\varepsilon]},{\bm{\mathcal{Z}}}^{(u)}_{n}|_{[u,1-\varepsilon]}\right)\xrightarrow[n\to\infty]{d}\left(({\bm{\mathcal{E}}}_{\rho}-{\bm{\mathcal{E}}}_{\rho}(u))|_{[u,1-\varepsilon]},{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}|_{[u,1-\varepsilon]}\right).$
(201)

Using two times Prokorov’s theorem, as already done in 4.4, we have that the
sequence

$\left({\bm{\mathcal{W}}}_{n},\,\,\left(({{\bm{\mathcal{W}}}}_{n}-{\bm{\mathcal{W}}}_{n}(u))|_{[u,1-\varepsilon]},{{\bm{\mathcal{Z}}}}^{(u)}_{n}|_{[u,1-\varepsilon]}\right)_{\varepsilon\in\mathbb{Q}\cap(0,u\wedge
1-u)}\right)$ (202)

is tight and its limit in distribution must be

$\left({\bm{\mathcal{E}}}_{\rho},\,\,\left(({\bm{\mathcal{E}}}_{\rho}-{\bm{\mathcal{E}}}_{\rho}(u))|_{[u,1-\varepsilon]},{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}|_{[u,1-\varepsilon]}\right)_{\varepsilon\in\mathbb{Q}\cap(0,u\wedge
1-u)}\right)$ (203)

because ${\bm{Z}}^{(u)}_{n}|_{[u,1-\varepsilon]}$ is a measurable functional
of $(\bm{W}_{n}(k)-\bm{W}_{n}(\lfloor\varepsilon
n\rfloor))_{\lfloor\varepsilon n\rfloor\leq
k\leq\lfloor(1-\varepsilon)n\rfloor}$ and the restriction operation
${\bm{\mathcal{W}}}_{n}\mapsto({\bm{\mathcal{W}}}_{n}-{\bm{\mathcal{W}}}_{n}(u))|_{[u,1-\varepsilon]}$
is continuous. Therefore we obtain convergence in distribution of the sequence
in Eq. 202 to the limit in Eq. 203. By Skorokhod’s theorem we can now assume
that a.s., we have uniform convergence on $[0,1]$ of
${{\bm{\mathcal{W}}}}_{n}$ to ${\bm{\mathcal{E}}}_{\rho}$, and uniform
convergence on $[u,1-\varepsilon]$ of ${\bm{\mathcal{Z}}}^{(u)}$ to
${\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ for every
$\varepsilon>0,\varepsilon\in\mathbb{Q}$. This is enough to conclude the
proof. ∎

###### Proof of 4.9.

Fix $u_{1},\ldots,u_{k}\in(0,1)$. The fact that
${\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ is a measurable functions of
${\bm{\mathcal{E}}}_{\rho}$ and tightness, give that convergence in
distribution in Eq. 101 holds jointly for $u\in\\{u_{1},\ldots,u_{k}\\}$.
Therefore, for every continuous bounded
$\varphi:\mathcal{C}([0,1],\mathbb{R}^{2})\times(\mathcal{C}([0,1],\mathbb{R})\times\mathcal{C}([0,1),\mathbb{R}))^{\mathbb{Z}_{>0}}\to\mathbb{R}$,
it holds that

$\mathbb{E}\left[\varphi\left({{\bm{\mathcal{W}}}}_{n},\left({{\bm{\mathcal{Z}}}}^{(u_{i})}_{n}\right)_{i\in[k]}\right)\right]\to\mathbb{E}\left[\varphi\left({\bm{\mathcal{E}}}_{\rho},\left({{\bm{\mathcal{Z}}}}^{(u_{i})}_{\rho,q},\right)_{i\in[k]}\right)\right].$
(204)

Using dominated convergence theorem, it is possible to integrate this over
$u_{1},\ldots,u_{k}\in[0,1]$. Therefore, using Fubini–Tonelli’s theorem, we
can conclude that

$\mathbb{E}\left[\varphi\left({{\bm{\mathcal{W}}}}_{n},\left({{\bm{\mathcal{Z}}}}^{(\bm{u}_{i})}_{n}\right)_{i\in[k]}\right)\right]\to\mathbb{E}\left[\varphi\left({\bm{\mathcal{E}}}_{\rho},\left({{\bm{\mathcal{Z}}}}^{(\bm{u}_{i})}_{\rho,q}\right)_{i\in[k]}\right)\right].$
(205)

Noting that $k$ is arbitrary, we obtain convergence in distribution in the
product topology. ∎

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††thanks: These two authors contributed equally to this work

# Strong coupling of antiferromagnetic resonance with sub-THz cavity fields

M. Białek$\dagger$<EMAIL_ADDRESS>Institute of Physics, École
Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland  J.
Zhang  Fert Beijing Institute, School of Microelectronics, Beijing Advanced
Innovation Center for Big Data and Brain Computing, Beihang University,
Beijing 100191, China  H. Yu<EMAIL_ADDRESS>Fert Beijing Institute,
School of Microelectronics, Beijing Advanced Innovation Center for Big Data
and Brain Computing, Beihang University, Beijing 100191, China  J.-Ph.
Ansermet  Institute of Physics, École Polytechnique Fédérale de Lausanne
(EPFL), 1015 Lausanne, Switzerland

###### Abstract

Strong coupling of electromagnetic cavity fields with antiferromagnetic spin
waves in hematite ($\alpha$-Fe2O3) was achieved above room temperature. A cube
of hematite was placed in a metallic tube and transmission was measured, using
a continuous-wave THz spectrometer. Spectra, collected as a function of
temperature, reveal the formation of magnetic polaritons.

##  I Introduction

The hybrid nature of strongly coupled light-matter states, were the
dissipation rate is lower than the exchange rate (Rabi frequency) [1], has
attracted a lot of attention since the late 1980s [2]. In the 2000s decade,
strong light-matter coupling was investigated in the solid state [3], where
the coupling of light interacting with $N$ resonators increases by a factor of
$\sqrt{N}$ [4]. This regime might lead to device elements that may play a role
in quantum devices [5, 6, 1, 7, 8, 9]. In the 2010s, interest turned toward
magnon-photon coupling in ferromagnetic materials [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22], as spins benefit from a relatively low coupling to
their environment [6, 23]. Achieving strong coupling with antiferromagnets has
the advantage of operating at frequencies in the terahertz (THz) range [24,
25] that brings a lot of interesting physics unreachable in ferromagnets [26,
27, 28]. There is a large variety of antiferromagnets available, and many of
them show magnetic ordering above room temperature. The absence of stray
fields could in principle allow for extremely dense packing of elements [29,
24]. However, there are reports only on a few examples of weak magnon-photon
coupling in antiferromagnets [30, 31]. Strong coupling was only achieved at
low frequencies [32], in bulk samples [33, 34] or via an indirect coupling
[35, 36], while direct strong cavity-magnon coupling had not been shown so far
[37]. One of the technical difficulties is constructing high-frequency
cavities that are well coupled to wave guides. Here, we present a relatively
simple implementation that allows strong coupling of the antiferromagnetic
resonance and cavity electromagnetic modes.

In this letter, we show strong coupling between electromagnetic fields of a
3-dimensional cavity and spin waves in hematite ($\alpha$-Fe2O3), a very
common room-temperature antiferromagnet. Hematite crystallizes in
approximately hexagonal structure with space group R$\overline{3}$c. Precise
measurements show that the actual symmetry is monoclinic C2/c or C2’/c’ [38].
Below the Néel temperature $T_{N}\approx 955$ K [39], the Fe3+ magnetic
moments orient in an antiferromagnetic order. Above the spin-reorientation
transition (Morin phase transition) at about $T_{M}\approx 260$ K [40], the
superexchange Dzialoshinskii-Moriya interaction leads to a canting of the two
sublattices that gives rise to a net magnetisation $\mathbf{m}$ [41], i.e.
making this material a weak ferromagnet. Owing to the spin canting, the
antiferromagnetic resonance has two modes, one of which is at a high
frequency. This quasi-antiferromagnetic resonance (qAFMR) mode is excited by
dynamical magnetic field $\mathbf{h}$ parallel to the magnetization
($\mathbf{h}\parallel\mathbf{m}$). At temperatures above room temperature the
width of qAFMR is about 1 GHz only, and its frequency rises sharply with
temperature [40].

##  II Experimental

Figure 1:  Plane electromagnetic wave (blue wiggles) is focused into a
cylinder cavity containing a hematite cube. The cavity magnetic field (blue
curve) interacts with the resonance (violet circles) of two antiferromagnetic
sublattices (red arrows). Measured: transmitted linear polarization parallel
to that of the incident beam.

Thanks to the development of frequency extenders for vector network analyzers
(VNA), continuous-wave spectroscopic measurements up to 1.5 THz can be rapidly
conducted with a high frequency resolution and with a very high dynamic range
[42, 43, 44, 31, 45].

Out of a bulk natural single crystal, we cut a cube of $l_{s}=0.2$ mm edge
length, with faces in the $a$, $b$ and $c$ crystalline directions. The cube
was placed inside a tube of $l_{c}=2$ mm length and $2r_{c}=0.58$ mm internal
diameter, cut out of a gold-plated stainless steel needle. The needle was
wrapped with Teflon tape to prevent the cube from dropping out of the tube.
The tube with the cube in it was inserted in a copper holder composed of two
cones (cone angle, about 60∘) which focused the THz beam into the cavity (Fig.
1). The THz beam propagated between the source, sample holder and detector in
oversized metallic waveguides, 8 mm in diameter, matching the larger diameter
of the cones. The holder was placed between Peltier elements that controlled
its temperature, which was monitored with a K-type thermocouple inserted in a
hole on its side. The detector measured the transmitted power and phase of the
THz electric field. Temperature scans started from the highest temperature
with a step of $\Delta T=-0.25$ K. This temperature step was so chosen that
the frequency change was smaller than the line width of the resonance. After
stabilizing the sample temperature $T$, we measured transmission as a function
of radiation frequency $f$.

The source emits linearly polarized beam of THz radiation and the detector
detects only radiation of linear polarization that matches its waveguide, i.e.
it acts like a polarization filter. Rotation of the polarization plane is
possible by rotating source and detector about the optical axis. We define the
polarization angle $\delta$ as the angle of the radiation $h$ field with
respect to the plane of the optical table (Fig. 1). We measured transmission
in a polarization angle range of $\delta=-90^{\circ}$ to $+90^{\circ}$.

##  III Results and analysis

Figure 2:  Normalized transmission magnitude at polarization angle
$\delta=30^{\circ}$. (a) Measured spectra (violet lines) and fit using Eq. 1
(green lines). (b) Map of the spectra, green dashed lines show interacting and
non-interacting modes fitted with Eq. 3. (c) Map of spectra fitted with Eq. 1.

In Fig. 2a we can see avoided crossing of the cavity mode at $f_{1}\approx
242.5$ GHz with the qAFMR mode, the frequency of which $f_{r}(T)$ rises
approximately linearly with temperature (see appendix). This result was
obtained at polarization angle $\delta=30^{\circ}$. We fitted the observed
spectra using the equation developed in the framework of input-output theory
[10, 46]:

$S_{21}=1+\frac{a_{1}}{i(f-f_{1}(T))-\frac{\kappa_{1}}{2}+\frac{G_{1}^{2}}{i(f-f_{r}(T))-g/2}},$
(1)

where, $a_{1}=(0.32-i0.02)$ GHz is a complex parameter describing the coupling
of the cavity with the source and the detector (see Appendix). The parameter
$f_{1}(T)\approx 242.5$ GHz is the observed frequency of the 1st cavity mode
and $\kappa_{1}=2.1$ GHz describes its width, $2G_{1}=6.2$ GHz is the minimum
splitting between upper and lower polariton branches and $g=0.5$ GHz describes
qAFMR width. With these parameters, we can determine the cooperativity factor

$C=\frac{4G_{1}^{2}}{\kappa_{1}g}\approx 40,$ (2)

that is, the square of a ratio of the Rabi splitting ($2G_{1}$) and the losses
of the polariton states. This means that the splitting was about 6 times
larger than the linewidth. According to the harmonic coupling model, the upper
and lower polariton frequencies are given by [47, 12],

$f_{\pm}=\frac{1}{2}\left(f_{1}+f_{r}\pm\sqrt{(f_{1}-f_{r})^{2}+4g_{i}^{2}f_{1}}\right).$
(3)

The coupled mode frequencies $f_{pm}$ are drawn on Fig. 2b as green dotted
lines, with $2g_{i}\sqrt{f_{1}}=2G_{1}=6.2$ GHz, as well as the non-
interacting mode frequencies $f_{r}(T)$ and $f_{1}(T)$. Clearly, this
prediction matches that of Eq. 1 (Fig. 2b). The discrepancies between fits
using Eq. 1 and data (Fig. 2a) may arise from interactions with other cavity
modes, as we discuss in further sections of this communication

###  III.1 Microscopic model

The coupling strength can be calculated using a microscopic model [12]

$g_{i}=\frac{g_{s}\mu_{B}}{2h}\sqrt{\frac{\mu_{0}h}{2}\rho\frac{V_{s}}{V_{c}}},$
(4)

where, $g_{s}=2$, $\rho$ is density of resonators in the hematite cube of a
volume $V_{s}=l_{s}^{3}$ and $V_{c}$, the volume of the metal tube, $V_{c}=\pi
r_{c}^{2}l_{c}\approx 0.528$ mm3. In Eq. 4, each magnon is coupled to the
electromagnetic cavity mode with a coupling strength $g_{s}\mu_{B}B_{0}/2h$
[48], where $B_{0}=\sqrt{\mu_{0}hf_{1}/2V_{c}}\approx 4.4\cdot 10^{-10}$ T is
the magnetic component of vacuum fluctuations [49] (here $f_{1}=242.5$ GHz).
This is a few orders of magnitude smaller than the amplitude of the THz field
in our experiment. The collective coupling strength of $N=\rho V_{s}$
oscillators is increased by a factor $\sqrt{N}$ [4, 48, 12, 14, 35], thus
$g_{i}$ depends only on the oscillators density $\rho$ and the ratio of the
crystal volume to the cavity volume [12, 35], that is $V_{s}/V_{c}\approx
1.51\times 10^{-2}$.

We assumed the density of oscillators in hematite is $\rho=5\rho_{Fe}$ [50,
51], where $\rho_{Fe}=3.987\times 10^{28}$ m-3 [52] and factor $5$ comes from
the magnetic moment of Fe3+ ions [53]. Density of iron atoms in hematite is
quite high compared to that of many common antiferromagnets. Hence, it is a
good material to achieve strong coupling. Taking this into account gives
$g_{i}=1.57\times 10^{4}$ Hz1/2, i.e. the splitting is predicted to be
$2G_{p}=2g_{i}\sqrt{f_{1}}\approx 15.4$ GHz for the cavity mode at
$f_{1}=242.5$ GHz. The observed splitting $2G_{1}=6.2$ GHz is about 40% of
that value. This discrepancy results from an imperfect matching of the
electromagnetic mode with the antiferromagnetic resonance in the hematite cube
which is excited by fields parallel to the weak ferromagnetic moment. As
confirmed in the simulation discussed below (Fig. 5), the cavity magnetic
field in the volume of the cube does not excite all the spins with full
amplitude. We calculated that in the 1st mode, the magnetic field component
$h_{z}^{2}$ makes about 78% of all the integrated magnetic field energy in the
cube volume (Tab. 1). The remaining difference between the predicted and the
observed splitting must result from the magnetic moment of the cube being
titled away from the $z$ axis.

###  III.2 Angular dependence

Figure 3:  (a, b) Angular dependence of $a_{1}$ (Eq. 1) in two configurations
of the experiment. (c) Normalized $|S_{21}|$ at $\delta=-75^{\circ}$ in (a)
configuration. (e) Normalized $|S_{21}|$ at $\delta=-60^{\circ}$ and (g)
$\delta=70^{\circ}$, both in configuration (b). (e, f, g) Fits using Eq. 1.

We found that the parameter $a_{1}$ strongly depends on the polarization angle
$\delta$ (Fig. 3a). This parameter describes the coupling between the cavity
and the rest of the spectrometer, i.e. constructive or destructive
interferences between the cavity and the experimental setup. The result
presented in Fig. 1a, obtained at $\delta=30^{\circ}$, shows a minimum in
transmission that is accounted for with $a_{1}$ having a small imaginary part
and a positive real part. The dependence of $a_{1}$ on polarization angle
(Fig. 3a) is not an intrinsic property of the cavity, but depends on the
coupling of the cavity with the experimental setup. As a consequence, after it
was rearranged, we obtained a slightly different value for $a_{1}$ (Fig. 1b).
Thus, the result obtained at $\delta=-75^{\circ}$ (Fig. 3c) under (a)
conditions shows a dispersive lineshape that is accounted for by a negative
imaginary $a_{1}$ (Fig. 3e). In contrast, the result obtained at
$\delta=-60^{\circ}$ (Fig. 3d) under (b) conditions shows constructive
interference that is explained by an almost entirely real, negative $a_{1}$
(Fig. 3f). Under the same conditions, at $\delta=70^{\circ}$ a weak inversed
dispersive lineshape was observed (Fig. 3f) that is accounted for with a small
positive imaginary $a_{1}$.

###  III.3 Higher cavity modes

Raw data for the spectra are dominated by multiple interferences that are not
strongly interacting with the cavity and are weakly temperature-dependent. We
normalized both the measured spectra and fitted functions to a base frequency
that is temperature-dependent and passes though the middle of the interaction.
This reduced artifacts at the cavity mode frequency that would show if
instead, we normalized for example to the first recorded spectrum. We could
eliminate this interference background even better by calculating temperature-
derivative spectra, i.e. by subtracting from one another successive spectra,
thus having amplitude derivatives:

$\frac{d|S_{21}|}{dT}[dB]=\frac{20}{\Delta T}\log_{10}\frac{|S_{21}(f,T+\Delta
T)|}{|S_{21}(f,T)|},$ (5)

and phase derivatives:

$\frac{d(\arg{S_{21}})}{dT}=\frac{\arg S_{21}(f,T+\Delta T)-\arg
S_{21}(f,T)}{\Delta T}.$ (6)

Figure 4:  Temperature-derivative spectra: (a) magnitude and (c) phase. (b and
d) Corresponding fits using Eq. 7.

The advantage of looking at temperature-differential spectra is that it is
possible to see small features, like a weak middle line passing through the
interaction region (Fig. 4a and c) that is almost invisible in normalized
spectra. This middle line can be qualitatively explained by taking into
account the 2nd cavity mode at $f_{2}=258.4$ GHz. We can take it into account
by assuming that both modes interact independently with the magnetic resonance
mode:

$S_{21}=1+\sum_{j=1}^{2}\frac{a_{j}}{i(f-f_{j}(T))-\frac{\kappa_{j}}{2}+\frac{G_{j}^{2}}{i(f-f_{r}(T))-g/2}},$
(7)

In Fig. 4b we present temperature-differential of Eq. 7, where we put
$a_{2}=(4.5-i1.1)$ GHz, $G_{2}=0.56$ GHz and $\kappa_{2}=14.7$ GHz. These
values are consistent with our idea of a weak interaction with the 2nd cavity
mode that has a lower quality factor that the 1st mode. This simple model is
not fully sufficient above 250 GHz. We think that a more precise estimate
might require taking into account other weakly interacting cavity modes or
interaction between cavity modes that are mediated by the magnetic resonance.
We found that the observed phase temperature-differential spectra (Fig. 4c)
are reproduced by the same set of parameters (Fig. 4d), using Eq. 7 and Eq. 6.

###  III.4 Electrodynamic simulations of cavity modes

Figure 5:  Model of the cavity drawn to-scale in cross-section. Arrows show
strength of magnetic field of the 1st cavity mode on the $x$ plane.  Figure 6:
Temperature-derivative spectra over a broad temperature range. Arrows mark
expected frequencies of cavity modes and circles mark observed interactions.

We identified the cavity modes using a numerical electrodynamics field
simulation software (CST Microwave Studio). We modeled the cavity as a
metallic cylinder containing an isotropic dielectric cube (Fig. 5a). We
assumed that the dielectric constant of hematite was 19.1 at 350 K, as
determined from our measurements on bulk samples [54]. We calculated the
dependence of modes on the size and position of the cube inside the cylinder
(see appendix). The actual size of the cube was the smallest we could achieve
by our cutting process.

We give in Tab. 1 the expected frequencies of the first five modes and
dominant directions of the electric field in the cavity, as well as the
dominant direction of the magnetic field in the cube. Only the 5th mode has a
similar symmetry to that of the 1st mode, and is able to excite
antiferromagnetic resonance in the sample. This suggest that the magnetization
vector $\mathbf{m}$ in the cube was aligned along the $z$ axis of the cavity
during the experiment. Under this assumption, modes 2–4 do not excite the
magnetic resonance and thus, should only weakly hybridize with it. This agrees
with our observation in Fig. 6a, that only the 5th mode produces some strong
coupling, though not as clear as the coupling to the 1st mode. This may be due
to a more complex distribution of the cavity magnetic field in the cube, i.e.
the ratio of $h_{z}/|h_{z}|\approx 0.53$ is lower than in the case of the 1st
mode (Tab. 1). This means that some of the spins in the cube are excited out
of phase.

|  | cavity | mean $h$-field in the cube [arb.u.]
---|---|---|---
$j$ |  $f_{j}$ [GHz] |  $e$-field | $h_{x}$ | $h_{y}$ | $h_{z}$ | $|h_{x}|$ | $|h_{y}|$ | $|h_{z}|$ | $|\mathbf{h}|$
$1$ | $247$ | $x$ | 0.01 | -0.01 | 2.29 | 0.42 | 0.66 | 2.39 | 2.70
$2$ | $258$ | $y$ | 0.12 | 0.00 | 0.00 | 0.96 | 0.09 | 1.30 | 1.76
$3$ | $272$ | $z$ | 2.51 | 0.00 | -0.01 | 2.61 | 0.75 | 0.63 | 3.05
$4$ | $297$ | - | 0.00 | 2.30 | 0.01 | 0.72 | 2.51 | 0.90 | 3.15
$5$ | $329$ | $x$ | 0.00 | 0.00 | -1.00 | 0.54 | 0.43 | 1.87 | 2.19
Table 1: Predicted modes frequencies, selection rules and mean magnetic field
in the cube.

##  IV Summary

In conclusion, we observed strong coupling between the quasi-antiferromagnetic
resonance (qAFMR) of a hematite cube ($\alpha$-Fe2O3) located in a cavity near
room temperature. We showed a method of achieving a strong light-matter
coupling regime in antiferromagnets which is applicable to a broad range of
these materials The avoided crossing, which is characteristic of polariton
dynamics, occurred near 243 GHz. The cooperativity $C$, which is a measure of
the coupling strength with respect to both the magnetic resonance and the
cavity line widths (Eq. 2), was estimated at 40.

###### Acknowledgements.

We would like to thank Claude Amendola for help in fabricating samples.
Support by the Sino-Swiss Science and Technology Cooperation (SSSTC) grant no.
EG-CN_02_032019 is gratefully acknowledged. The VNA and frequency extenders
were funded by EPFL and the SNF R’Equip under Grant No. 206021_144983.

*

##  Appendix A A: Temperature dependence of the antiferromagnetic resonance
and the 1st cavity mode

Figure 7:  Close-up of the first cavity mode in an over-saturated scale. The
green line shows the assumed linear temperature dependence of the first mode
frequency.

Fig. 7 shows the observed temperature dependence of the 1st mode, that we
determined as

$f_{1}(T)=f_{1}^{345}+f_{1}^{T}(T-T_{1}),$

were $f_{1}^{345}=243$ GHz, $T_{1}=345$ K and $f_{1}^{T}=-2.1\times 10^{-2}$
GHz/K. This dependence is caused by the dielectric constant of hematite
increasing with rising temperature. For the cube, this dependence on the
dielectric constant is predicted to be small (Fig. 8f).

In the configuration (a) (Fig. 2 and Fig. 3d) the temperature dependence of
the antiferromagnetic was fitted with a linear function

$f_{r}(T)=f_{r}^{330}+f_{r}^{T}(T-T_{r}),$

were $f_{r}^{330}=230.7$ GHz, $T_{r}=330$ K and $f_{r}^{T}=1.176$ GHz/K. In
the configuration (b) (Fig. 3fh), we fitted: $f_{r}^{330}=220.1$ GHz, and
$f_{r}^{T}=1.410$ GHz/K. In Figs. 4 and 6, where we show temperature
dependence on a broader scale, it was necessarily to assume that the frequency
of qAFMR is described by a parabolic function

$f_{r}(T)=f_{r}^{330}+f_{r}^{T}(T-T_{r})+f_{r}^{T2}(T-T_{r})^{2},$

where $f_{r}^{330}=218.4$ GHz, $T_{r}=330$ K, $f_{r}^{T}=1.596$ GHz/K and
$f_{r}^{T2}=-5.15\times 10^{-3}$ GHz/K2. These differences of $f_{r}(T)$
result from different placements of the thermocouple in the holder, and thus,
a different temperature calibration.

*

##  Appendix B B: Frequency dependence of modes on cavity geometry

Figure 8:  Dependence of the modes frequencies on (a) displacement of the cube
from the center of the tube, (b) rotation of the cube about $y$ axis, (c)
length of the tube, (d) diameter of the tube, (e) cube edge length, (f)
dielectric constant of the cube. In (a)(c)(d)(e) and (f) vertical lines mark
nominal values of these parameters.

The size of the cube $l_{s}$ was limited by our cutting process. At this
smallest size, the mode frequencies are governed mostly by the size of the
cube $l_{s}$ (Fig. 8e) and, to a smaller extend, by the diameter of the tube
$2r_{c}$ (Fig. 8d). The length of the tube $l_{c}$ is irrelevant in the range
of about 1.5–2.5 mm (Fig. 8c). For a given cube size, we chose such a diameter
of the tube that the 1st and 2nd modes are well split (Fig. 8d). We found,
that a random position of the cube in the needle spans a mode frequency range
of at most 15 GHz (Fig. 8ab). In other calculations (Fig. 8c–f), the cube was
assumed to be in the center of the needle ($Z_{off}=0$) and edges parallel to
its axis ($\beta=0$). The calculated frequency of the 1st mode is a few GHz
higher than the observed one. This might be due to inaccuracy in the cube
dimensions. We observed the biggest discrepancy with the predicted and
observed frequencies of the 5th mode (Fig. 6). This could be related to the
fact that it is the mode which shows the strongest dependence on displacement
of the cube from the cavity center (Fig. 8a).

Figure 9:  (a) Model of the cavity drawn to-scale in cross-section. The cube is rotated about the $y$ axis by an angle $\beta=45^{\circ}$. Rotation about $x$ axis is bound by the condition of the cube touching the tube by its three corners. Arrows show direction and strength of magnetic field of the 1st cavity mode on the $x$ plane. (b) Field in the second mode the cavity on the $x$ plane.  |  | cavity | mean $h$-field in the cube [arb.u.]
---|---|---|---
$j$ |  $f_{j}$ [GHz] |  $e$-field | $h_{x}$ | $h_{y}$ | $h_{z}$ | $|h_{x}|$ | $|h_{y}|$ | $|h_{z}|$ | $|\mathbf{h}|$
$1$ | $258$ | $x$ | 0.00 | -0.02 | 2.20 | 0.43 | 0.62 | 2.32 | 2.62
$2$ | $262$ | $y$ | 0.33 | 0.00 | 0.01 | 0.93 | 0.31 | 1.24 | 1.74
$3$ | $278$ | $z$ | -2.55 | 0.00 | 0.00 | 2.68 | 0.63 | 0.67 | 3.06
$4$ | $292$ | - | 0.00 | 2.45 | 0.00 | 0.68 | 2.64 | 0.77 | 3.15
$5$ | $322$ | $x$ | 0.00 | 0.00 | 1.29 | 0.58 | 0.69 | 2.28 | 2.71
Table 2: Predicted modes frequencies for the cube rotated by
$\beta=45^{\circ}$ (Fig. 9a). Selection rules and mean magnetic field in the
cube are very close to that of $\beta=0^{\circ}$ (Tab. 1).

*

##  Appendix C C: 2-port analysis

Let us represent a cavity as a device with 2 ports. At each port, we
distinguish incoming and outgoing waves. The total voltage and current at each
port are given by

$V_{i}=V_{i}^{+}+V_{i}^{-}\qquad I_{i}=I_{i}^{+}+I_{i}^{-}\qquad(i=1,2)$ (8)

One way of characterizing the cavity is to specify the parameters $A,B,C,D$
defined by

$\begin{pmatrix}V_{1}\\\ I_{1}\end{pmatrix}=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\begin{pmatrix}V_{2}\\\ I_{2}\end{pmatrix}$ (9)

For a two-port device in which each port sees the common impedance $Z_{3}$,
the $A,B,C,D$ parameters are given by [55]

$A=1\quad B=0\quad C=\frac{1}{Z_{3}}\quad D=1$ (10)

The scattering matrix is defined by

$\begin{pmatrix}V_{1}^{-}\\\
V_{2}^{-}\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\\
S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}V_{1}^{+}\\\ V_{2}^{+}\end{pmatrix}$
(11)

In particular, the $S_{21}$ parameter is given by

$S_{21}=\left.\frac{V_{2}^{-}}{V_{1}^{+}}\right|_{V_{2}^{+}=0}$ (12)

It is straightforward to calculate that,

$S_{21}=\frac{1}{1+Z_{0}/(2Z_{3})}$ (13)

where $Z_{0}$ is the transmission line (or wave guide) characteristic
impedance at the ports.

Let us apply this formalism to analyze a magnetic material located inside the
cavity. We represent the cavity by discrete $R,L,C$ components in series. We
assume that the measurement setup is such that away from a resonance of the
cavity or from a resonance of the material in it, the signal propagates
through the cavity without loss, i.e. $S_{21}=1$. We have,

$Z_{3}=i\omega L+\frac{1}{i\omega C}+R$ (14)

We assume that the $RLC$ circuit is designed so that the transmission is
nearly one at all frequencies, i.e. $Z_{3}\gg Z_{0}$. Then (13) yields,

$S_{21}\approx 1-\frac{Z_{0}}{2Z_{3}}$ (15)

We take into account the presence of the material in the cavity by assuming
that the inductance $L$ is filled entirely with a medium of an effective
permittivity $\mu$ written in the form of a Lorentzian line shape, so as to
account for the magnetic resonance in the material

$L=\mu
L_{0}=(1+\frac{\Delta\mu\,\omega_{m}^{2}}{\omega_{m}^{2}-\omega^{2}-j\omega\gamma_{m}})L_{0},$
(16)

where $L_{0}$ is an inductance of an empty cavity, $\Delta\mu$ is a unitless
parameter that characterizes the coupling of the material to the cavity,
$\omega_{m}$ is its frequency and $\gamma_{m}$ defines its width. Using
$Z_{3}$ given by (14) and (16), (15) gives,

$S_{21}=1-\frac{i\omega Z_{0}C/2}{1+i\omega
RC-\omega^{2}CL_{0}\left(1+\frac{\Delta\mu\,\omega_{m}^{2}}{\omega_{m}^{2}-\omega^{2}-i\omega\gamma_{m}}\right)}$
(17)

Let us call $\omega_{c}^{2}=1/(L_{0}C)$ in reference to the resonance of the
empty cavity. We consider a situation when $\omega$, $\omega_{c}$ and
$\omega_{m}$ are close, so we can approximate

$\omega^{2}-\omega_{c}^{2}=(\omega-\omega_{c})(\omega+\omega_{c})\approx
2\omega_{c}(\omega-\omega_{c})$ (18)

Since the resonances are sharp, $\omega$ can be replaced by $\omega_{c}$ when
$\omega$ is not in a term that leads to a divergence (in the absence of
damping). Thus, we can write as in [10]

$S_{21}=1+\frac{a}{i(\omega-\omega_{c})-\dfrac{\kappa}{2}+\dfrac{G^{2}}{i(\omega-\omega_{m})-\gamma_{m}/2}}$
(19)

where $a=\omega_{c}^{2}Z_{0}C/4$, $G^{2}=\omega_{c}\omega_{m}\Delta\mu/4$ and
$\kappa=\omega_{c}^{2}RC$.

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