$\operatorname{\mathbb{E}}\left[\underset{a\in\left(\Pi\times\Pi\right)\setminus\Delta}{\sum}|f(a;\Pi)|\;\pi_{a}\right]<\infty\,.$

Suppose $f$ is non-negative and admits a factorization as in Lemma 26;

$f((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi)\quad=\quad
f_{\text{invar}}((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi\setminus{\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0}})\;\times\;\\\
\;\times\;f_{\text{speed}}(v(\operatorname{\mathcal{L}}_{0}))\;\times\;f_{\text{config}}(\operatorname{dist}_{\text{signed}}(\operatorname{\mathcal{L}}_{-}\cap\operatorname{\mathcal{L}}_{0},\operatorname{\mathcal{L}}_{x,0}\cap\operatorname{\mathcal{L}}_{0}),\operatorname{dist}_{\text{signed}}(\operatorname{\mathcal{L}}_{0},\operatorname{\operatorname{\bf
o}}),\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{0},{\operatorname{\mathcal{L}}_{*}^{0}})))\,,$

where $f_{\text{invar}}$ is similarity-invariant. In particular, if
$\phi_{-}=\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{-},{\operatorname{\mathcal{L}}_{0}})$
and
$v(\operatorname{\mathcal{L}}_{-})=e^{t_{-}}v(\operatorname{\mathcal{L}}_{0})$
then we write

$f_{\text{invar}}((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi\setminus\\{\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0}\\})\;=\;f_{\text{invar}}\left(\frac{v(\operatorname{\mathcal{L}}_{-})}{v(\operatorname{\mathcal{L}}_{0})},\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{-},{\operatorname{\mathcal{L}}_{0}}),S_{(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0})}^{-1}(\Pi\setminus\\{\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0}\\})\right)\\\
\quad=\quad f_{1}(e^{t_{-}},\phi_{-},\Psi^{(0)}_{0})\,.$

Suppose further that the bounded $g$ is itself similarity-invariant: we write

$g(\widetilde{b};\Pi)\quad=\quad
g((\operatorname{\mathcal{L}}_{+},\operatorname{\mathcal{L}}_{0});\Pi)\quad=\quad
g_{\text{invar}}((\operatorname{\mathcal{L}}_{+},\operatorname{\mathcal{L}}_{0});\Pi\setminus\\{\operatorname{\mathcal{L}}_{0},\operatorname{\mathcal{L}}_{+}\\})\,.$

Since the dynamics of $Z$ are similarity-invariant, this means that

$\operatorname{\mathbb{E}}\left[g(\widetilde{Z}_{1};\Pi)|Z_{0}=a,\Pi\right]\quad=\quad\operatorname{\mathbb{E}}\left[g(S\widetilde{Z}_{1};S\Pi)|SZ_{0}=a,S\Pi\right]$

for any similarity $S$, so
$\operatorname{\mathbb{E}}\left[g(\widetilde{Z}_{1};\Pi)|Z_{0}=a,\Pi\right]$
is similarity-invariant as a function of $a$ and $\Pi\setminus a$.
Consequently we may apply Lemma 26 to

$\xi((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi)\quad=\quad
f((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi)\operatorname{\mathbb{E}}\left[g(\widetilde{Z}_{1};\Pi)|Z_{0}=(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0}),\Pi\right]\quad=\quad\\\
f_{\text{invar}}((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi\setminus\\{\operatorname{\mathcal{L}}_{0},\operatorname{\mathcal{L}}_{-}\\})\times\operatorname{\mathbb{E}}\left[g_{\text{invar}}(\widetilde{Z}_{1};\Pi\setminus\\{\operatorname{\mathcal{L}}_{0},\operatorname{\mathcal{L}}_{+}\\})\;|\;Z_{0}=(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0}),\Pi\right]\times\\\
\times
f_{\text{speed}}(v(\operatorname{\mathcal{L}}_{0}))\;\times\;f_{\text{config}}(\operatorname{dist}_{\text{signed}}(\operatorname{\mathcal{L}}_{-}\cap\operatorname{\mathcal{L}}_{0},\operatorname{\mathcal{L}}_{x,0}\cap\operatorname{\mathcal{L}}_{0}),\operatorname{dist}_{\text{signed}}(\operatorname{\mathcal{L}}_{0},\operatorname{\operatorname{\bf
o}}),\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{0},{\operatorname{\mathcal{L}}_{*}^{0}})))\,.$

We deduce

$B(f,g)\quad=\quad\operatorname{\mathbb{E}}\left[\underset{\operatorname{\mathcal{L}}_{-}\neq\operatorname{\mathcal{L}}_{0}\in\Pi}{\sum\sum}\pi_{(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0})}f((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi)\operatorname{\mathbb{E}}\left[g(\widetilde{Z}_{1};\Pi)|Z_{0}=(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0}),\Pi\right]\right]\\\
\quad=\quad\left(\frac{\gamma-1}{2}\right)^{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\pi}f_{\text{config}}(s,r,\theta){\,\operatorname{d}}\theta{\,\operatorname{d}}s{\,\operatorname{d}}r\;\times\;\int_{0}^{\infty}f_{\text{speed}}(v)v^{\alpha-2\gamma+1}{\,\operatorname{d}}{v}\;\times\;\\\
\int_{-\infty}^{\infty}\int_{0}^{\pi}\min\\{e^{(\alpha-(\gamma-1))t_{-}},e^{-(\gamma-1)t_{-}}\\}\times\\\
\times\operatorname{\mathbb{E}}\left[f_{\text{invar}}((\operatorname{\mathcal{L}}_{*}^{\phi_{-}}(e^{t_{-}}),\operatorname{\mathcal{L}}_{*}^{0});\Pi)\operatorname{\mathbb{E}}\left[g_{\text{invar}}(\widetilde{Z}_{1};\Pi)|Z_{0}=(\operatorname{\mathcal{L}}_{*}^{\phi_{-}}(e^{t_{-}}),\operatorname{\mathcal{L}}_{*}^{0}),\Pi\right]\right]\sin\phi_{-}{\,\operatorname{d}}{\phi_{-}}\,{\,\operatorname{d}}{t_{-}}\,,$

where the line $\operatorname{\mathcal{L}}_{*}^{\phi_{-}}(e^{t_{-}})$ meets
the unit-speed ($x$-axis) $\operatorname{\mathcal{L}}_{*}^{0}$ at
$\operatorname{\operatorname{\bf o}}$, making an angle $\phi_{-}$, and has
speed $e^{t_{-}}$.

We may deduce the following by choosing
$f_{\text{config}}(s,r,\theta)=\operatorname{\mathbb{I}}\left[s^{2}+r^{2}\leq
c^{2}\right]$ and
$f_{\text{speed}}(v)=\operatorname{\mathbb{I}}\left[v>c\right]$ for fixed
$c>0$. Sample the speed-marked line process $\Pi$, and sample
$\operatorname{\mathcal{L}}_{0}$ uniformly at random from the set of lines of
$\Pi$ lying within $c$ of $\operatorname{\operatorname{\bf o}}$ and with speed
exceeding $c$. Then sample $\operatorname{\mathcal{L}}_{-}$ from the lines of
$\Pi$ intersecting $\operatorname{\mathcal{L}}_{0}$ and such that (i) the
intersection point is within $c$ of $\operatorname{\operatorname{\bf o}}$,
using sampling weights
$\min\\{1,(v(\operatorname{\mathcal{L}}_{1})/v(\operatorname{\mathcal{L}}_{0}))^{\alpha}\\}$
If there is no such line $\operatorname{\mathcal{L}}_{0}$, or there turn out
to be no such intersections, then re-sample $\Pi$ and repeat till successful.
Use the resulting
$(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0})$ as the
initial point of the SIRSN-RRF $Z$. Then the resulting relative environment
process is associated with the following Dirichlet form:

$B^{\text{relative}}(f_{\text{invar}},g_{\text{invar}})\;=\;\operatorname{\mathbb{E}}\left[\min\left\\{1,\left(\frac{v(\operatorname{\mathcal{L}}_{-})}{v(\operatorname{\mathcal{L}}_{0})}\right)^{\alpha}\right\\}f_{\text{invar}}((\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0});\Pi)\operatorname{\mathbb{E}}\left[g_{\text{invar}}(\widetilde{Z}_{1};\Pi)|Z_{0}=(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0}),\Pi\right]\right]\\\
\;=\;\int_{-\infty}^{\infty}\int_{0}^{\pi}\operatorname{\mathbb{E}}\Bigg{[}f_{1}(e^{t_{-}},\phi_{-},\Pi)\sum_{\operatorname{\mathcal{L}}_{+}\in\Pi}\omega^{\Pi\setminus\\{\operatorname{\mathcal{L}}_{+}\\}}_{\operatorname{\operatorname{\bf
o}},(\operatorname{\mathcal{L}}_{*}^{0}\cap\operatorname{\mathcal{L}}_{+})}\min\\{1,v(\operatorname{\mathcal{L}}_{+})^{\alpha}\\}\times\\\
\times
g_{1}(v(\operatorname{\mathcal{L}}_{+}),\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{*}^{0},\operatorname{\mathcal{L}}_{+}),\Pi)\Bigg{]}\min\\{e^{(\alpha-(\gamma-1))t_{-}},e^{-(\gamma-1)t_{-}}\\}\sin\phi_{-}{\,\operatorname{d}}{\phi_{-}}\,{\,\operatorname{d}}{t_{-}}\,.$

Here

$\omega^{\Pi\setminus\\{\operatorname{\mathcal{L}}_{+}\\}}_{\operatorname{\operatorname{\bf
o}},(\operatorname{\mathcal{L}}_{*}^{0}\cap\operatorname{\mathcal{L}}_{+})}\quad=\quad\mathop{\underset{\operatorname{\mathcal{L}}\in\Pi\setminus\\{\operatorname{\mathcal{L}}_{+}\\}\text{
separating}}{\prod}}_{\operatorname{\operatorname{\bf o}}\text{ and
}(\operatorname{\mathcal{L}}_{*}^{0}\cap\operatorname{\mathcal{L}}_{+})}\Big{(}1-s_{\operatorname{\mathcal{L}}_{0}^{*},\operatorname{\mathcal{L}}}\Big{)}\,.$

One further application of the Slivnyak-Mecke theorem (Theorem 20), using the
representation (2) for
$\nu({\,\operatorname{d}}{\operatorname{\mathcal{L}}_{+}})$, now yields

$B^{\text{relative}}(f_{\text{invar}},g_{\text{invar}})\quad=\quad\\\
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\pi}\;\int_{-\infty}^{\infty}\int_{0}^{\pi}\operatorname{\mathbb{E}}\Bigg{[}f_{1}(e^{t_{-}},\phi_{-},\Pi)\omega^{\Pi}_{[\operatorname{\operatorname{\bf
o}},s]}\min\\{1,e^{\alpha
t_{+}}\\}g_{1}(e^{t_{+}},\phi_{+},\Pi)\Bigg{]}\times\\\
\times\min\\{e^{(\alpha-(\gamma-1))t_{-}},e^{-(\gamma-1)t_{-}}\\}\sin\phi_{-}{\,\operatorname{d}}{\phi_{-}}\,{\,\operatorname{d}}{t_{-}}\;e^{-(\gamma-1)t_{+}}\sin\phi_{+}{\,\operatorname{d}}{\phi_{+}}\,{\,\operatorname{d}}{t_{+}}{\,\operatorname{d}}{s_{+}}\\\
\quad=\quad\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{0}^{\pi}\;\int_{-\infty}^{\infty}\int_{0}^{\pi}\operatorname{\mathbb{E}}\left[f_{1}(e^{t_{-}},\phi_{-},\Pi)g_{1}(e^{t_{+}},\phi_{+},\Pi)\;\omega^{\Pi}_{[\operatorname{\operatorname{\bf
o}},s}\right]\times\\\
\times\min\\{e^{(\alpha-(\gamma-1))t_{-}},e^{-(\gamma-1)t_{-}}\\}\min\\{e^{(\alpha-(\gamma-1))t_{+}},e^{-(\gamma-1)t_{+}}\\}\times\\\
\times\sin\phi_{-}{\,\operatorname{d}}{\phi_{-}}\sin\phi_{+}{\,\operatorname{d}}{\phi_{+}}\,{\,\operatorname{d}}{t_{-}}\,{\,\operatorname{d}}{t_{+}}{\,\operatorname{d}}{s_{+}}\,,$
(31)

where $[\operatorname{\operatorname{\bf o}},s]$ is short-hand for the interval
along $\operatorname{\mathcal{L}}_{*}^{0}$ with one endpoint given by
$\operatorname{\operatorname{\bf o}}$ and with the signed length $s$.

The invariance of $f_{\text{invar}}$, $g_{\text{invar}}$, and
$f_{\text{invar}}$ collectively imply that the Dirichlet form
$B^{\text{relative}}(f_{\text{invar}},g_{\text{invar}})$ is symmetric. This in
turn implies that the relative environment process $\Psi$ is stationary, with
an invariant measure which is a probability measure which makes the three
coordinates of log relative speed $t$ of previous line, angle $\phi$ between
previous and current lines, and reduced environment $\Psi^{(0)}$ independent
with joint distribution as follows

  1. 1.

corresponding to the log relative speed $t$, a (possibly asymmetric) Laplacian
density over $\mathbb{R}$ (Johnson et al., 1995, ch. 24), given by

$f_{\alpha,\gamma}(y)\quad=\quad\begin{cases}\frac{(\gamma-1)(\alpha-(\gamma-1))}{\alpha}e^{-(\gamma-1)|y|}&\text{
when }y\geq 0\,,\\\
\frac{(\gamma-1)(\alpha-(\gamma-1))}{\alpha}e^{-(\alpha-(\gamma-1))|y|}&\text{
when }y<0\,;\end{cases}$ (32)

  2. 2.

corresponding to $\phi$, a half-sine density over $[0,\pi)$, given by
$\tfrac{1}{2}\sin\phi$;

  3. 3.

corresponding to the reduced relative environment $\Psi^{(0)}$, a distribution
which agrees with that of the underlying SIRSN candidate $\Pi$.

This completes the proof. ∎

We are actually interested in the log-relative speed of the current line with
respect to the previous line. The distribution of this in stationary state is
readily computed directly from Theorem 27, bearing in mind that if $Z$ is at
$(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0})$ then its
next state is
$(\operatorname{\mathcal{L}}_{0},\operatorname{\mathcal{L}}_{+})$, where
$\operatorname{\mathcal{L}}_{+}$ is drawn from the reduced relative
environment $\Psi^{(0)}$, such that
$(\operatorname{\mathcal{L}}_{0},\operatorname{\mathcal{L}}_{+})$ is the first
intersection along $\operatorname{\mathcal{L}}_{0}$ is the chosen direction
which is accepted by the rule summarized by the acceptance probability (24).
We obtain

######  Corollary 28.

In the situation of Theorem 27, when $\Psi$ is stationary, the distribution of
the log of the speed of the current line relative to the speed of the previous
line has density given by the (possibly asymmetric) Laplacian density
prescribed by (32), with mean value given by

$\int_{-\infty}^{\infty}yf_{\alpha,\gamma}(y){\,\operatorname{d}}{y}\quad=\quad\frac{1}{\gamma-1}-\frac{1}{\alpha-(\gamma-1)}\quad=\quad\frac{\alpha-2(\gamma-1)}{(\gamma-1)(\alpha-(\gamma-1))}\,.$
(33)

###### Proof.

Leaving $(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0})$, $Z$
encounters intersections with lines of $\Psi^{(0)}$ according to a speed-
marked Poisson process with intensity measure
$v^{-\gamma}{\,\operatorname{d}}v{\,\operatorname{d}}s$, where $v$ is now the
speed of the new line relative to the speed of
$\operatorname{\mathcal{L}}_{0}$ and $s$ is the scaled distance along
$\operatorname{\mathcal{L}}_{0}$ from
$(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0})$. Note that
the mark measure does not have finite mass. We have to thin this marked
Poisson process with retention probability $\min\\{1,v^{\alpha}\\}$, according
to the acceptance probability (24) (bearing in mind that $v$ is the relative
speed of the new line), so the mark distribution of the retained lines does
have finite mass. Accordingly the mark distribution of retained lines, and
thus the relative speed of the new line, is proportional to (hence equal to)
(32). ∎

In particular the log-relative-speed density has zero mean in the symmetric
case, when $\alpha=2(\gamma-1)$, in which case the log-relative-speed
stationary distribution is is symmetric and is given by the Laplace or double-
headed exponential distribution, with rate parameter $\gamma-1$, and density

$f_{2(\gamma-1),\gamma}(y)\quad=\quad\frac{\gamma-1}{2}e^{-(\gamma-1)|y|}\,.$
(34)

Finally note that the reduced relative environment process, and hence the
relative environment process, is very far from being irreducible. Indeed, any
particular realization of the state $\Psi$ of the reduced environment process
defines a countable set of intersection angles
$A=\\{\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{1},\operatorname{\mathcal{L}}_{2})):(\operatorname{\mathcal{L}}_{1},\operatorname{\mathcal{L}}_{2})\in(\Psi\times\Psi)\setminus\Delta\\}$
_which remains time-constant under the evolution of $\Psi$_. But $\Psi$ has
equilibrium distribution given by the SIRSN candidate $\Pi$, and there will be
zero probability of _any_ of the countably many intersections of lines from
$\Pi$ having an angle belonging to a fixed countable set. Indeed, by the
Slivnyak-Mecke theorem (Theorem 20) we know that for any fixed angle $\phi$ we
must have

$\operatorname{\mathbb{E}}\left[\sum_{(\operatorname{\mathcal{L}}_{1},\operatorname{\mathcal{L}}_{2})\in\left(\Pi\times\Pi\right)\setminus\Delta}\operatorname{\mathbb{I}}\left[\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{1},\operatorname{\mathcal{L}}_{2})=\phi\right]\right]\quad=\quad\int\int\operatorname{\mathbb{I}}\left[\mathord{<\mspace{-9.0mu}\mathrel{)}\mspace{2.0mu}}(\operatorname{\mathcal{L}}_{1},\operatorname{\mathcal{L}}_{2})=\phi\right]\nu({\,\operatorname{d}}{\operatorname{\mathcal{L}}_{1}})\nu({\,\operatorname{d}}{\operatorname{\mathcal{L}}_{2}})\quad=\quad
0\,.$

Thus conventional Markovian arguments cannot be applied. However, as we will
see in the next section, ergodic theory allows us to prove the results we
need.

##  5 Long-term behaviour of SIRSN-RRF speed

If $Z=(Z_{0},Z_{1},\ldots)$ is the (discrete-time) SIRSN-RRF and if
$V=(V_{0},V_{1},\ldots)$ yields the corresponding sequence of speeds for the
current line, then the relative speed change $V_{n}/V_{n-1}$ can be determined
using only $\Psi_{n}$ the relative environment (relativized by centering,
rotating, and scaling). Theorem 27 implies that the log-relative speed-changes
$U_{n}=\log(V_{n}/V_{n-1})$ form a stationary sequence, if the initial
relative environment is given the stationary distribution discussed in the
previous section, and thus $U_{0}$ is given the equilibrium density specified
in Equation (32). We may therefore apply the non-ergodic part of Birkhoff’s
ergodic theorem (see for example Kallenberg, 2002, Theorem 10.6) to show

$\frac{1}{n}\log(V_{n}/V_{0})\quad=\quad\frac{1}{n}\sum_{m=1}^{n}U_{m}\quad\to\quad\operatorname{\mathbb{E}}\left[U_{0}|\mathcal{I}\right]\,,$
(35)

where $\mathcal{I}$ is the $\sigma$-algebra of shift-invariant events for the
random process $U$, and convergence is both almost sure and in $L^{1}$.

It immediately follows that, away from the critical case $\alpha=2(\gamma-1)$,
the speed $V_{n}$ has at least a positive chance of either diverging to
$+\infty$ exponentially fast as $n\to\infty$ (if $\alpha>2(\gamma-1)$) or
converging to $0$ exponentially fast as $n\to\infty$ (if
$\alpha<2(\gamma-1)$). We may therefore rule out non-critical cases
($\alpha\neq 2(\gamma-1)$) in our search for an example which is speed-
neighbourhood-recurrent.

Suppose it can be shown that $\Psi$ (and therefore $U$) is ergodic, so that we
can replace $\operatorname{\mathbb{E}}\left[U_{0}|\mathcal{I}\right]$ by
$\operatorname{\mathbb{E}}\left[U_{0}\right]$ in (35). In the non-critical
cases discussed above, this means we can replace “has at least a positive
chance” by “will almost surely end up“. But in the critical case
$\operatorname{\mathbb{E}}\left[U_{0}\right]=0$ is not sufficient in itself to
guarantee neighborhood-recurrence for $U$. However in the ergodic case
neighborhood-recurrence actually follows rather simply from the celebrated
(but unpublished) Kesten, Spitzer and Whitman range theorem (described by
Spitzer, 1976, page 38). In its original form the range theorem implies
concerns recurrence for integer-valued stationary ergodic processes of zero
mean. The real-valued / neighbourhood recurrent case is a simple variation on
the original idea:

######  Theorem 29.

(Kesten-Spitzer-Whitman, real-valued case.) Suppose $U_{1}$, …, $U_{n}$, …form
a stationary ergodic sequence, with
$\operatorname{\mathbb{E}}\left[U_{1}\right]=0$. Set
$W_{n}=U_{1}+\ldots+U_{n}$. Then for all $\varepsilon>0$ it is the case that

$\operatorname{\mathbb{P}}\left[|W_{n}-W_{0}|\leq\varepsilon\text{ infinitely
often in }n\right]\quad=\quad 1\,.$

###### Proof.

Birkhoff’s ergodic theorem guarantees that $W_{n}/n\to 0$ almost surely, hence

$n^{-1}\sup\\{|W_{1}|,\ldots,|W_{n}|\\}\quad\to\quad 0\qquad\text{ almost
surely.}$ (36)

Set $A_{n}=[|W_{m}-W_{n}|>\varepsilon\text{ for all }m>n]$.

Applying Birkhoff’s ergodic theorem again,

$n^{-1}(\operatorname{\mathbb{I}}\left[A_{1}\right]+\ldots+\operatorname{\mathbb{I}}\left[A_{n}\right])\quad\to\quad\operatorname{\mathbb{P}}\left[A_{0}\right]\,.$
(37)

If $m\neq n$ then $|W_{m}-W_{n}|>\varepsilon$ on $A_{n}\cap A_{m}$, and
therefore a simple packing argument shows that

$n^{-1}(\operatorname{\mathbb{I}}\left[A_{1}\right]+\ldots+\operatorname{\mathbb{I}}\left[A_{n}\right])\quad\leq\quad(n\varepsilon)^{-1}\sup\\{|W_{1}|,\ldots,|W_{n}|\\}\,.$
(38)

Letting $n\to\infty$ in (38) and using (36), we can deduce from (37) that
$\operatorname{\mathbb{P}}\left[A_{0}\right]=0$. Consequently

$\operatorname{\mathbb{P}}\left[|W_{n}-W_{0}|\leq\varepsilon\text{ at least
once for }n>0\right]\quad=\quad 1\,.$

The same argument applies for the sub-sampled process
$(W_{0},W_{m},W_{2m},\ldots)$, for any sub-sampling gap $m>0$. Therefore the
event $\bigcap_{m}\bigcup_{n\geq m}[|W_{n}-W_{0}|\leq\varepsilon]$ happens
almost surely. Consequently it is almost sure that $W_{n}$ returns to within
$\varepsilon$ of $W_{0}$ for infinitely many $n$ and the result follows. ∎

Accordingly speed-neighborhood-recurrence is established in the critical case
$\alpha=2(\gamma-1)$ if we can show that the reduced environment process
$\Psi$ is ergodic. This is the main result of this paper:

######  Theorem 30.

Given a SIRSN-RRF $Z$ on a Poisson line SIRSN or SIRSN candidate $\Pi$,
parametrized by $\alpha>\gamma-1$ where $\gamma$ is the Poisson line SIRSN
parameter, the relative environment process $\Psi$ for the SIRSN-RRF $Z$ is
ergodic.

###### Proof.

The key part of the argument is a variation on an argument of Kozlov (1985,
Lemma 1, page 82).

Firstly, consider the process $\Psi$ of the relative environment viewed from
the particle. Let $h$ be a bounded harmonic function on the relative
environment state-space (harmonic with respect to the process $\Psi$). It
suffices to show that $h(\Psi_{n})$ is almost surely constant in (discrete)
time $n$.

Now consider

$\operatorname{\mathbb{E}}\left[(h(\Psi_{0})-h(\Psi_{1}))^{2}\right]\quad=\quad
2\,\operatorname{\mathbb{E}}\left[h(\Psi_{0})^{2}\right]-2\,\operatorname{\mathbb{E}}\left[h(\Psi_{0})h(\Psi_{1})\right]\quad=\quad
0\,,$

where the first step follows from stationarity and the second because
$h(\Psi)$ is a martingale. Consequently
$\operatorname{\mathbb{P}}\left[h(\Psi_{1})=h(\Psi_{0})\right]=1$.

Secondly, using $\Psi$ to explore the network represented by $\Pi$, we see
that there is a $\Pi$-measurable random variable $H=H(\Pi)$ such that
$h(\Psi_{n})=H(\Pi)$ for all $n$, for environment $\Pi$. Moreover $H(\Pi)$
inherits similarity-invariance from $\Psi$. It follows from the ergodicity of
$\Pi$ (Theorem 21) that $H(\Pi)$ must be non-random. ∎

This together with Theorem 29 implies speed-neighborhood-recurrence for the
RRF $Z$, as required. It also shows that in non-critical cases the speed will
either almost surely diverge to infinity or almost surely converge to zero.
Accordingly a version of Conjecture 1 holds for the randomly-broken local
$\Pi$-geodesics formed by a critical SIRSN-RRF: in the critical case
$\alpha=2(\gamma-1)$ the RRF provides a “randomly-broken local
$\Pi$-geodesic”, which avoids slowing down to zero speed (or speeding up to
infinite speed).

We conclude this section with a formal statement of the speed-neighborhood-
recurrence result.

######  Theorem 31.

Let $\Pi$ be a Poisson line SIRSN or SIRSN candidate with parameter
$\gamma\geq 2$. Then there exists a (discrete-time) SIRSN-RRF $Z$ on $\Pi$ (an
RRF with similarity-invariant dynamics with zero defect) such that the speed
process $V$ (given by $V_{n}=v(\operatorname{\mathcal{L}}_{0})$ when
$Z_{n}=(\operatorname{\mathcal{L}}_{-},\operatorname{\mathcal{L}}_{0})$)
almost surely returns infinitely often to any neighbourhood of $V_{0}$.

###### Proof.

Bearing in mind the characterization (Theorem 22) of such SIRSN-RRF by index
$\alpha>\gamma-1$, we choose the SIRSN-RRF with critical index
$\alpha=2(\gamma-1)$. Considering the relative environment process $\Psi$ run
in stationarity, and noting that in stationarity the mean log-relative speed
$U=\log(V)$ has mean zero (Theorem 27), and forms an ergodic process (Theorem
30), the desired speed-neighborhood-recurrence result follows from the adapted
Kesten-Spitzer-Whitman Theorem 29. ∎

##  6 Conclusion

This paper defines and characterises SIRSN-RRF (similarity-equivariant
discrete-time Rayleigh random flights taking place on scale invariant random
spatial networks) using an axiomatic approach to scattering processes. It is
shown that the relative environment viewed from the SIRSN-RRF is ergodic
stationary, and that there exists a critical SIRSN-RRF whose speed process is
neighborhood-recurrent. We offer this as evidence in favour of Conjecture 1,
that $\Pi$-geodesics in a Poisson line SIRSN never come to a complete halt,
and therefore can be constructed using doubly infinite sequences of segments
taken from the Poisson line SIRSN.

We note that the abstract approach to scattering set out in Section 2 merits
further exploration in its own right.

In conclusion, we briefly discuss some points going beyond the question of
speed-neighbourhood recurrence in the critical case.

A little more can be said concerning the two non-critical cases. Bearing in
mind Corollary 28, if $\alpha<2(\gamma-1)$, so that the log-relative-speed
distribution of the next line relative to the current line for the SIRSN-RRF
has negative mean, then ergodicity of the relative environment means that the
SIRSN-RRF process itself must almost surely converge to a limiting random
point as time tends to infinity. This is because its speed must tend to zero,
and so almost surely it must eventually get trapped in a cell of the proper
Poisson line tessellation formed by $\Pi_{\geq\varepsilon}$ (recall
$\Pi_{\geq\varepsilon}$ is the part of $\Pi$ for which speeds are higher than
some $\varepsilon>0$). The trapping occurs as $\varepsilon\to 0$, since
$\Pi_{\geq\varepsilon}$ is a proper Poisson line tessellation, increasing
monotonically as a random set as $\varepsilon\to 0$, with intensity tending to
infinity and with cells shrinking down to zero size. Since the SIRSN-RRF has
positive chance of not escaping from $\Pi\setminus\Pi_{\geq\varepsilon}$ onto
$\Pi_{\geq\varepsilon}$ for large time, and has a positive chance of moving
freely within the current connected component of
$\Pi\setminus\Pi_{\geq\varepsilon}$ (by considerations akin to those of the
irreducibility Corollary 24), it follows that the limiting point of the SIRSN-
RRF must indeed be random. We call this case the _converging case_.

On the other hand, if $\alpha>2(\gamma-1)$, then the log-relative-speed
distribution of the next line relative to the current line for the SIRSN-RRF
has positive mean, and so the SIRSN-RRF process must almost surely diverge to
infinity. This is because its speed must tend to infinity (using again
ergodicity of the relative environment), and so almost surely the process must
get trapped on $\Pi_{\geq v}$ for any $v>0$. The divergence occurs as
$v\to\infty$, since $\Pi_{\geq v}$ is a proper Poisson line tessellation,
decreasing monotonically as a random set as $v\to\infty$, with intensity
tending to zero, and therefore with cells blowing up to arbitrarily large size
with cell boundaries almost surely being eventually arbitrarily far from the
origin $\operatorname{\operatorname{\bf o}}$. We call this case the _diverging
case_.

Whether the case is critical, diverging, or converging, the discrete-time
process is defined for all time (since it will take an infinite number of
jumps for the speed to exceed all bounds, or to reduce to zero). Consequently
the continuous-time variant (defined by interpolating between scatterings
using top-speed linear motion) is defined for all time in the critical case
$\alpha=2(\gamma-1)$. More generally, under stationarity consider Palm-
conditioning on the current line at the $n^{\text{th}}$ scattering instant.
The marginal distribution of the distance $D_{n}$ travelled till next
scattering must be exponentially distributed, because the pattern of speed-
marked intersections on the current line is Poisson. Note that
$T_{n}=D_{n}/V_{n}$ is the time till the next scattering. Now
$D_{n}/V_{n}^{\gamma-1}$ is a function of the relative environment $\Psi_{n}$
(using the scaling transformation (3)) and therefore forms an ergodic
sequence. Considering independent thinning of all lines for which scattering
fails, one calculates the (conditioned) exponential rate of
$D_{n}/V_{n}^{\gamma-1}=T_{n}/V_{n}^{\gamma-2}$ to be
$\tfrac{\alpha}{\alpha-(\gamma-1)}$. By the ergodic theorem it follows that
$\tfrac{1}{n}\sum_{r=0}^{n}T_{r}/V_{r}^{\gamma-2}$ converges almost surely to
$\tfrac{\alpha-(\gamma-1)}{\alpha}$. Thus in the non-SIRSN case of $\gamma=2$
it follows that scattering happens at a constant rate in time, and thus the
continuous process will be defined for all time.

In the SIRSN case of $\gamma>2$, if the diverging case holds (so
$\alpha>2(\gamma-1)$) then $V_{n}$ will eventually tend to $\infty$ at a
geometric rate. Thus in that case almost sure convergence of
$\tfrac{1}{n}\sum_{r=0}^{n}T_{r}/V_{r}^{\gamma-2}$ to a positive constant
forces us to conclude that $\sum_{r=0}^{n}T_{r}$ diverges to $\infty$, and
therefore again the continuous process will be defined for all time.

In contrast, in the converging case $\alpha<2(\gamma-1)$ a similar argument
shows that the continuous-time process will reach zero-speed in finite time,
trading off the asymptotically linear decrease of the log-speed against the
asymptotically linear increase of $\sum_{r=0}^{n}T_{r}/V_{r}^{\gamma-2}$.

In the diverging case $\alpha>2(\gamma-1)$ it is natural to ask whether the
(discrete or continuous time) scattering process achieves a limiting direction
as viewed from $\operatorname{\operatorname{\bf o}}$. In fact it will not do
so. This follows by an argument involving:

  * •

an ergodic theorem for $\Pi$ under scaling symmetries: (this is proved in the
same manner as Theorem 21 but using the $r$-$\theta$ coordinatization used in
Equation (1) for the intensity measure $\nu$ of $\Pi$, instead of the
$s$-$\phi$ parametrization used for Equation (2));

  * •

a demonstration that if $C_{v}$ is the zero-cell for $\Pi_{\geq v}$ (the
_Crofton cell_ containing the origin for the corresponding tessellation) then
there is $p>0$ such that if $Z_{0}$ lies in $C_{v}$ then $p$ is a lower bound
for the probability that Z makes a complete circuit of $\partial C_{v}$ when
first hitting $\partial C_{v}$. (This is established by noting that by scaling
it suffices to consider $v=1$; then the relevant probability can be estimated
by thinning $\Pi_{<1}$ such that lines are only retained if they hit $\partial
C_{1}$ and additionally will not scatter $Z$ on the two occasions when its
circuit encounters the line in question.) In fact Calka (2002) gives
distributional bounds on the out-radius of $C_{1}$, though here we need only
use the fact that $C_{1}$ is stochastically bounded;

  * •

finally combining these two to show $\operatorname{\mathbb{P}}\left[Z\text{
makes a circuit of }C_{n}\text{ for infinitely many }n\right]=1$.

Since $C_{n}$ will intersect any fixed line for large enough $n$, it follows
that $Z$ will eventually visit any fixed line, and therefore cannot be
eventually confined within any wedge, and therefore cannot possess a limiting
direction.

We conclude with some questions for further work.

In the critical case $\alpha=2(\gamma-1)$ it is an open question whether the
(discrete or continuous time) process (as opposed to its speed) is positive-
recurrent on neighborhoods of the origin $\operatorname{\operatorname{\bf
o}}$. Note that simple arguments imply that positive-recurrence on
neighbourhoods would force the conclusion that the process was _point_
-recurrent; if $Z$ will always eventually return to a neighbourhood $A$ of the
origin then it may (and therefore eventually will) move on to the intersection
$A\cap\Pi_{\geq v}$ between the neighbourhood and the proper Poisson line
process $\Pi_{\geq v}$ (choosing the positive speed $v$ depending on $\Pi$ so
that $A\cap\Pi_{\geq v}\neq\emptyset$); irreducibility (Lemma 24) then implies
that $Z$ has positive chance of visiting any specified point on
$A\cap\Pi_{\geq v}$, and therefore will succeed in doing so eventually after
repeated visits to $A\cap\Pi_{\geq v}$. However there is some evidence that in
fact $Z$ is transient: $Z$ is caricatured by the two-dimensional integrated
Brownian motion $\int V{\,\operatorname{d}}s$ (for $V$ a $2$-dimensional
Brownian motion), which can be shown almost surely to have only finite total
length of path within any bounded neighbourhood of its starting point, and
thus to be transient in the sense of almost surely eventually leaving this
starting point never to return.

It is natural to ask whether some kind of central limit behaviour can be
established. Certainly this question makes sense for the log-speed process, as
this is produced by partial sums of the stationary ergodic process of log-
relative speeds. We leave this question to further work, but note that the
approach to this will depend a great deal on whether or not the scattering
process itself is point-recurrent.

Central limit behaviour for the scattering process itself is complicated by
the fact that in the true SIRSN case $\gamma>2$ the times between scattering
have statistics depending monotonically on the current speed (see remarks
earlier in this section). However it may be possible to formulate the process
as being approximated by a constructed process based on a Brownian motion,
using the coupling techniques of Kendall and Westcott (1987).

It has been noted that the SIRSN-RRF defined here cannot exist on high-
dimensional SIRSN (with $\gamma>d>2$), for the simple reason that lines of
Poisson line processes in space of dimension $3$ or higher will almost surely
never intersect. However it _does_ make sense to ask whether this construction
can be generalized to line patterns in $3$-space formed by a Poisson process
of planes. However it would first be necessary to extend the results of
Kendall (2017) and Kahn (2016) to this situation. Finally, it would be an
interesting exercise to establish similar results for Rayleigh random flights
on Aldous’s (2014) binary hierarchy SIRSN.

Acknowledgements. The author acknowledges the support of the Isaac Newton
Institute for Mathematical Sciences, Cambridge, under EPSRC grant EP/K032208
(“Random Geometry” programme), by the Alan Turing Institute under EPSRC grant
EP/N510129, and also by EPSRC grants EP/K013939 and EP/R022100 for the
author’s research.

This is a theoretical research paper and, as such, no new data were created
during this study.

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The integrals involving $(l^{0})^{3}$ and $(l^{0})$ vanish as they are odd
integrals. We subdivide $I_{13}^{(1)}$ into three more integrals.

$\displaystyle I_{13a}^{(1)}$ $\displaystyle=960D\int
d^{D}l\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}dxdydrds\delta(x+y+r+s-1)\frac{xy(l^{0})^{4}}{(l^{2}-\Delta)^{6}}$
(287) $\displaystyle=960iD\int_{0}^{\infty}dl_{E}\int\cdots\int
d\varphi_{1}d\varphi_{2}\cdots
d\varphi_{D-1}\sin^{D-2}\varphi_{1}\sin^{D-3}\varphi_{2}\cdots\sin\varphi_{D-2}$
$\displaystyle\quad\times\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}dxdydrds\delta(x+y+r+s-1)\frac{xyl_{E}^{D-1}(l_{E}\sin\varphi_{1}\sin\varphi_{2}\cdots\sin\varphi_{D-2}\cos\varphi_{D-1})^{4}}{(l_{E}^{2}+\Delta)^{6}}$
$\displaystyle=960iD\int_{0}^{\pi}\sin^{D+2}\varphi_{1}d\varphi_{1}\int_{0}^{\pi}\sin^{D+1}\varphi_{2}d\varphi_{2}\int_{0}^{\pi}\sin^{D}\varphi_{3}d\varphi_{3}\cdots\int_{0}^{\pi}\sin^{5}\varphi_{D-2}\int_{0}^{2\pi}\cos^{4}\varphi_{D-1}d\varphi_{D-1}$
$\displaystyle\quad\times\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}dxdydrds\delta(x+y+r+s-1)\int_{0}^{\infty}dl_{E}\frac{xyl_{E}^{D+3}}{(l_{E}^{2}+\Delta)^{6}}$
$\displaystyle-\frac{3i(D-6)(D-4)(D-2)D^{2}(D+2)\pi^{D/2+1}}{32\sin\big{(}\frac{\pi
D}{2}\big{)}m_{f}^{8-D}}\bigg{(}\prod_{k=1}^{D-2}\frac{\Gamma\big{(}\frac{D-k+4}{2}\big{)}}{\Gamma\big{(}\frac{D-k+5}{2}\big{)}}\bigg{)}\times$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\int_{0}^{1}dr\int_{0}^{1-r}ds\int_{0}^{1-r-s}dy\frac{(1-y-r-s)y}{(r+s)^{4-\frac{D}{2}}}$
$\displaystyle=-\frac{3i\pi^{D/2+1}D(D-6)}{\sin\big{(}\frac{\pi
D}{2}\big{)}\Gamma\big{(}\frac{D+4}{2}\big{)}m_{f}^{8-D}}$

for $4<D<8$. Next, we have

$\displaystyle I_{13b}^{(1)}$
$\displaystyle=5760iDp^{2}\int_{0}^{\pi}\sin^{D}\varphi_{1}\int_{0}^{\pi}\sin^{D-1}\varphi_{2}\cdots\int_{0}^{\pi}\sin^{3}\varphi_{D-2}d\varphi_{D-2}\int_{0}^{2\pi}\cos\varphi_{D-1}d\varphi_{D-1}$
(288)
$\displaystyle\quad\times\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}dxdydrds\delta(x+y+r+s-1)\int_{0}^{\infty}dl_{E}\frac{xy(y+s)^{2}l_{E}^{D+1}}{(l_{E}^{2}+\Delta)^{6}}$
$\displaystyle=\frac{72i\pi^{D/2+1}D(D-8)p^{2}}{\sin\big{(}\frac{\pi
D}{2}\big{)}\Gamma\big{(}\frac{D+2}{2}\big{)}m_{f}^{10-D}}$

for $6<D<10$.

Finally, we have

$\displaystyle I_{13c}^{(1)}$ $\displaystyle=960iDp^{4}\int\cdots\int
d\varphi_{1}d\varphi_{2}\cdots
d\varphi_{D-1}\sin^{D-2}\varphi_{1}\sin^{D-3}\varphi_{2}\cdots\sin\varphi_{D-2}$
(289)
$\displaystyle\times\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}dxdydrds\delta(x+y+r+s-1)\int
dl_{E}\frac{xy(y+s)^{4}l_{E}^{D-1}}{(l^{2}_{E}+\Delta)^{6}}$
$\displaystyle=-\frac{4i\pi^{D/2+1}p^{4}(D-10)(D^{4}+20D^{2}+240D+2304)}{5(D+2)(D+4)(D+6)\sin\big{(}\frac{\pi
D}{2}\big{)}\Gamma\big{(}\frac{D}{2}\big{)}m_{f}^{12-D}}\,.$

for $8<D<12$.

Therefore, we have completed all the calculations of the integrals. The total
amplitude would be

$\mathcal{M}^{(n)}=-\frac{m_{f}^{2}}{v_{n}^{2}(2\pi)^{D}}\sum\mathrm{Integrals}\,.$
(290)

Since from the above calculations, each integral corresponds to a specific
range of $D$ or specific value of $D$ such that the integral converges, we see
that there is no common $D$ for all the integrals to be convergent. The rotor
mechanism can only save the divergence for each term of the calculation with
higher dimension $D$ one by one in the $W$-boson self-energy correction
diagram.

##  9 Discussion on non-Unitarity of Quantum Field Theory under Rotor
Mechanism

In normal quantum field theory, the S-matrix satisfies unitarity, subsequently
this leads to the optical theorem. However, high-order derivative quantum
field theory is infamous for dynamic instability, where the Hamiltonian is
unbounded below [48, 49, 50, 51, 30]. This causes the non-unitarity of
S-matrix of the theory. The non-unitarity issue also appears in quantizing
high-order derivative gravitational theories. High-order derivative field
theory has a great advantage of renormalizability, but so often if suffers
from the unitary problem. The quantum field theory of rotor mechanism is not
exceptional too.

Under rotor mechanism, the S-matrix is given by the time-ordered exponential
of interaction Hamiltonian of the $n$-th rotored fields in the interaction
picture,

$S=\mathrm{T}\exp\bigg{(}i\int
d^{D}x\hat{H}_{\mathrm{int}}[\Box^{n}\text{fields}(x)]\bigg{)}\,,$ (291)

where $\mathrm{T}$ is the time-ordered operator. The generic quantum amplitude
is given by

$\mathcal{M}=\langle
f|S|i\rangle=\langle\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}|S|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle\,,$
(292)

where
$|i\rangle=|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle$
is the initial incoming states and
$|f\rangle=|\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}\rangle$
is the final outgoing states. Explicitly, these states can be expressed as the
excitation of the vacuum, giving

$|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle=\prod_{i=1}^{m}\sqrt{2E_{i}}\,\hat{a}_{\boldsymbol{\mathrm{k}}_{i}}^{\dagger}|0\rangle\quad\text{and}\quad|\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}\rangle=\prod_{j=1}^{m}\sqrt{2E_{j}}\,\hat{a}_{\boldsymbol{\mathrm{p}}_{j}}^{\dagger}|0\rangle\,.$
(293)

The normalization is given by

$\langle\boldsymbol{\mathrm{k}}_{i}|\boldsymbol{\mathrm{k}}_{j}\rangle=2E_{i}(2\pi)^{D-1}\delta^{D-1}(\boldsymbol{\mathrm{k}}_{i}-\boldsymbol{\mathrm{k}}_{j})\,.$
(294)

Now the quantum amplitude reads

$\mathcal{M}=\prod_{i=1}^{m}2E_{i}(2\pi)^{D-1}\delta^{D-1}(\boldsymbol{\mathrm{p}}_{i}-\boldsymbol{\mathrm{k}}_{i})\,+\,\langle\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}|\sum_{n=1}^{\infty}\frac{i^{n}}{n!}\bigg{(}\int
d^{D}x\hat{H}_{\mathrm{int}}[\Box^{n}\text{fields}(x)]\bigg{)}^{n}|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle\,,$
(295)

The S matrix can be written as

$S=1+iT\,,$ (296)

where $T$ is the transfer matrix. If the unitarity condition of the S-matrix
is satisfied, it demands

$S^{\dagger}S=1\,,$ (297)

giving

$-i(T-T^{\dagger})=T^{\dagger}T\,.$ (298)

Then, this would follow

$\displaystyle\quad\langle\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}|T^{\dagger}T|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle$
(299)
$\displaystyle=\sum_{n}\prod_{i=1}^{n}\int\frac{d^{D-1}\boldsymbol{\mathrm{q}}_{i}}{(2\pi)^{D-1}}\frac{1}{2E_{i}}\langle\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}|T^{\dagger}|\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n}\rangle\langle\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n}|T|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle\,,$

where we have imposed the completeness relation of

$\int\frac{d^{D-1}\boldsymbol{\mathrm{q}}_{i}}{(2\pi)^{D-1}}\frac{1}{2E_{i}}|\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n}\rangle\langle\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n}|=\boldsymbol{1}\,.$
(300)

As

$\langle\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}|T^{\dagger}|\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n}\rangle=(2\pi)^{D}\delta^{D}\bigg{(}\sum_{i=1}^{m}p_{i}-\sum_{i=1}^{n}q_{i}\bigg{)}\mathcal{M}^{\dagger}(\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}\rightarrow\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n})\,.$
(301)

Thus (299) is evaluated to be

$\displaystyle\quad\langle\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}|T^{\dagger}T|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle$
(302)
$\displaystyle=\sum_{n}\prod_{i=1}^{n}\int\frac{d^{D-1}\boldsymbol{\mathrm{q}}_{i}}{(2\pi)^{D-1}}\frac{1}{2E_{i}}(2\pi)^{2D}\delta^{D}\bigg{(}\sum_{i=1}^{m}p_{i}-\sum_{i=1}^{m}q_{i}\bigg{)}$
$\displaystyle\quad\times\mathcal{M}^{\dagger}(\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}\rightarrow\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n})\mathcal{M}(\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rightarrow\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n})\,.$

Using (298), take the bra states of
$\langle\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}|$
and ket states of
$|\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rangle$
it follows that

$\displaystyle\quad-i\big{(}\mathcal{M}(\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rightarrow\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{n})-\mathcal{M}^{\dagger}(\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rightarrow\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{n})\big{)}$
(303)
$\displaystyle=\sum_{n}\prod_{i=1}^{n}\int\frac{d^{D-1}\boldsymbol{\mathrm{q}}_{i}}{(2\pi)^{D-1}}\frac{1}{2E_{i}}(2\pi)^{D}\mathcal{M}^{\dagger}(\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}\rightarrow\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n})\mathcal{M}(\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rightarrow\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n})\,.$

Therefore, the optical theorem is obtained if unitary is satisfied

$\displaystyle\quad
2\mathrm{Im}\mathcal{M}(\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rightarrow\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{n})$
(304)
$\displaystyle=\sum_{n}\prod_{i=1}^{n}\int\frac{d^{D-1}\boldsymbol{\mathrm{q}}_{i}}{(2\pi)^{D-1}}\frac{1}{2E_{i}}(2\pi)^{D}\mathcal{M}^{\dagger}(\boldsymbol{\mathrm{p}}_{1}\boldsymbol{\mathrm{p}}_{2}\cdots\boldsymbol{\mathrm{p}}_{m}\rightarrow\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n})\mathcal{M}(\boldsymbol{\mathrm{k}}_{1}\boldsymbol{\mathrm{k}}_{2}\cdots\boldsymbol{\mathrm{k}}_{m}\rightarrow\boldsymbol{\mathrm{q}}_{1}\boldsymbol{\mathrm{q}}_{2}\cdots\boldsymbol{\mathrm{q}}_{n})\,.$

However, in the following, we will show that in fact the quantum field theory
under rotor mechanism resonates a similar problem with higher-derivative
quantum gravity. For example, in Stelle theory, where second-order derivative
is concerned, the action reads [52]

$S_{\mathrm{QG}}=\int d^{D}x\sqrt{-g}\bigg{(}2\Lambda_{C}+\zeta
R+\alpha\bigg{(}R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^{2}\bigg{)}-\frac{\xi}{6}R^{2}\bigg{)}\,,$
(305)

where $R_{\mu\nu}$ is the Ricci tensor, $R=g^{\mu\nu}R_{\mu\nu}$ is the Ricci
scalar and $\Lambda_{C}$ is the cosmological constant. Upon quantization, this
action will give Feynman propagators in form of

$\frac{1}{p^{2}-m_{1}^{2}}\frac{1}{p^{2}-m_{2}^{2}}=\frac{1}{m_{1}^{2}-m_{2}^{2}}\bigg{(}\frac{1}{p^{2}-m_{1}^{2}}-\frac{1}{p^{2}-m_{2}^{2}}\bigg{)}\,.$
(306)

The dangerous minus sign in front of the second term causes the issue of non-
unitarity, for which the completeness relation in (300) cannot be satisfied.
Similarly for our rotor model case, we have

$\frac{i}{p^{4n}(p^{2}-m^{2})}\,.$ (307)

For example for $n=1$, we have

$\frac{i}{p^{4}(p^{2}-m^{2})}=\frac{i}{m^{4}(p^{2}-m^{2})}-\frac{i}{m^{4}p^{2}}-\frac{i}{m^{2}p^{4}}\,,$
(308)

where the minus sign for the second and the third term causes the problem. The
is the origin of non-unitarity such that the optical theorem in (304) cannot
be satisfied . To rescue the unitary problem, one can follow the approach by
introducing ‘fakeons’, which is a fake particle that provides a fake degree of
freedom that propagates inside the Feynman diagrams [53, 54]. Fakeons can make
a high-order derivative quantum field theory unitary [53, 54]. However, the
full analysis would be beyond the scope of this paper and requires further
studies in future.

##  10 Conclusion

In conclusion, we have applied the rotor mechanism and quantization to the
standard model of particle physics, which naturally generates high-order
derivative quantum fields in the standard model’s Lagrangian. Upon path
integral quantization, we develop feynman propagators of scalar particles,
gauge bosons and Dirac fermions under rotor model. When the rotor index is
$n=0$, this restores to the original standard model case. Then we explicitly
calculate the quantum amplitudes of the one-loop self-energy correction
diagrams of the Higgs Boson under rotor mechanism. This includes the
correction by the Higgs-self interaction, $W$-boson and fermion respectively.
We find that the rotor mechanism can generally remove the UV divergences,
however, IR divergence is arisen at the same time. We discover that the rotor
model is able to remove infinities (both UV and IR), or suppress divergences
arise from the case of the Higgs-self interaction. We find that the minimum
spacetime dimension $D$ for $n=1$ rotor index is $9$, and for $n=2$ is $17$,
and so on with a general formal of $D_{\mathrm{min}}=8n+1$. This suggests that
the rotor mechanism can remove infinities (both UV and IR) arise from simple
integral calculation, thus giving a new way to partially solve the Hierarchy
problem. However, for diagrams with more complicated integrals, such as the
$W$-boson loop correction and fermion loop correction, due to specific
dimension range arise from each integral term, there does not exist a general
$D$ that can cure the divergence all at once for specific $n$. More future
work has to be done. Finally, it is understood that a renormalizable high-
order derivative quantum field theory breaks unitarity, and yet can be rescued
by the the ‘fakeons’ approach, but detailed analysis requires rigorous work in
the future.

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