# Laplacians and Random Walks on CW Complexes

Tim Höpfner111This work is part of the author’s doctorial thesis at the
University of Göttingen.

###### Abstract

We construct random walks taking place on the $k$-cells of free $G$-CW
complexes of finite type. These random walks define operators acting on the
cellular $k$-chains that relate nicely to the (upper) cellular $k$-Laplacian.
As an application, we use this relation to show that the Novikov-Shubin
invariants of a free $G$-CW complex $X$ of finite type can be recovered from
quantities related to return probabilities of the random walks on the cells of
$X$.

###### Contents

  1. 1 Introduction
  2. 2 Random Walks on $k$-Cells
    1. 2.1 Degree $k$-Upper Random Walks
    2. 2.2 Lazy Degree $k$-Upper Random Walks
  3. 3 Relationship to Laplace Operators
  4. 4 Application to $L^{2}$-Invariants
    1. 4.1 Computing Novikov-Shubin Invariants
    2. 4.2 Example: Degree 1-Upper Random Walk on $\mathbb{R}^{2}$

##  1 Introduction

In this paper we generalise a classical result that relates the cellular
Laplace operator acting on the $0$-chains of a locally finite CW complex $X$
to a random walk taking place on the $0$-cells of the complex. This classical
result has proven to be a useful and powerful tool, allowing us to study the
Laplace operator and its spectrum by stochastic means. Indeed, if the
$1$-skeleton $X^{(1)}$ is a $d$-regular graph, and we denote the propagation
operator of the uniform nearest neighbour random walk on $X^{(1)}$ by $P$,
then we can interpret $P$ as an operator acting on
$L^{2}C^{\mathrm{cell}}_{0}(X)$ and it satisfies
$P=\operatorname{Id}-\frac{1}{d}\Delta_{0}$. Thus, we get a direct
correspondence between the spectrum of $P$ and the spectrum of $\Delta_{0}$.
While the assumption of pure degree makes the relation between $P$ and
$\Delta_{0}$ particularly nice, the arguments extend without too much trouble
to the case of graphs that are not of pure degree. Recently, work of O.
Parzanchevski and R. Rosenthal in [PR17] generalises this kind of results to
finite simplicial complexes. They constructed a random walk taking place on
the $k$-simplices such that the corresponding propagation operator again
relates nicely to the upper $k$-Laplacian
$\Delta^{\textrm{up}}_{k}=d_{k+1}^{*}d_{k+1}$ (see also R. Rosenthal’s paper
[Ros14] and S. Mukherjee and J. Steenbergen’s paper [MS16] for further
details). Using this, they deduce information about the homology of $X$ and
the spectral gap of $\Delta_{k}$ from the random walk. We will generalise
their results further, constructing random walks that take place on free
$G$-CW complexes of finite type for some group $G$ and relate the resulting
random walk to $\Delta^{\textrm{up}}_{k}$. This builds heavily on the work of
O. Parzanchevski and R. Rosenthal but some additional problems arise. In fact,
we will define a family of random walks on the oriented $k$-cells of $X$
depending on a parameter $q\in[0,1]$, such that the corresponding propagation
operators $P_{q}$ of this random walk induces, in a natural way, an operator
$B_{q}$ acting on the cellular $k$-chains of $X$. We will show that these
operators satisfy the equation

$B_{q}\circ M_{1,q}=\operatorname{Id}-\Delta^{\textrm{up}}_{k}\circ M_{2,q},$

where $M_{1,q}$ and $M_{2,q}$ are multiplication operators given explicitly in
terms of the local glueing information of the $k$\- and $(k+1)$-cells of $X$.
Using this, we can recover spectral information of $\Delta^{\textrm{up}}_{k}$
from this random walk.

For finite CW complexes, the classical Betti numbers measure the size of the
kernels of the Laplace operators. Passing to infinite CW complexes, these
Betti numbers tend to be infinite and provide little information. In these
cases, we can use some extra structure, for example coming in the form of a
cocompact free proper group action, to define a normalised version called the
$L^{2}$-Betti numbers as well as the so-called Novikov-Shubin invariants that
measure the content of subspaces close to the kernel of the Laplace operators.
As an application of the constructed random walk, we prove that the $k$th
$L^{2}$-Betti number $b^{(2)}(d_{k+1}^{*})$ and the $k$th Novikov-Shubin
invariant $\alpha_{k}(X)$ (measuring the spectrum of
$\Delta^{\textrm{up}}_{k}$ at zero and close to zero respectively) of a free
$G$-CW complex of finite type can be described by a quantity $p_{q}(n)$ of the
random walk that is closely related to return probabilities. Concretely, we
show that if $X$ is regular enough so that $M_{i,q}\equiv C_{i,q}$ are
constant, then for $q<1$ large enough,

$C_{1,q}^{-n}\left(b^{(2)}(d_{k+1}^{*})+C^{-1}n^{-\alpha_{k}(X)/2}\right)\
\leq\ p_{q}(n)\ \leq\
C_{1,q}^{-n}\left(b^{(2)}(d_{k+1}^{*})+Cn^{-\alpha_{k}(X)/2}\right).$

For $k=0$, this recovers a classical result of N. Th. Varopoulos [Var84],
stating that the return probability $p$ of the uniform nearest neighbour
random walk on $X^{(1)}$ satisfies $p(2n)\sim n^{-\alpha_{0}(X)/2}$ as
$n\to\infty$. This is a key result, as it allows us to compute the zeroth
Novikov-Shubin invariant easily by using that these random walks have the same
asymptotic behaviour in terms of the growth rate $N(G)$ of the group $G$,
meaning that $\alpha_{0}(X)=N(G)$. In contrast, there are currently only few
results for Novikov-Shubin invariants in higher degrees.

Recent work in the same direction by L. Bérnard, Y. Chaubet, N. V. Dang and T.
Schick [BCDS23] uses such random walks on higher dimensional skeleta to
compute $L^{2}$-Betti numbers and linking numbers via random walks.

##  2 Random Walks on $k$-Cells

###  2.1 Degree $k$-Upper Random Walks

Before we define the random walk, we introduce some quantities that will be
useful later on. These quantities capture the local structure of the CW
complex described by the incidence numbers of the CW complex. For this, let
$X$ be a free $G$-CW complex of finite type222Let $G$ be a group and $X$ a CW
complex with cellular left action $G\curvearrowright X$. Then $X$ is called a
free $G$-CW complex of finite type if the projection
$X\twoheadrightarrow{\left.\raisebox{-1.49994pt}{$G$}\middle\backslash\raisebox{0.50003pt}{$X$}\right.}$
is a regular covering and
${\left.\raisebox{-1.49994pt}{$G$}\middle\backslash\raisebox{0.50003pt}{$X$}\right.}$
is a finite CW complex. In particular, the finite CW complex
${\left.\raisebox{-1.49994pt}{$G$}\middle\backslash\raisebox{0.50003pt}{$X$}\right.}$
comes with a CW structure such that there are only finitely many $k$-cells in
${\left.\raisebox{-1.49994pt}{$G$}\middle\backslash\raisebox{0.50003pt}{$X$}\right.}$,
exactly one for $G$-type of $k$-cells in $X$.. We denote the set of $k$-cells
of $X$ by $I_{k}$. On each $k$-cell $\alpha\in I_{k}$ we fix (arbitrarily) one
of the two possible orientations. We denote by $\alpha=\alpha_{+}$ the cell
$\alpha$ equipped with this preferred orientation and by $-\alpha=\alpha_{-}$
the cell $\alpha$ with the other, reversed orientation. Given a pair of
$(k+1)$-cell $\beta\in I_{k+1}$ and $k$-cell $\alpha\in I_{k}$, the incidence
number $[\beta:\alpha]\in\mathbb{Z}$ is given by the mapping degree of the map
$\chi_{\beta,\alpha}$,

${\partial\beta_{+}}$${X^{(k)}}$${\frac{X^{(k)}}{X^{(k)}\setminus\\{\alpha\\}}}$${\frac{\alpha_{+}}{\partial\alpha_{+}}}$${S^{k}}$${S^{k},}$${\simeq}$$\scriptstyle{\chi_{\beta}}$$\scriptstyle{\simeq}$${\simeq}$$\scriptstyle{\chi_{\beta,\alpha}}$

where $\chi_{\beta}$ denotes the attaching map of $\beta$ to $X^{(k)}$. Note
that the sign of $[\beta:\alpha]$ depends on the orientations, where
$[\beta_{\nu}:\alpha_{\nu^{\prime}}]=\nu\nu^{\prime}[\beta:\alpha]$ for signs
$\nu,\nu^{\prime}\in\\{\pm\\}$.

######  Definition 2.1.

Let $X$ be a free $G$-CW complex of finite type. For $\alpha\in I_{k}$ and
$\beta\in I_{k+1}$ we define the quantities

$\displaystyle\begin{split}d_{+,2}(\alpha)&=\sum_{\beta^{\prime}\in
I_{k+1}}[\beta^{\prime}:\alpha]^{2},\\\ d_{+}(\alpha)&=\sum_{\beta^{\prime}\in
I_{k+1}}|[\beta^{\prime}:\alpha]|,\\\
d_{-}(\beta;\alpha)&=\begin{cases}\sum\limits_{\alpha\neq\alpha^{\prime}\in
I_{k}}|[\beta:\alpha^{\prime}]|&\text{ if }[\beta:\alpha]\neq 0,\\\ 0&\text{
if }[\beta:\alpha]=0,\end{cases}\\\ d_{-}(\alpha)&=\max_{\beta^{\prime}\in
I_{k+1}}d_{-}(\beta;\alpha),\end{split}$ (1)

where the maximum in the definition of $d_{-}(\alpha)$ exists since
$d_{-}(\beta;\alpha)$ is invariant under the
$G$-action,333$d_{-}(\beta;\alpha)=d_{-}(g.\beta;g.\alpha)$ for all $g\in G$.
This, and the application in mind, are the reason we restrict to these
complexes. The construction makes sense as long as this maximum exists. so
that it can only assume finitely many different values. Note also that these
quantities are independent of the orientations chosen on $\alpha$ and
$\beta$.444Since in the following, orientations on $(k+1)$-cells will never
play a role, we will only take care of orientations on the $k$-cells and work
with the preferred orientation on $(k+1)$-cells throughout.

These quantities generalise the idea of the degree of a vertex to $k$-cells,
with $d_{+}$ being the incoming degree and $d_{-}$ the (maximal) outgoing
degree. They capture the local structure of the CW complex around the $k$-cell
$\alpha\in I_{k}$.

We also introduce the notation

$d(\alpha,\alpha^{\prime},\beta)=-[\beta:\alpha][\beta:\alpha^{\prime}],$ (2)

measuring how well connected $\alpha$ is to $\alpha^{\prime}$ along
$\beta$.555 Here, we introduce an extra minus sign to mirror what happens in
the case of graphs. There, random walkers can walk from a vertex $v_{1}$ along
an (oriented) edge $e=(v_{1},v_{2})$ to the vertex $v_{2}$, where $e$ is an
(outgoing) edge for $v_{1}$ with $[e:v_{1}]=-1$ and an (incoming) edge for
$v_{2}$ with $[e:v_{2}]=1$, so that $[e:v_{1}][e:v_{2}]=-1$ if
$e=\\{v_{1},v_{2}\\}$. With this extra minus sign, the quantity
$d(v_{1},v_{2},e)=1$ is positive in this case. Note that
$d(\alpha,\alpha^{\prime},\beta)=d(\alpha^{\prime},\alpha,\beta)$.

######  Remark 2.2.

Recall that on the cellular $k$-chains, the upper $k$-Laplacian
$\Delta^{\textrm{up}}_{k}$ is given by the formula

$\displaystyle\begin{split}\Delta^{\textrm{up}}_{k}&\left(\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\cdot\alpha\right)\\\ &=\sum_{\alpha\in
I_{k}}\left[\sum_{\beta\in
I_{k+1}}[\beta:\alpha]^{2}\lambda_{\alpha}-\sum_{\alpha\neq\alpha^{\prime}\in
I_{k}}\sum_{\beta\in
I_{k+1}}-[\beta:\alpha][\beta:\alpha^{\prime}]\lambda_{\alpha^{\prime}}\right]\cdot\alpha\\\
&=\sum_{\alpha\in
I_{k}}\left[d_{+,2}(\alpha)\lambda_{\alpha}-\sum_{\alpha\neq\alpha^{\prime}\in
I_{k}}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)\lambda_{\alpha^{\prime}}\right]\cdot\alpha,\end{split}$
(3)

where, for $k=0$, $d_{2,+}(\alpha)=\deg(\alpha)$ is the degree of the vertex
$\alpha$ (ignoring loops) and $d(\alpha,\alpha^{\prime},\beta)=1$ if $\beta$
is an edge from $\alpha$ to $\alpha^{\prime}$ (again, ignoring loops) and zero
otherwise. Note that $\Delta^{\mathrm{up}}_{k}$ sees the incidence numbers
between $k$\- and $(k+1)$-cells in $X$. Hence, they have to appear in the
definition of our random walk. Furthermore, both the sign and the size of the
incidence numbers have to play a role.

######  Definition 2.3.

Consider the set of oriented $k$-cells
$I_{k}^{\pm}=I_{k}\cup\left\\{\alpha_{-}\>\middle|\>\alpha\in I_{k}\right\\}.$
We call two oriented $k$-cells $\alpha_{\nu},\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}$, where $\nu,\nu^{\prime}\in\\{\pm\\}$ denote orientations,
(upper) neighbours along $\beta\in I_{k+1}$, and write
$\alpha_{\nu}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}$ if

$\alpha_{\nu}\neq-\alpha^{\prime}_{\nu^{\prime}}\quad\text{ and }\quad
d(\alpha_{\nu},\alpha^{\prime}_{\nu^{\prime}},\beta)=\nu\nu^{\prime}d(\alpha,\alpha^{\prime},\beta)>0.$

Note that this condition is independent of the orientation chosen on $\beta\in
I_{k+1}$.

######  Definition 2.4.

The random walk $\mathfrak{R}^{k}=\mathfrak{R}^{k}(X)$ is given by the state
space $I_{k}^{*}=I_{k}^{\pm}\cup\\{\Theta\\}$, where $\Theta$ is an auxiliary,
absorbing state together with the following moving probabilities:

  * •

The moving probabilities starting from the absorbing state $\Theta$ are given
by

$\displaystyle\mathbb{P}(\Theta\to\Theta)$
$\displaystyle=1\quad\text{and}\quad\mathbb{P}(\Theta\to\alpha_{\pm})=0\quad\text{for
all }\alpha_{\pm}\in I_{k}^{\pm}.$

  * •

To define the moving probabilities starting from $\alpha_{\nu}\in
I_{k}^{\pm}$, we define first for $\alpha^{\prime}_{\nu^{\prime}}\in
I^{\pm}_{k}$ and $\beta\in I_{k+1}$ the quantities

$\displaystyle\mathbb{P}(\alpha_{\nu}\nearrow\beta)$
$\displaystyle=\frac{|[\beta:\alpha_{\nu}]|}{d_{+}(\alpha)}\qquad\text{and}\qquad\mathbb{P}_{\alpha}(\beta\searrow\alpha^{\prime}_{\nu^{\prime}})=\frac{|[\beta:\alpha^{\prime}_{\nu^{\prime}}]|}{d_{-}(\alpha)}$

These probabilities can be seen as an intermediate step of moving first from
$\alpha$ to $\beta$ and then from $\beta$ to $\alpha^{\prime}$ (keeping in
mind that we started at $\alpha$). In this sense, we define

$\displaystyle\mathbb{P}(\alpha_{\nu}\xrightarrow{\beta}\alpha^{\prime}_{\nu^{\prime}})$
$\displaystyle=\begin{cases}\mathbb{P}(\alpha_{\nu}\nearrow\beta)\mathbb{P}_{\alpha}(\beta\searrow\alpha^{\prime}_{\nu^{\prime}})=\frac{-[\beta:\alpha_{\nu}][\beta:\alpha^{\prime}_{\nu^{\prime}}]}{d_{+}(\alpha)d_{-}(\alpha)}=\frac{d(\alpha_{\nu},\alpha^{\prime}_{\nu^{\prime}},\beta)}{d_{+}(\alpha)d_{-}(\alpha)}>0&\text{if
}\alpha_{\nu}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}},\\\
0&\text{else,}\end{cases}$

the probability of moving from $\alpha$ along $\beta$ to $\alpha^{\prime}$.
Recall here that $\alpha\not\sim\pm\alpha$ by definition, so that
$\mathbb{P}(\alpha\xrightarrow{\beta}\pm\alpha)=0$. Finally, we set

$\displaystyle\mathbb{P}(\alpha_{\nu}\to\alpha^{\prime}_{\nu^{\prime}})$
$\displaystyle=\sum_{\beta\in
I_{k+1}}\mathbb{P}(\alpha_{\nu}\xrightarrow{\beta}\alpha^{\prime}_{\nu^{\prime}})=\sum_{\stackrel{{\scriptstyle\beta\in
I_{k+1}}}{{\alpha\overset{\beta}{\sim}\alpha^{\prime}}}}\frac{d(\alpha_{\nu},\alpha^{\prime}_{\nu^{\prime}},\beta)}{d_{+}(\alpha)d_{-}(\alpha)}$
$\displaystyle=\frac{1}{{d_{+}(\alpha)d_{-}(\alpha)}}\sum_{\stackrel{{\scriptstyle\beta\in
I_{k+1}}}{{\alpha_{\nu}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}}}}{d(\alpha_{\nu},\alpha^{\prime}_{\nu^{\prime}},\beta)}$

The moving probabilities would add to one if we use $d_{-}(\beta;\alpha)$ in
place of $d_{-}(\alpha)$, however it will be important later on that we can
pull the factor $d_{-}(\alpha)^{-1}$ out of the sum as it depends only on the
starting cell $\alpha$. Consequently, their sum needs not to be one but can be
smaller since we might make the denumerator bigger for some of the summands.

  * •

The complementary probability will be the probability of moving to $\Theta$,
that is

$\mathbb{P}(\alpha_{\nu}\to\Theta)=1-\sum_{\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}}\mathbb{P}(\alpha_{\nu}\to\alpha^{\prime}_{\nu^{\prime}}).$

This defines a random walk on $I_{k}^{\ast}$. The propagation operator $P$
with entries $P_{s,s^{\prime}}=\mathbb{P}(s^{\prime}\to s)$ for
$s,s^{\prime}\in I_{k}^{*}$ acts on $\ell^{2}(I_{k}^{*})=\left\\{\sum_{s\in
I_{k}^{*}}\lambda_{s}\cdot s\>\middle|\>\sum_{s\in
I_{k}^{*}}|\lambda_{s}|^{2}<\infty\right\\}$ by

$P\left(\sum_{s\in I_{k}^{*}}\lambda_{s}\cdot s\right)=\sum_{s\in
I_{k}^{*}}\left[\sum_{s^{\prime}\in
I_{k}^{*}}P_{s,s^{\prime}}\lambda_{s^{\prime}}\right]\cdot s=\sum_{s\in
I_{k}^{*}}\left[\sum_{s^{\prime}\in I_{k}^{*}}\mathbb{P}(s^{\prime}\to
s)\lambda_{s^{\prime}}\right]\cdot s.$

######  Example 2.5.

Let us consider a small example to see how we can find the moving
probabilities. For this, let $X$ be a CW complex and $\alpha\in I_{k}$ some
$k$-cell. In order to find the $k$-cells we can move to from $\alpha=+\alpha$,
we proceed as follows:

  1. 1.

First we consider all $(k+1)$-cells $\beta\in I_{k+1}$ and find those, that
have non-zero incidence number $[\beta:\alpha]\neq 0$. For this example, let
us say there are two such $(k+1)$-cells $\beta_{1}$ and $\beta_{2}$ with
$[\beta_{1}:\alpha]=1$ and $[\beta_{2}:\alpha]=-2$.

  2. 2.

Then we consider all other $k$-cells $\alpha\neq\alpha^{\prime}\in I_{k}$ that
have non-zero incidence numbers with at least one of the $(k+1)$-cells above,
that is $[\beta_{1}:\alpha^{\prime}]\neq 0$ or
$[\beta_{2}:\alpha^{\prime}]\neq 0$. For this example, let us say there are
three such $k$-cells, $\alpha_{1}$ with $[\beta_{1}:\alpha_{1}]=1$,
$\alpha_{2}$ with $[\beta_{1}:\alpha_{2}]=2$ and $[\beta_{2}:\alpha_{2}]=4$
and $\alpha_{3}$ with $[\beta_{2}:\alpha_{3}]=-2$.

We visualise this with the following diagram:

${\beta_{1}}$${\beta_{2}}$${\in
I_{k+1}}$${\alpha_{1}}$${\alpha_{2}}$${\alpha}$${\alpha_{2}}$${\alpha_{3}}$${\in
I_{k}}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{4}$$\scriptstyle{-2}$$\scriptstyle{1}$$\scriptstyle{-2}$

Next, we first change the orientations on the $(k+1)$-cells such that the
incidence numbers with $\alpha$ are negative, so here we change the
orientation on $\beta_{1}$.666This orientation of $\beta_{1}$ has no impact on
the formulas in the end and is thus suppressed in the formal definition. Then
we change the orientations on the $\alpha_{i}$, $i\in\\{1,2,3\\}$, for each of
the $\beta_{j}$ independently such that the incidence numbers with the
$(k+1)$-cells become positive. This changes our diagram as follows:

${-\beta_{1}}$${\beta_{2}}$${\in
I_{k+1}^{\pm}}$${-\alpha_{1}}$${-\alpha_{2}}$${\alpha}$${\alpha_{2}}$${-\alpha_{3}}$${\in
I_{k}^{\pm}}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{4}$$\scriptstyle{2}$$\scriptstyle{-1}$$\scriptstyle{-2}$

We now introduce the auxiliary state $\Theta$. For each of the $(k+1)$-cells,
we sum the outgoing incidence numbers. Here, we get
$d_{-}(\beta_{1};\alpha)=1+2=3$ for $\beta_{1}$ and
$d_{-}(\beta_{2};\alpha)=2+4=6$ for $\beta_{2}$. The maximum is therefore $6$,
and we add connections from each of the $(k+1)$-cells to the new state
$\Theta$ until the sum of outgoing edges is equal to this maximum, i.e.,
$-\beta_{1}\xrightarrow{3}\Theta$ in this case:

${-\beta_{1}}$${\beta_{2}}$${\in
I_{k+1}^{\pm}}$${\Theta}$${-\alpha_{1}}$${-\alpha_{2}}$${\alpha}$${\alpha_{2}}$${-\alpha_{3}}$${\in
I_{k}^{*}}$$\scriptstyle{3}$$\scriptstyle{1}$$\scriptstyle{2}$$\scriptstyle{4}$$\scriptstyle{2}$$\scriptstyle{-1}$$\scriptstyle{-2}$

The moving probabilities can now be read to be proportional to the annotations
of the arrows. For the first intermediate step,
$\mathbb{P}(\alpha\nearrow-\beta_{1})=\nicefrac{{1}}{{3}}$ and
$\mathbb{P}(\alpha\nearrow\beta_{2})=\nicefrac{{2}}{{3}}$. For the second
step,

$\mathbb{P}(-\beta_{1}\searrow\Theta)=\nicefrac{{3}}{{6}},\qquad\mathbb{P}(-\beta_{1}\searrow-\alpha_{1})=\nicefrac{{1}}{{6}}\quad\text{and}\quad\mathbb{P}(-\beta_{1}\searrow-\alpha_{2})=\nicefrac{{2}}{{6}}$

and for $\beta_{2}$ we have

$\mathbb{P}(\beta_{2}\searrow\alpha_{2})=\nicefrac{{4}}{{6}}\quad\text{and}\quad\mathbb{P}(\beta_{2}\searrow-\alpha_{3})=\nicefrac{{2}}{{6}}.$

The introduction of $\Theta$ guaranties that the denominator of these
(unreduced) fractions is the same everywhere and the moving probabilities are
proportional to the incidence numbers even if the first intermediate step
leads to different $\beta_{i}$s. Multiplying these accordingly, we find

$\displaystyle\mathbb{P}(\alpha\xrightarrow{-\beta_{1}}-\alpha_{1})=\nicefrac{{1}}{{3}}\cdot\nicefrac{{1}}{{6}},\quad$
$\displaystyle\mathbb{P}(\alpha\xrightarrow{-\beta_{1}}-\alpha_{2})=\nicefrac{{1}}{{3}}\cdot\nicefrac{{2}}{{6}},\quad$
$\displaystyle\mathbb{P}(\alpha\xrightarrow{-\beta_{1}}\Theta)=\nicefrac{{1}}{{3}}\cdot\nicefrac{{3}}{{6}},$
$\displaystyle\mathbb{P}(\alpha\xrightarrow{\beta_{2}}\alpha_{2})=\nicefrac{{2}}{{3}}\cdot\nicefrac{{4}}{{6}},$
$\displaystyle\mathbb{P}(\alpha\xrightarrow{-\beta_{1}}-\alpha_{3})=\nicefrac{{2}}{{3}}\cdot\nicefrac{{2}}{{6}}.$

Here, every oriented $k$-cell can be reached only via one $(k+1)$-cell,
otherwise we would have to sum over all $(k+1)$-cells, that is
$\mathbb{P}(\alpha\to s)=\sum_{\pm\beta\in
I_{k+1}^{\pm}}\mathbb{P}(\alpha\xrightarrow{\pm\beta}s)$ for $s\in I_{k}^{*}$.
Therefore, in this example, a random walker starting at $\alpha$ has the
following possible moves, with annotations denoting the probabilities:

${\alpha}$${-\alpha_{1}}$${-\alpha_{2}}$${\alpha_{2}}$${-\alpha_{3}}$${\Theta}$$\scriptstyle{\nicefrac{{1}}{{18}}}$$\scriptstyle{\nicefrac{{1}}{{9}}}$$\scriptstyle{\nicefrac{{4}}{{9}}}$$\scriptstyle{\nicefrac{{2}}{{9}}}$$\scriptstyle{\nicefrac{{1}}{{6}}}$

Note that the cell $\alpha_{2}$ can be reached with both possible
orientations. If we start at the cell $-\alpha$ with reversed orientation, we
obtain the same moving probabilities as for $\alpha$, but now leading to the
same cells but with flipped orientations instead.

We now define an operator $B$ acting directly on the unoriented $k$-skeleton
$I_{k}$, which is closely related to this random walk.

######  Definition 2.6.

We define the projection operator
$T\colon\ell^{2}(I_{k}^{*})\to\ell^{2}(I_{k})$ by

$T\left(\sum_{s\in I_{k}^{*}}\lambda_{s}\cdot s\right)=\sum_{\alpha\in
I_{k}}(\lambda_{\alpha_{+}}-\lambda_{\alpha_{-}})\cdot\alpha$

and the inclusion operator $I\colon\ell^{2}(I_{k})\to\ell^{2}(I_{k}^{*})$,
using that $I_{k}=I_{k}^{+}\subset I_{k}^{*}$, by

$I\left(\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\cdot\alpha\right)=\sum_{\alpha_{+}\in
I_{k}^{+}}\lambda_{\alpha_{+}}\cdot\alpha_{+}.$

Lastly, we define the operator $B\colon\ell^{2}(I_{k})\to\ell^{2}(I_{k})$ by

$B\left(\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\cdot\alpha\right)=\sum_{\alpha\in
I_{k}}\left[\sum_{\alpha\neq\alpha^{\prime}\in
I_{k}}\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)\lambda_{\alpha^{\prime}}\right]\cdot\alpha.$

With
$B_{\alpha,\alpha^{\prime}}=\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\beta\in
I_{k+1}}d(\alpha_{\nu},\alpha^{\prime}_{\nu^{\prime}},\beta)$ for
$\alpha\neq\alpha^{\prime}$ and $B_{\alpha,\alpha}=0$,

$B\left(\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\cdot\alpha\right)=\sum_{\alpha\in
I_{k}}\left[\sum_{\alpha^{\prime}\in
I_{k}}B_{\alpha,\alpha^{\prime}}\lambda_{\alpha^{\prime}}\right]\cdot\alpha.$

This is captured by the diagram

${\ell^{2}(I_{k}^{*})}$${\ell^{2}(I_{k}^{*})}$${\ell^{2}(I_{k})}$${\ell^{2}(I_{k}).}$$\scriptstyle{P}$$\scriptstyle{T}$$\scriptstyle{T}$$\scriptstyle{B}$$\scriptstyle{I}$$\scriptstyle{I}$

This operator $B$ does not describe a random walk since
$B_{\alpha,\alpha^{\prime}}$, the “probability of moving from
$\alpha^{\prime}$ to $\alpha$”, may even be negative. However, this operator
is closely related to the random walk described by $P$. Indeed, using the
operators $T$ and $I$ we can see that $B$ describes the process that arises
from the random walk if we consider a random walker arriving at a cell
$\alpha_{-}$ (equipped with the reversed orientation) as the inverse of a
random walker at $\alpha_{+}$ — that is we allow random walkers at
$\alpha_{+}$ and $\alpha_{-}$ to cancel each other out.777While it is not
clear which, if any, physical process this operator $B$ describes,
cancellation between different objects does happen in physics. For example,
studying fermions via the Dirac equation suggests that for every particle
there is a corresponding anti-particle, such as electrons and positrons. If
they meet, they will annihilate each other.

######  Lemma 2.7.

The operators $T$, $I$ and $B$ defined above satisfy the equations

$BT=TP,\qquad B=TPI\quad\text{and}\quad B^{n}=TP^{n}I.$

###### Proof.

We check these equalities by direct computation. For $BT$ we obtain

$\displaystyle BT\left(\sum_{s\in I_{k}^{*}}\lambda_{s}\cdot s\right)$
$\displaystyle=B\left(\sum_{\alpha\in
I_{k}}(\lambda_{\alpha_{+}}-\lambda_{\alpha_{-}})\cdot\alpha\right)$
$\displaystyle=\sum_{\alpha\in I_{k}}\left[\sum_{\alpha^{\prime}\in
I_{k}}\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)\cdot(\lambda_{\alpha^{\prime}_{+}}-\lambda_{\alpha^{\prime}_{-}})\right]\cdot\alpha$

and for $TP$ we compute (omitting the coefficient of $\Theta$ as it disappears
in the next step) that

$\displaystyle TP$ $\displaystyle\left(\sum_{s\in
I_{k}^{*}}\lambda_{s}s\right)$ $\displaystyle=T\left(\sum_{\alpha_{\nu}\in
I_{k}^{\pm}}\left[\sum_{\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}}\frac{1}{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}\\!\\!\sum_{\alpha_{\nu}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}}\\!\\!-[\beta:\alpha_{\nu}][\beta:\alpha^{\prime}_{\nu^{\prime}}]\lambda_{\alpha^{\prime}_{\nu^{\prime}}}\right]{\alpha_{\nu}}+\cdots\Theta\right)$
$\displaystyle=\sum_{\alpha\in
I_{k}}\left[\sum_{\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}}\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\alpha_{+}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}}-[\beta:\alpha_{+}][\beta:\alpha^{\prime}_{\nu^{\prime}}]\lambda_{\alpha^{\prime}_{\nu^{\prime}}}\right]\cdot\alpha$
$\displaystyle\quad-\sum_{\alpha\in
I_{k}}\left[\sum_{\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}}\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\alpha_{-}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}}-[\beta:\alpha_{-}][\beta:\alpha^{\prime}_{\nu^{\prime}}]\lambda_{\alpha^{\prime}_{\nu^{\prime}}}\right]\cdot\alpha.$

Now we use that $[\beta:\alpha]=[\beta:\alpha_{+}]=-[\beta:\alpha_{-}]$ and
$[\beta:\alpha][\beta:\alpha^{\prime}]\neq 0$ only if either
$\alpha_{\pm}\overset{\beta}{\sim}\alpha^{\prime}_{\pm}$ or
$\alpha_{\pm}\overset{\beta}{\sim}\alpha^{\prime}_{\mp}$ together with
$-[\beta:\alpha_{\nu}][\beta:\alpha^{\prime}_{\nu^{\prime}}]=\nu\nu^{\prime}d(\alpha,\alpha^{\prime},\beta)$
to find that

$\displaystyle TP\left(\sum_{s\in I_{k}^{*}}\lambda_{s}\cdot s\right)$
$\displaystyle=\sum_{\alpha\in
I_{k}}\left[\sum_{\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}}\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\alpha_{+}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}}d(\alpha,\alpha^{\prime},\beta)\cdot\nu^{\prime}\lambda_{\alpha^{\prime}_{\nu^{\prime}}}\right]\cdot\alpha$
$\displaystyle\quad+\sum_{\alpha\in
I_{k}}\left[\sum_{\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}}\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\alpha_{-}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}}d(\alpha,\alpha^{\prime},\beta)\cdot\nu^{\prime}\lambda_{\alpha^{\prime}_{\nu^{\prime}}}\right]\cdot\alpha$
$\displaystyle=\sum_{\alpha\in I_{k}}\left[\sum_{\alpha\neq\alpha^{\prime}\in
I_{k}}\frac{1}{{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)\cdot(\lambda_{\alpha^{\prime}_{+}}-\lambda_{\alpha^{\prime}_{-}})\right]\cdot\alpha$

showing the first equality. For the second equality, we have
$TI=\operatorname{Id}$, hence $TPI=BTI=B$ and lastly $B^{n}=B^{n}TI=TP^{n}I$.
∎

######  Corollary 2.8.

For the operators $B$ and $P$ defined above, $n\in\mathbb{N}$ and all
$\alpha,\alpha^{\prime}\in I_{k}$,

$\langle B^{n}(\alpha),\alpha^{\prime}\rangle=\langle
P^{n}(\alpha_{+}),\alpha^{\prime}_{+}\rangle-\langle
P^{n}(\alpha_{+}),\alpha^{\prime}_{-}\rangle.$

###### Proof.

Using $B^{n}=TP^{n}I$, we compute the coefficient $\langle
B^{n}(\alpha),\alpha^{\prime}\rangle$ of $\alpha^{\prime}$ in $B(\alpha)$ by

$\displaystyle\langle B^{n}(\alpha),\alpha^{\prime}\rangle$
$\displaystyle=\langle TP^{n}I(\alpha),\alpha^{\prime}\rangle=\langle
TP^{n}(\alpha_{+}),\alpha^{\prime}\rangle=\langle
P^{n}(\alpha_{+}),\alpha^{\prime}_{+}\rangle-\langle
P^{n}(\alpha_{+}),\alpha^{\prime}_{-}\rangle.\qed$

In particular, we can define the following quantities generalising the idea of
return probabilities.

######  Definition 2.9.

For the random walk described by $P$ and $\alpha\in I_{k}$, we define the
return probabilities $p_{\alpha,+}$ and the probabilities of returning with
reversed orientation $p_{\alpha,-}$ respectively by

$p_{\alpha,+}(n)=\langle
P^{n}(\alpha_{+}),\alpha_{+}\rangle\quad\text{and}\quad
p_{\alpha,-}(n)=\langle P^{n}(\alpha_{+}),\alpha_{-}\rangle.$

For the process described by $B$ we define

$p_{\alpha}(n)=\langle B^{n}(\alpha),\alpha\rangle.$

Note that all three quantities are independent of the choice of preferred
orientations. Notice also that
$p_{\alpha}(n)=p_{\alpha,+}(n)-p_{\alpha,-}(n)$, hence, after summing over all
$G$-types of $k$-cells we obtain the following expression for the von Neumann
trace of $B^{n}$.

######  Corollary 2.10.

For $n\in\mathbb{N}$, the von Neumann trace of $B^{n}$ is given by

$\operatorname{tr}_{\mathcal{N}G}(B^{n})=\sum_{\alpha\in I_{k}(G\setminus
X)}p_{\alpha}(n)=\sum_{\alpha\in I_{k}(G\setminus
X)}p_{\alpha,+}(n)-p_{\alpha,-}(n).$

Before moving on and relating this random walk to the upper $k$-Laplacian, we
will introduce one extra parameter that will prove useful later on when
studying the spectra of the operators.

###  2.2 Lazy Degree $k$-Upper Random Walks

Given a random walk $\mathfrak{R}=(V,P)$ with state space $V$ and propagation
operator $P$ and $q\in[0,1]$, the $q$-lazy version $\mathfrak{R}_{q}$ of the
random walk $\mathfrak{R}$ is given by
$\mathfrak{R}_{q}=(V,q\operatorname{Id}+(1+q)P)$, that is the random walk on
$V$ that stays put with probability $q$ and moves according to the random walk
$\mathfrak{R}$ with probability $1-q$. In particular, the moving probabilities
for the $q$-lazy version $\mathfrak{R}^{k}_{q}(X)$ of the previously defined
random walk on $I_{k}^{*}$ are given as follows:

  * •

For the absorbing state $\Theta$,
$\qquad\mathbb{P}_{q}(\Theta\to\Theta)=1,\qquad\mathbb{P}_{q}(\Theta\to\alpha_{\nu})=0.$

  * •

For $\alpha_{\nu}\in I_{k}^{\pm}$ (that is $\alpha\in I_{k}$ and
$\nu\in\\{\pm\\}$),

$\displaystyle\mathbb{P}_{q}(\alpha_{\nu}\to\alpha_{\nu})=q,\qquad\mathbb{P}_{q}(\alpha_{\nu}\to-\alpha_{\nu})=0.$

  * •

For $\alpha_{\nu},\alpha^{\prime}_{\nu^{\prime}}\in I_{k}^{\pm}$ with
$\alpha_{\nu}\neq\pm\alpha^{\prime}_{\nu^{\prime}}$,

$\mathbb{P}_{q}(\alpha_{\nu}\to\alpha^{\prime}_{\nu^{\prime}})=\frac{1-q}{{d_{+}(\alpha)d_{-}(\alpha)}}\sum_{\stackrel{{\scriptstyle\beta\in
I_{k+1}}}{{\alpha_{\nu}\overset{\beta}{\sim}\alpha^{\prime}_{\nu^{\prime}}}}}d(\alpha_{\nu},\alpha_{\nu^{\prime}},\beta)=(1-q)\mathbb{P}(\alpha_{\nu}\to\alpha^{\prime}_{\nu^{\prime}}).$

  * •

Lastly,

$\mathbb{P}_{q}(\alpha_{\nu}\to\Theta)=1-\sum_{\alpha^{\prime}_{\nu^{\prime}}\in
I_{k}^{\pm}}\mathbb{P}_{q}(\alpha_{\nu}\to\alpha^{\prime}_{\nu^{\prime}})=(1-q)\mathbb{P}(\alpha_{\nu}\to\Theta).$

In the same spirit, we define $B_{q}\curvearrowright\ell^{2}I_{k}$ by
$B_{q}=q\operatorname{Id}+(1-q)B$. This operator is given by

$\displaystyle B_{q}\left(\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\cdot\alpha\right)$ $\displaystyle=\sum_{\alpha\in
I_{k}}\left[q\lambda_{\alpha}+\sum_{\alpha\neq\alpha^{\prime}\in
I_{k}}\frac{1-q}{d_{+}(\alpha^{\prime})d_{-}(\alpha^{\prime})}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)\lambda_{\alpha^{\prime}}\right]\cdot\alpha.$

######  Corollary 2.11.

These operators satisfy $B_{q}T=TP_{q}$, $B_{q}=TP_{q}I$ and
$B_{q}^{n}=TP_{q}^{n}I$.

###### Proof.

This follows from the previous equalities since
$B_{q}T=qT+(1-q)BT=qT+(1-q)TP=TP_{q}$ and
$TP_{q}I=qTI+(1-q)TPI=q+(1-q)B=B_{q}$. ∎

As before, we consider the probabilities of returning to the same $k$-cell
with the same orientation or the reversed orientation respectively.

######  Definition 2.12.

For $\alpha\in I_{k}$, we define the quantities

$\displaystyle p_{q,\alpha,+}(n)$ $\displaystyle=\langle
P_{q}^{n}(\alpha_{+}),\alpha_{+}\rangle,$ $\displaystyle p_{q,\alpha,-}(n)$
$\displaystyle=\langle P_{q}^{n}(\alpha_{+}),\alpha_{-}\rangle,$
$\displaystyle p_{q,\alpha}(n)$ $\displaystyle=\langle
B_{q}^{n}(\alpha),\alpha\rangle$

Again, $p_{q,\alpha}(n)=p_{q,\alpha,+}(n)-p_{q,\alpha,-}(n)$, hence we can
compute the von Neumann trace of $B_{q}^{n}$ using the probabilities of the
random walk. We define

$p_{q}(n)=\operatorname{tr}_{\mathcal{N}G}(B_{q}^{n})=\sum_{\alpha\in
I_{k}(G\setminus X)}p_{q,\alpha,+}(n)-p_{q,\alpha,-}(n)$

##  3 Relationship to Laplace Operators

We now compare the operator $B_{q}$ to the upper Laplacian
$\Delta^{\mathrm{up}}_{k}$. Recall from Equation (3) that
$\Delta^{\mathrm{up}}_{k}$ acts on $\ell^{2}I_{k}$ by

$\displaystyle\Delta^{\mathrm{up}}_{k}\left(\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\cdot\alpha\right)$ $\displaystyle=\sum_{\alpha\in
I_{k}}\left[\sum_{\beta\in
I_{k+1}}[\beta:\alpha]^{2}\lambda_{\alpha}-\sum_{\alpha^{\prime}\neq\alpha\in
I_{k}}\sum_{\beta\in
I_{k+1}}-[\beta:\alpha][\beta:\alpha^{\prime}]\lambda_{\alpha^{\prime}}\right]\cdot\alpha$
$\displaystyle=\sum_{\alpha\in
I_{k}}\left[d_{+,2}(\alpha)\lambda_{\alpha}-\sum_{\alpha^{\prime}\neq\alpha\in
I_{k}}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)\lambda_{\alpha^{\prime}}\right]\cdot\alpha$

######  Theorem 3.1.

Let $B_{q}\curvearrowright\ell^{2}I_{k}=\ell^{2}C^{\mathrm{cell}}_{k}(X)$ be
the operator $B_{q}=TP_{q}I$ defined as above. Then

$\displaystyle B_{q}\circ
M_{1,q}=\operatorname{Id}-\Delta^{\mathrm{up}}_{k}\circ M_{2,q},$

where $M_{1,q},M_{2,q}\curvearrowright\ell^{2}I_{k}$ are the non-negative
multiplication operators given by

$\displaystyle M_{1,q}=$
$\displaystyle\frac{d_{+}d_{-}}{qd_{+}d_{-}+(1-q)d_{+,2}},$
$\displaystyle\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\alpha\mapsto\sum_{\alpha\in
I_{k}}\frac{d_{+}(\alpha)d_{-}(\alpha)}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}\cdot\lambda_{\alpha}\alpha,$
$\displaystyle M_{2,q}=$ $\displaystyle\frac{1-q}{qd_{+}d_{-}+(1-q)d_{+,2}},$
$\displaystyle\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\alpha\mapsto\sum_{\alpha\in
I_{k}}\frac{1-q}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}\cdot\lambda_{\alpha}\alpha.$

###### Proof.

For $\alpha,\alpha^{\prime}\in I_{k}$ we compare the contributions
$(B_{q}\circ M_{1,q})_{\alpha^{\prime},\alpha}$ and
$(\operatorname{Id}-\Delta^{\mathrm{up}}_{k}\circ
M_{2,q})_{\alpha^{\prime},\alpha}$ coming from the coefficient of $\alpha$ in
the argument to the coefficient of $\alpha^{\prime}$ in the image.888In the
sense that both are operators
$\Xi\curvearrowright\ell^{2}C_{k}^{\mathrm{cell}}(X)$ that can be written as
$\Xi\colon\sum_{\alpha\in
I_{k}}\lambda_{\alpha}\alpha\mapsto\sum_{\alpha^{\prime}\in
I_{k}}\left[\sum_{\alpha\in
I_{k}}\Xi_{\alpha^{\prime},\alpha}\lambda_{\alpha}\right]\cdot{\alpha^{\prime}}.$
For $\alpha\neq\alpha^{\prime}$ these contributions are given by

$\displaystyle(B_{q}\circ M_{1,q})_{\alpha^{\prime},\alpha}$
$\displaystyle=\frac{d_{+}(\alpha)d_{-}(\alpha)}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}\cdot(B_{q})_{\alpha^{\prime},\alpha}$
$\displaystyle=\frac{d_{+}(\alpha)d_{-}(\alpha)}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}\cdot\frac{1-q}{d_{+}(\alpha)d_{-}(\alpha)}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)$
$\displaystyle=\frac{1-q}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)$
$\displaystyle=0-\frac{1-q}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}\cdot\left(-\sum_{\beta\in
I_{k+1}}d(\alpha,\alpha^{\prime},\beta)\right)$
$\displaystyle=(\operatorname{Id}-\Delta^{\mathrm{up}}_{k}\circ
M_{2,q})_{\alpha^{\prime},\alpha}$

and for $\alpha=\alpha^{\prime}$ by

$\displaystyle(B_{q}\circ M_{1,q})_{\alpha,\alpha}$
$\displaystyle=q\cdot\frac{d_{+}(\alpha)d_{-}(\alpha)}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}$
$\displaystyle=1-\frac{(1-q)d_{+,2}(\alpha)}{qd_{+}(\alpha)d_{-}(\alpha)+(1-q)d_{+,2}(\alpha)}=(\operatorname{Id}-\Delta^{\mathrm{up}}_{k}\circ
M_{2,q})_{\alpha,\alpha}.$

Since all these coefficients agree, the claim follows. ∎

######  Remark 3.2.

The construction here generalises the one given by O. Parzanchevski and R.
Rosenthal on simplicial complexes in [PR17] and the previous theorem
generalises Proposition 2.8 (1) of their paper. Considering the random walk of
O. Parzanchevski and R. Rosenthal in degree $k=(d-1)$, the incidence numbers
of a simplicial complex (viewed as a CW complex) are given as
$[\beta:\alpha]\in\\{0,\pm 1\\}$, where $\pm 1$ occurs if the $(d-1)$-simplex
$\alpha$ is in the boundary of the $d$-simplex $\beta$, with sign depending on
orientations. Therefore,

$d_{+}(\alpha)=d_{+,2}(\alpha)=\deg(\alpha),\quad d_{-}(\alpha)=d,$

where $\deg(\alpha)$ denoted the number of $d$-simplices $\beta\in I_{d}$
containing $\alpha$. Hence,

$\frac{d}{q(d-1)+1}B_{q}=\operatorname{Id}-\frac{1-q}{q(d-1)+1}\cdot\frac{\Delta^{\mathrm{up}}_{d-1}}{\deg(\alpha)},$

where
$\Delta^{\mathrm{up}}_{w}=\frac{\Delta^{\mathrm{up}}_{d-1}}{\deg(\alpha)}$ is
the weighted upper Laplacian used by O. Parzanchevski and R. Rosenthal,
defined by using a weighted scalar product on $\ell^{2}I_{k}$. Note that in
this case, the diagonal operator $M_{1,q}$ is given by multiplication by a
constant (depending on $q$ and $d$ but not on $\alpha\in I_{d-1}$).

##  4 Application to $L^{2}$-Invariants

As a short reminder on $L^{2}$-invariants, we briefly recall some important
definitions. For a proper introduction we refer to [Kam19] and [Lüc02].

Let $M$ be a manifold with cocompact free and proper action by a (discrete)
group $G$. We denote by $\mathcal{N}G$ the von Neumann algebra of $G$. On
$\ell^{2}G$, we define the von Neumann trace by
$\operatorname{tr}_{\mathcal{N}G}(f)=\langle f(e),e\rangle_{\ell^{2}G}$. This
notion can be extended to Hilbert $\mathcal{N}G$-modules, that is, Hilbert
spaces $V$ with isometric left $G$-action that have an isometric $G$-embedding
$V\hookrightarrow H\otimes\ell^{2}G$ for some Hilbert space $H$.

######  Definition 4.1.

Given two Hilbert $\mathcal{N}G$-modules $U,V$ and a closed densely defined
$G$-equivariant operator $f\colon U\to V$, we define the spectral density
function $F(f)$ of $f$ using the family of $G$-equivariant spectral projectors
of the self-adjoint operator $f^{*}f$
$\left\\{E^{f^{*}f}_{\lambda}\right\\}_{\lambda\geq 0}$ by

$F(f)\colon\mathbb{R}_{\geq 0}\to[0,\infty],\qquad
F(f)(\lambda)=\operatorname{tr}_{\mathcal{N}G}\left(E^{f^{*}f}_{\lambda^{2}}\right).$

We call $f$ Fredholm if there is $\lambda>0$ with $F(f)(\lambda)<\infty$.

This function measures the content of the spectrum of the operator $f$ close
to zero in terms of the parameter $\lambda$. Notice that $F$ is monotonously
increasing and right-continuous. The value at zero, $F(f)(0)$, measures the
size of the kernel of $f$ and is called the $L^{2}$-Betti number.
Additionally, we define the Novikov-Shubin invariants to measure the spectral
content close to zero.

######  Definition 4.2.

Let $F$ be monotonously increasing and right-continuous. We define the
$L^{2}$-Betti number $b^{(2)}(F)=F(0)$ and the Novikov-Shubin invariant

$\alpha(F)=\liminf_{\lambda\searrow
0}\frac{\log(F(\lambda)-F(0))}{\log(\lambda)}.$

If $F(f)$ is the spectral density function of $f$, we also write
$b^{(2)}(f)=b^{(2)}(F(f))$ and $\alpha(f)=\alpha(F(f))$.

Here, $\alpha(F)$ measures the asymptotic behaviour of $F$ close to zero.
Indeed, if asymptotically $F(\lambda)-F(0)\sim\lambda^{\alpha}$ as
$\lambda\searrow 0$, then $\alpha=\alpha(F)$. In particular, we only care for
the asymptotic behaviour of $F$ close to zero. This motivates the following
equivalence relation defined on such functions.

######  Definition 4.3.

Let $F,F^{\prime}\colon\mathbb{R}_{\geq 0}\to[0,\infty]$ be two monotonously
increasing right-continuous functions. We call $F$ and $F^{\prime}$
dilatationally equivalent, and write $F\sim F^{\prime}$, if there are
constants $C>0$ and $\lambda_{0}>0$ such that

$F^{\prime}(C^{-1}\lambda)\leq F(\lambda)\leq F^{\prime}(C\lambda)\qquad\text{
for all }\lambda\in[0,\lambda_{0}].$

Novikov-Shubin invariants are invariant under dilatational equivalence. Given
a free $G$-CW complex of finite type, we use the differentials of the cellular
$L^{2}$-chain complex,
$d_{k}\colon\ell^{2}C^{\mathrm{cell}}_{k}(X;\mathcal{N}G)\to\ell^{2}C^{\mathrm{cell}}_{k-1}(X;\mathcal{N}G)$,
and the cellular Laplace operators,
$\Delta_{k}=d_{k}d_{k}^{*}+d_{k}^{*}d_{k}\curvearrowright\ell^{2}C^{\mathrm{cell}}_{k}(X;\mathcal{N}G)$,
to associate $L^{2}$-Betti numbers and Novikov-Shubin invariants to $X$.

######  Definition 4.4.

Let $X$ be a free $G$-CW complex of finite type. We define the $L^{2}$-Betti
numbers of $X$ by $b^{(2)}_{k}(X)=b^{(2)}(\Delta_{k})$ and the Novikov-Shubin
invariants by $\alpha_{k}(X)=\alpha(d_{k}).$

Here, we use $d_{k}$ instead of $\Delta_{k}$ in the definition of the Novikov-
Shubin invariants, as they provide a more detailed picture. Indeed,
$\alpha(\Delta_{k})=\nicefrac{{1}}{{2}}\cdot\min\\{\alpha_{k}(X),\alpha_{k+1}(X)\\}$.

###  4.1 Computing Novikov-Shubin Invariants

We now study the connection between the random walk
$\mathfrak{R}^{k}_{q}=\mathfrak{R}^{k}_{q}(X)$ and the Novikov-Shubin
invariant $\alpha_{k}(X)$ for a free $G$-CW complex $X$ of finite type. In
degree zero it is reasonable to consider only connected spaces (since the
$\ell^{2}$-spaces and the Laplace operator split as a direct sum with one
summand for each connected component). By the same reasoning, we can assume
without loss of generality the following analogue in degree $k$.

######  Definition 4.5.

Let $X$ be a CW complex. We call $X$ upper $k$-connected if $|I_{k}|\geq 2$
and for all $\alpha,\alpha^{\prime}\in I_{k}$ there are

$\alpha=\alpha_{0},\alpha_{1},\dots,\alpha_{n-1},\alpha_{n}=\alpha^{\prime}\in
I_{k}\quad\text{ and }\quad\beta_{1},\dots,\beta_{n}\in I_{k+1}$

such that $[\beta_{i},\alpha_{i-1}]\neq 0$ and $[\beta_{i}:\alpha_{i}]\neq 0$
for all $1\leq i\leq n$, that is $\beta_{i}$ is attached non-trivially to
$\alpha_{i-1}$ and $\alpha_{i}$.

This condition implies for $\mathfrak{R}^{k}_{q}$ that a random walker can
move from any $k$-cell $\alpha_{\pm}\in I^{\pm}_{k}$ to any other (unoriented)
$k$-cell $\alpha^{\prime}$ (that is, to one of the oriented $k$-cells
$\alpha^{\prime}_{+}$ or $\alpha^{\prime}_{-}$). Furthermore, we get bounds on
the quantities from Definition 2.1.

######  Lemma 4.6.

Let $X$ be an upper $k$-connected free $G$-CW complex of finite type. Then
there exists $D\geq 1$ such that

$D\geq d_{+,2},d_{+},d_{-}\geq 1.$

In particular, if $q\in[0,1)$ then the operators $M_{1,q}$ and $M_{2,q}$ are
positive multiplication operators bounded from below by

$M_{1,q}\geq D^{-2}>0\quad\text{and}\quad M_{2,q}=(1-q)D^{-2}>0.$

###### Proof.

Let $\alpha\in I_{k}$ be arbitrary and let $\alpha\neq\alpha^{\prime}\in
I_{k}$ be any other $k$-cell. Since $X$ is upper $k$-connected, by definition
there is a sequence of $k$-cells $\alpha_{i}\in I_{k}$ and $(k+1)$-cells
$\beta_{i}\in I_{k+1}$ connecting $\alpha$ to $\alpha^{\prime}$. In
particular, there exists a $(k+1)$-cell $\beta_{1}\in I_{k+1}$ such that
$[\beta_{1}:\alpha]\neq 0$ and a $k$-cell $\alpha\neq\alpha_{1}\in I_{k}$ such
that $[\beta_{1}:\alpha_{1}]\neq 0$. Hence

$\displaystyle d_{+,2}(\alpha)$ $\displaystyle\geq[\beta_{1}:\alpha]^{2}\geq
1,$ $\displaystyle d_{+}(\alpha)$ $\displaystyle\geq|[\beta_{1}:\alpha]|\geq
1,$ $\displaystyle d_{-}(\alpha)$ $\displaystyle\geq
d_{-}(\beta_{1};\alpha)\geq|[\beta_{1}:\alpha_{1}]|\geq 1.$

Since $X$ is of finite type and these quantities depend only on the $G$-type
of $\alpha$, there exists

$D=\sup_{\alpha\in
I_{k}}\\{d_{+,2}(\alpha),d_{+}(\alpha),d_{-}(\alpha)\\}=\max_{\alpha\in
I_{k}(G\setminus X)}\\{d_{+,2}(\alpha),d_{+}(\alpha),d_{-}(\alpha)\\}\geq 1.$

It follows, therefore, that

$\displaystyle M_{1,q}$
$\displaystyle=\frac{d_{+}d_{-}}{qd_{+}d_{-}+(1-q)d_{+,2}}\geq\frac{1}{qD^{2}+(1-q)D}\geq\frac{1}{D^{2}}>0,$
$\displaystyle M_{2,q}$
$\displaystyle=\frac{1-q}{qd_{+}d_{-}+(1-q)d_{+,2}}\geq\frac{1-q}{qD^{2}+(1-q)D}\geq\frac{1-q}{D^{2}}>0,$

and the claim follows. ∎

Generalising the notion of regular graphs, we introduce the following notion
of upper $k$-regular free $G$-CW complexes.

######  Definition 4.7.

Let $X$ be a free $G$-CW complex of finite type. We call $X$ upper $k$-regular
if $X$ is upper $k$-connected and $d_{+}d_{-}=d_{+}(\alpha)d_{-}(\alpha)$ and
$d_{+,2}=d_{+,2}(\alpha)$ are independent of the cell $\alpha\in I_{k}$.

In this case, also the multiplication operators $M_{1,q}$ and $M_{2,q}$ are
just multiplication with a constant. Hence, the formula connecting $B_{q}$ and
$\Delta^{\mathrm{up}}_{k}$ simplifies further.

######  Example 4.8.

This happens, for example, in the case where $X$ is a regular graph for $k=0$
or in the case studied in [BCDS23] for $k=\dim(X)-1$. While we can compute the
Novikov-Shubin invariants in the latter case by Poincaré duality, we can use
their values to obtain informations about the random walk considered.

######  Corollary 4.9.

Let $X$ be an upper $k$-regular $G$-CW complex of finite type and $q\in[0,1]$.
Then

$C_{1,q}B_{q}=\operatorname{Id}-C_{2,q}\Delta^{\mathrm{up}}_{k},$

where $C_{1,q}$ and $C_{2,q}$ are positive constants given as

$C_{1,q}=\frac{d_{+}d_{-}}{qd_{+}d_{-}+(1-q)d_{+,2}}>0\qquad\text{and}\qquad
C_{2,q}=\frac{1-q}{qd_{+}d_{-}+(1-q)d_{+,2}}>0.$

######  Definition 4.10.

Let $X$ be an upper $k$-regular free $G$-CW complex of finite type. We define

$\displaystyle\widetilde{B_{q}}$ $\displaystyle=C_{1,q}B_{q}\quad\text{ and
}\quad\widetilde{\Delta^{\mathrm{up}}_{q,k}}=C_{2,q}\Delta^{\mathrm{up}}_{k}$

so that we have the equality
$\widetilde{B_{q}}=\operatorname{Id}-\widetilde{\Delta^{\mathrm{up}}_{q,k}}.$

We now can derive bounds on the spectrum
$\sigma\left(\widetilde{\Delta^{\mathrm{up}}_{q,k}}\right)$ of the operator
$\widetilde{\Delta^{\mathrm{up}}_{q,k}}$. It is well-known that
$\Delta^{\textrm{up}}_{k}$ is bounded,

######  Lemma 4.11.

Let $X$ be a free $G$-CW complex of finite type, then
$\Delta^{\textrm{up}}_{k}\curvearrowright\ell^{2}C^{\mathrm{cell}}_{k}(X)$ is
bounded and $\sigma(\Delta^{\textrm{up}}_{k})\subset[0,S_{k}]$ for some
$S_{k}<\infty$.

While the precise bounds won’t matter for this paper, we can find explicit
bounds by straight-forward computations, for example,

$S_{k}=\max_{\alpha\in I_{k}(G\backslash X)}\left\\{\sum_{\beta\in
I_{k+1}}\sum_{\alpha^{\prime}\in
I_{k}}|d(\alpha,\alpha^{\prime},\beta)|\right\\}<\infty.$

Using this, we can prove the following lemma.

######  Lemma 4.12.

Let $X$ be an upper $k$-regular free $G$-CW complex of finite type. Then there
exists $q_{0}\in(0,1)$ such that for all $q_{0}\leq q\leq 1$ the spectrum of
$\widetilde{\Delta^{\mathrm{up}}_{q,k}}$ is contained in the unit interval,
$\sigma(\widetilde{\Delta^{\mathrm{up}}_{q,k}})\subset[0,1].$

###### Proof.

By Lemma 4.11, $\sigma(\Delta^{\mathrm{up}}_{k})\subset[0,S]$ for some $S>0$,
hence $\sigma(\widetilde{\Delta^{\mathrm{up}}_{q,k}})\subset[0,C_{2,q}S]$.
Note that $d_{+}d_{-}\geq 1$ and $d_{+,2}\geq 1$ so
$C_{2,q}=\frac{1-q}{qd_{+}d_{-}+(1-q)d_{+,2}}$ is continuous in $q\in(0,1)$
and converges to $0$ as $q\nearrow 1$. In particular, there is $q_{0}\in(0,1)$
such that $0<C_{2,q}\leq S^{-1}$ for all $q_{0}\leq q\leq 1$. ∎

######  Corollary 4.13.

Let $X$ be an upper $k$-regular free $G$-CW complex of finite type and
$q\in[q_{0},1)$. Let $\widetilde{d}_{k+1}=\sqrt{C_{2,q}}d_{k+1}$. Then
$\widetilde{d}_{k+1}^{*}=\sqrt{C_{2,q}}d_{k+1}^{*}$ and

$\widetilde{\Delta^{\mathrm{up}}_{q,k}}=\widetilde{d}_{k+1}\widetilde{d}_{k+1}^{*}$

is a self-adjoint positive operator with
$\sigma(\widetilde{\Delta^{\mathrm{up}}_{q,k}})\subset[0,1]$.

######  Remark 4.14.

Recall that for $q\in[q_{0},1)$, since $d$ and $\widetilde{d}$ differ only by
a constant factor $\sqrt{C_{2,q}}$, their spectral density functions are
dilatationally equivalent and hence their Novikov-Shubin invariants agree,
that is,
$\alpha_{k}(X)=\alpha(d_{k+1})=\alpha(\widetilde{d}_{k+1})=\alpha({\widetilde{d}_{k+1}}^{*})$.

######  Lemma 4.15.

Let $\chi_{I}$ denote the indicator function of the interval $I$, then

$\operatorname{tr}_{\mathcal{N}G}(\chi_{[1-\lambda,1]}(\widetilde{B_{q}}))=F(\widetilde{d}_{k+1}^{*})(\sqrt{\lambda}).$

###### Proof.

Recall that
$\widetilde{B_{q}}=\operatorname{Id}-\widetilde{d}_{k+1}\widetilde{d}_{k+1}^{*}$,
hence

$\displaystyle\operatorname{tr}_{\mathcal{N}G}\left(\chi_{[1-\lambda,1]}\left(\widetilde{B_{q}}\right)\right)$
$\displaystyle=\operatorname{tr}_{\mathcal{N}G}\left(\chi_{[0,\lambda]}\left(\widetilde{d}_{k+1}\widetilde{d}_{k+1}^{*}\right)\right)$
$\displaystyle=\operatorname{tr}_{\mathcal{N}G}\left(E_{\lambda}^{\widetilde{d}_{k+1}\widetilde{d}_{k+1}^{*}}\right)=F\left(\widetilde{d}_{k+1}^{*}\,\right)\left(\sqrt{\lambda}\right).\qed$

We can now proceed in the same way as in degree zero, compare Lück’s book
[Lüc02, §2.1.4].

######  Theorem 4.16.

Let $X$ be an upper $k$-regular free $G$-CW complex of finite type and
$q\in[q_{0},1)$, with $q_{0}$ given by Lemma 4.12. Then $\alpha_{k}(X)=2a$ if
and only if there is a constant $C>0$ such that for all $n\in\mathbb{N}$,

$C_{1,q}^{-n}\left(b^{(2)}(d_{k+1}^{*})+C^{-1}n^{-a}\right)\quad\leq\quad
p_{q}(n)\quad\leq\quad C_{1,q}^{-n}\left(b^{(2)}(d_{k+1}^{*})+Cn^{-a}\right).$

###### Proof.

Since by Lemma 4.12,
$\sigma(\widetilde{\Delta^{\mathrm{up}}_{q,k}})\subset[0,1]$ and by
construction
$\widetilde{\Delta^{\mathrm{up}}_{q,k}}=\operatorname{Id}-\widetilde{B_{q}}$,
it follows that also $\sigma(\widetilde{B_{q}})\subset[0,1]$. Therefore,

$(1-\lambda)^{n}\chi_{[1-\lambda,1]}(\widetilde{B_{q}})\quad\leq\quad\widetilde{B_{q}}^{n}\quad\leq\quad(1-\lambda)^{n}\chi_{[0,1-\lambda]}(\widetilde{B_{q}})+\chi_{[1-\lambda,1]}(\widetilde{B_{q}}).$

Taking traces using Lemma 4.15 and denoting
$\widetilde{p}_{q}(n)=\operatorname{tr}_{\mathcal{N}G}(\widetilde{B_{q}}^{n})$
yields

$(1-\lambda)^{n}F(\widetilde{d}_{k+1}^{*})(\sqrt{\lambda})\quad\leq\quad\widetilde{p}_{q}(n)\quad\leq\quad(1-\lambda)^{n}+F(\widetilde{d}_{k+1}^{*})(\sqrt{\lambda}).$

By rearranging these terms and taking logarithms, we obtain the inequalities

$\displaystyle\frac{\log\left(F(\widetilde{d}_{k+1}^{*})(\sqrt{\lambda})-b^{(2)}(d_{k+1}^{*})\right)}{\log\lambda}$
$\displaystyle\leq\frac{\log(\widetilde{p}_{q}(n)-(1-\lambda)^{n}b^{(2)}(d_{k+1}^{*}))}{\log\lambda}-n\cdot\frac{\log(1-\lambda)}{\log\lambda},$
(4)
$\displaystyle\frac{\log\left(F(\widetilde{d}_{k+1}^{*})(\sqrt{\lambda})-b^{(2)}(d_{k+1}^{*})\right)}{\log\lambda}$
$\displaystyle\geq\frac{\log(\widetilde{p}_{q}(n)-b^{(2)}(d_{k+1}^{*})-(1-\lambda)^{n})}{\log\lambda}.$
(5)

Using $b^{(2)}(d_{k+1}^{*})=b^{(2)}(\widetilde{d}_{k+1}^{*})$ and taking the
limit inferior for $\lambda\searrow 0$ on the left-hand-sides yields

$\displaystyle\liminf_{\lambda\searrow
0}\frac{\log\left(F(\widetilde{d})(\sqrt{\lambda})-b^{(2)}(d_{k+1}^{*})\right)}{\log\lambda}$
$\displaystyle=\frac{\alpha(\widetilde{d}_{k+1}^{*})}{2}=\frac{\alpha(\widetilde{d}_{k+1})}{2}=\frac{\alpha_{k}(X)}{2}.$

After substituting $p(n)=\widetilde{p}_{q}(n)-b^{(2)}(d_{k+1}^{*})$, the term
on the right-hand-side of Equation (5) agrees with the term in [Lüc02, Thm.
2.48], so by the same argument

$\alpha_{k}(X)\leq 2a\qquad\text{if }\widetilde{p}_{q}(n)\geq
b^{(2)}(d_{k+1}^{*})+Dn^{-a}\text{ for }n\geq 1,$

for some constant $D>0$.

For the right-hand-side of Equation (4), let $\varepsilon>0$ be arbitrarily
small and $n=n(\lambda)$ the largest integer such that
$n\leq\lambda^{-\varepsilon}$, that is
$n=\lfloor\lambda^{-\varepsilon}\rfloor$. If $\widetilde{p}(n)\geq
Cn^{-a}+b^{(2)}(d)$ for some constant $C>0$ and $n\geq 1$, we obtain

$\displaystyle\frac{\log(\widetilde{p}_{q}(n)-(1-\lambda)^{n}b^{(2)}(d_{k+1}^{*}))}{\log\lambda}-n\cdot\frac{\log(1-\lambda)}{\log\lambda}$
$\displaystyle\qquad\geq\frac{\log(Cn^{-a}+\left[1-(1-\lambda)^{n}\right]b^{(2)}(d_{k+1}^{*}))}{\log\lambda}-\frac{\log(1-\lambda)}{\lambda^{\varepsilon}\log\lambda}$
$\displaystyle\qquad\geq\frac{\log(Cn^{-a})}{\log\lambda}-\frac{\log(1-\lambda)}{\lambda^{\varepsilon}\log\lambda},$

where we use $\left[1-(1-\lambda)^{n}\right]b^{(2)}(d_{k+1}^{*})\geq 0$
(indeed, even
$\left[1-(1-\lambda)^{n}\right]b^{(2)}(d_{k+1}^{*})\xrightarrow{\lambda\searrow
0}1-e^{-\varepsilon}$). From here, we proceed precisely as in [Lüc02, Thm.
2.48] and find

$\alpha_{k}(X)\geq 2a\qquad\text{if }\widetilde{p}_{q}(n)\leq
b^{2}(d_{k+1}^{*})+Cn^{-a}\text{ for }n\geq 1,$

concluding the proof of the theorem. ∎

######  Remark 4.17.

This generalises the theorem in degree zero, since in degree zero we have
$d_{+}=d_{+,2}=|S|$, where $|S|$ is the size of a finite generating set of $G$
chosen in the construction of $\operatorname{Cayley}(G)$, and $d_{-}=1$. Thus
$C_{1,q}=1$ and the exponential decay factor $C_{1,q}^{-n}=1$ disappears.

######  Example 4.18.

Let $k\geq 2$ and let $G$ be a finitely generated group with Cayley graph
$\operatorname{Cayley}(G)$. Construct a $G$-CW complex $X$ in the following
way.

  * •

Start with $X^{(1)}=\operatorname{Cayley}(G)$.999If $k\geq 3$ and $G$ is
finitely presented, we can further glue in $2$-cells according to the
relations in $G$, so that $X^{(2)}$ is the Cayley complex of $G$. In that case
the constructed CW complex $X$ satisfies $\pi_{1}(X)=G$, see for example A.
Hatcher’s book [Hat02, p. 77].

  * •

For every $g\in G$ glue a $k$-cell $\alpha_{g}$ to $X^{(1)}$ by collapsing the
boundary of $\alpha_{g}$ to the vertex $v_{g}$ corresponding to $g\in G$ in
$\operatorname{Cayley}(G)$. This defines $X^{(k)}$.

  * •

For every edge $(g,gs)$ in the Cayley graph, glue one $(k+1)$-cell
$\beta_{g,gs}$ to $X^{(k)}$ by sending the boundary of $\beta_{g,gs}$ to
$\alpha_{g}\cup(g,gs)\cup\alpha_{gs}$ such that
$[\beta_{g,gs}:\alpha_{g}]=-[\beta_{g,gs}:\alpha_{gs}]\in\\{\pm 1\\}$. This
defines $X^{(k+1)}=X$.

On $X$, the degree $k$-upper random walk $\mathfrak{R}^{k}$ agrees with the
random walk $\mathfrak{R}$ on $\operatorname{Cayley}(G)$ when identifying the
state corresponding to $\alpha_{g}=(\alpha_{g})_{+}$ in $\mathfrak{R}^{k}$
with the state corresponding to $g$ in $\mathfrak{R}$. In particular, for
$\mathfrak{R}^{k}$ we have $p_{-}(n)\equiv 0$ so that $p(n)=p_{+}(n)$ is the
usual return probability. Further, the values $d_{+}=d_{2,+}=|S|$ and
$d_{-}=1$ agree with the values on $\operatorname{Cayley}(G)$, so that
$C_{1,q}=1$. Therefore, the previous theorem tells us that for $X$ we obtain
$\alpha_{k}(X)=\alpha_{0}(X)$.

Indeed, we can also see this in a different way because $d_{k+1}d_{k+1}^{*}$
and $d_{1}d_{1}^{*}$ are unitarely equivalent by identifying $\alpha_{g}$ with
$g$ and $\beta_{g,gs}$ with $(g,gs)$.

###  4.2 Example: Degree 1-Upper Random Walk on $\mathbb{R}^{2}$

Consider $\mathbb{R}^{2}$ as a $\mathbb{Z}^{2}$-CW complex of finite type as
shown on the right, with arrows indicating the chosen preferred orientation.
For notation’s sake, we will write $\mathbb{Z}^{2}$ as a multiplicative group
with unit element $1\in\mathbb{Z}^{2}$. Let $x$ and $y$ be two generators of
$\mathbb{Z}^{2}=\langle x,y\>|\>[x,y]=1\rangle$ and the
$\mathbb{Z}^{2}$-action on this CW complex be generated by $x$ shifting to the
right by one and $y$ shifting up by one. The red cells indicate
$\mathbb{Z}^{2}$-bases. We will denote the $0$-basis
$\mathcal{B}_{0}=\\{\gamma_{\bullet}\\}$, the $1$-basis
$\mathcal{B}_{1}=\\{\alpha_{\uparrow},\alpha_{\rightarrow}\\}$ and the
$2$-basis $\mathcal{B}_{2}=\\{\beta_{\circlearrowright}\\}$ in the way
suggested by the indices. Given a cell $c$ and
$g=x^{a}y^{b}\in\mathbb{Z}^{2}$, we denote by $gc$ the cell obtained by
translating $c$ by $g$, that is $a$ units to the right and $b$ units up.

$\circlearrowright$$\circlearrowright$$\circlearrowright$$\circlearrowright$$\circlearrowright$$\circlearrowright$$\circlearrowright$$\circlearrowright$$\circlearrowright$

The incidence numbers between a $2$-cell $\beta$ and a $1$-cell $\alpha$ are
given by $[\beta:\alpha]=0$ if $\beta$ and $\alpha$ do not touch and
$[\beta:\alpha]=\pm 1$ if the cells touch; with sign $+1$ if the orientation
$\beta$ induces on $\alpha$ agrees with the orientation on $\alpha$ and $-1$
otherwise. This is an upper $2$-regular CW complex with

$\displaystyle d_{+}=2,\ d_{+,2}=2,\ d_{-}=3,\quad
C_{1,q}=\frac{3}{2q+1},\quad C_{2,q}=\frac{1}{2}\frac{1-q}{2q+1},\quad
C_{1,q}^{-1}C_{2,q}=\frac{1-q}{6}.$

The upper Laplacian
$\Delta=\Delta^{\mathrm{up}}_{1}\curvearrowright\ell^{2}\left((\mathbb{R}^{2})^{(1)}\right)$
in degree one can be written, with respect to the basis $\mathcal{B}_{1}$, as
the $\mathbb{C}[\mathbb{Z}^{2}]$-valued matrix

$\Delta=2-\begin{pmatrix}x+x^{-1}&1-x-y^{-1}+xy^{-1}\\\
1-x^{-1}-y+x^{-1}y&y+y^{-1}\end{pmatrix}.$

For the non-lazy random walk on $1$-cells, on $\mathcal{B}_{1}$ the
propagation operator is given as described in Figure 1.

$6P$$6P$ Figure 1: Visual representation of propagation operator $P$

Accounting for changing orientations with signs, this means we can write the
corresponding operator $B=TPI$ with respect to $\mathcal{B}_{1}$ as the
$\mathbb{C}[\mathbb{Z}^{2}]$-valued matrix

$B=\frac{1}{6}\begin{pmatrix}x+x^{-1}&1-x-y^{-1}+xy^{-1}\\\
1-x^{-1}-y+x^{-1}y&y+y^{-1}\end{pmatrix}.$

We can readily verify that for $q\in[0,1]$ indeed

$\displaystyle B_{q}$
$\displaystyle=q\operatorname{Id}+(1-q)B=q\operatorname{Id}+\frac{1-q}{6}\left(2\operatorname{Id}-\Delta\right)=C_{1,q}^{-1}\operatorname{Id}-C_{1,q}^{-1}C_{2,q}\Delta,$

and thus $C_{1,q}B_{q}=\operatorname{Id}-C_{2,q}\Delta$. Looking at the
boundary of $\beta_{\circlearrowright}$ given by

$S=(1-x)\alpha_{\uparrow}+(y-1)\alpha_{\rightarrow},$

it is an eigenstate of $B$ with eigenvalue
$\frac{1}{6}\left(x+x^{-1}+y+y^{-1}-2\right)$, compare Figure 2.

$6P$$3\times$$3\times$$3\times$$3\times$ Figure 2: A visual representation of
$S$ and $BS$.

Here, $x+x^{-1}+y+y^{-1}=4P^{\mathbb{Z}^{2}}$, where $P^{\mathbb{Z}^{2}}$ can
formally also be interpreted as the propagation operator of the uniform
nearest neighbour random walk on the grid
$\operatorname{Cayley}(\mathbb{Z}^{2})$ (or, in this case, rather the
$2$-cells of $\mathbb{R}^{2}$ with this chosen CW structure). We denote
$\lambda=\frac{1}{6}(4P^{\mathbb{Z}^{2}}-2)$. A straight-forward computation
shows that $S$ is an eigenstate to $B_{q}$ with eigenvalue

$\lambda_{q}=C_{1,q}^{-1}C_{2,q}\left(4P^{\mathbb{Z}^{2}}+[C_{2,q}^{-1}-4]\right).$

Since $C_{2,q}(4+[C_{2,q}^{-1}-4])=1$, we can set
$q^{\prime}=1-4C_{2,q}=\frac{4q-1}{2q+1}$ and can formally interpret

$4C_{2,q}P^{\mathbb{Z}^{2}}+[1-4C_{2,q}]=P^{\mathbb{Z}^{2}}_{q^{\prime}}$

as the propagation operator of the corresponding $q^{\prime}$-lazy random walk
on $\operatorname{Cayley}(\mathbb{Z}^{2})$. Note that for
$q\in[\nicefrac{{1}}{{4}},1]$ we have $4C_{2,q}\in[0,1]$ and
$q^{\prime}\in[0,1]$ so this makes sense. In particular,

$\lambda_{q}=C_{1,q}^{-1}P^{\mathbb{Z}^{2}}_{q^{\prime}}\quad\text{ and }\quad
B_{q}S=C_{1,q}^{-1}P^{\mathbb{Z}^{2}}_{q^{\prime}}S.$

The return quantity $p_{q}(n)$ that we are interested in is given by
$p_{q}(n)=p_{q,\alpha_{\uparrow}}(n)+p_{q,\alpha_{\rightarrow}}(n)$, where
$p_{q,\alpha_{\uparrow}}(n)=\langle
B_{q}^{n}\alpha_{\uparrow},\alpha_{\uparrow}\rangle$ is the coefficient of
$1\alpha_{\uparrow}$ in $B_{q}^{n}\alpha_{\uparrow}$ and similarly for
$\alpha_{\rightarrow}$. By symmetry,
$p_{q,\alpha_{\uparrow}}(n)=p_{q,\alpha_{\rightarrow}}(n)$ so that

$p_{q}(n)=2p_{q,\alpha_{\uparrow}}(n)=2\langle
B_{q}^{n}\alpha_{\uparrow},\alpha_{\uparrow}\rangle.$

We note from Figure 1 that

$\displaystyle B_{q}(\alpha_{\uparrow})$
$\displaystyle=q\alpha_{\uparrow}+\frac{1-q}{6}\left(x^{-1}S-S+2\alpha_{\uparrow}\right)=C_{1,q}^{-1}\alpha_{\uparrow}+C_{1,q}^{-1}C_{2,q}(x^{-1}-1)S.$
(6)

Since the random walk is $\mathbb{Z}^{2}$-invariant, this yields

$\displaystyle B_{q}^{n}(\alpha_{\uparrow})$
$\displaystyle=C_{1,q}^{-1}B_{q}^{n-1}(\alpha_{\uparrow})+C_{1,q}^{-1}C_{2,q}(x^{-1}-1)B_{q}^{n-1}S$

Resolving this recursive formula we obtain

$\displaystyle B_{q}^{n}(\alpha_{\uparrow})$
$\displaystyle=C_{1,q}^{-n}\alpha_{\uparrow}+\sum_{k=0}^{n-1}C_{1,q}^{-n+k}C_{2,q}(x^{-1}-1)B_{q}^{k}S$
$\displaystyle=C_{1,q}^{-n}\left(\alpha_{\uparrow}+\sum_{k=0}^{n-1}C_{2,q}(x^{-1}-1)(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k}S\right)$

In order to find the coefficient of $1\alpha_{\uparrow}$, we notice that

$\displaystyle\langle S,1\alpha_{\uparrow}\rangle$
$\displaystyle=1,\qquad\langle x^{-1}S,1\alpha_{\uparrow}\rangle=-1\quad\text{
and }\quad\langle gS,1\alpha_{\uparrow}\rangle=0\quad\text{ for
}g\notin\\{1,x^{-1}\\}$

and therefore it follows that

$\displaystyle\langle(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k}S,1\alpha_{\uparrow}\rangle$
$\displaystyle=\langle(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k},1-x^{-1}\rangle,$
$\displaystyle\langle
x^{-1}(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k}S,1\alpha_{\uparrow}\rangle$
$\displaystyle=\langle(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k},x-1\rangle.$

Using this we obtain

$\displaystyle\frac{1}{2}p_{q}(n)$ $\displaystyle=\langle
B_{q}^{n}\alpha_{\uparrow},\alpha_{\uparrow}\rangle$
$\displaystyle=C_{1,q}^{-n}\left(1+\sum_{k=0}^{n-1}C_{2,q}\left\langle(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k},(x-1)-(1-x^{-1})\right\rangle\right)$
$\displaystyle=C_{1,q}^{-n}\left(1+\sum_{k=0}^{n-1}C_{2,q}\left\langle(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k},x+x^{-1}-2\right\rangle\right).$

By symmetry, the coefficients of $(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k}$ for
$x$ and $x^{-1}$ agree, hence

$\displaystyle\frac{1}{2}C_{1,q}^{n}p_{q}(n)$
$\displaystyle=1-2C_{2,q}\sum_{k=0}^{n-1}\left\langle(P^{\mathbb{Z}^{2}}_{q^{\prime}})^{k},1-x\right\rangle$
$\displaystyle=1-2C_{2,q}\sum_{k=0}^{n-1}\left(p^{\mathbb{Z}^{2}}_{q^{\prime}}(k)-p^{\mathbb{Z}^{2}}_{q^{\prime}}(e\xrightarrow{k}x)\right)$

where $p^{\mathbb{Z}^{2}}_{q^{\prime}}(k)$ is the return probability of the
$q^{\prime}$-lazy nearest neighbour random walk on $\mathbb{Z}^{2}$ after $k$
steps and $p^{\mathbb{Z}^{2}}_{q^{\prime}}(e\xrightarrow{k}x)$ the probability
of the random walk to be at the vertex $x$ after $k$ steps. If we write
$\mathbb{E}^{g}_{q}(n)$ for the expected number of visits of the vertex $g$ in
the first $n$ steps for the $q$-lazy nearest neighbour random walk on
$\mathbb{Z}^{2}$ (counting the starting position for $\mathbb{E}^{e}_{q}(n)$,
if $q=0$ we suppress it in notation), we can write this as

$\frac{1}{2}C_{1,q}^{n}p_{q}(n)=1-2C_{2,q}(\mathbb{E}^{e}_{q^{\prime}}(n-1)-\mathbb{E}^{x}_{q^{\prime}}(n-1)).$

Notice that $q^{\prime}=1-4C_{2,q}$ implies that
$2C_{2,q}=\frac{1-q^{\prime}}{2}$. Hence,

$\frac{1}{2}C_{1,q}^{n}p_{q}(n)=1-\frac{1-q^{\prime}}{2}(\mathbb{E}^{e}_{q^{\prime}}(n-1)-\mathbb{E}^{x}_{q^{\prime}}(n-1)).$

For $q<1$ large enough, we expect that

$\mathbb{E}^{e}_{q^{\prime}}(n-1)-\mathbb{E}^{x}_{q^{\prime}}(n-1)\sim
1-\Theta(n^{-1})\qquad\text{for }n\to\infty.$

Plugging this back into the equation above, this would imply that

$p_{q}(n)\sim C^{-n}_{1,q}\left(1+\Theta(n^{-1})\right)\qquad\text{for
}n\to\infty.$

Here, we can read off $b^{(2)}(d_{2}^{*})=1$, corresponding to the kernel of
$d_{2}^{*}$ of $\mathcal{N}G$-dimension one, and the Novikov-Shubin invariant
$\alpha_{1}(\mathbb{R}^{2})=\alpha(d_{2}^{*})=2$.

## References

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  * [Hat02] Allen Hatcher  “Algebraic topology”  Cambridge University Press, Cambridge, 2002, pp. xii+544
  * [Kam19] Holger Kammeyer  “Introduction to $\ell^{2}$-invariants” 2247, Lecture Notes in Mathematics  Springer, Cham, 2019, pp. viii+181  DOI: 10.1007/978-3-030-28297-4
  * [Lüc02] Wolfgang Lück  “$L^{2}$-invariants: theory and applications to geometry and $K$-theory” 44, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]  Springer-Verlag, Berlin, 2002, pp. xvi+595  DOI: 10.1007/978-3-662-04687-6
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# Constraining the growth rate on linear scales by combining SKAO and DESI
surveys

Simthembile Dlamini${}^{1}{}^{,}{}^{a}$, Sheean Jolicoeur${}^{1}{}^{,}{}^{b}$,
Roy Maartens${}^{1,2,3}{}^{,}{}^{c}$

###### Abstract

In the pursuit of understanding the large-scale structure of the Universe, the
synergy between complementary cosmological surveys has proven to be a powerful
tool. Using multiple tracers of the large-scale structure can significantly
improve the constraints on cosmological parameters. We explore the potential
of combining the Square Kilometre Array Observatory (SKAO) and the Dark Energy
Spectroscopic Instrument (DESI) spectroscopic surveys to enhance precision on
the growth rate of cosmic structures. We employ a multi-tracer Fisher analysis
to estimate precision on the growth rate when using pairs of mock surveys that
are based on SKAO and DESI specifications. The pairs are at both low and high
redshifts. For SKA-MID, we use the HI galaxy and the HI intensity mapping
samples. In order to avoid the complexities and uncertainties at small scales,
we confine the analysis to scales where linear perturbations are reliable. The
consequent loss of signal in each individual survey is mitigated by the gains
from the multi-tracer. After marginalising over cosmological and nuisance
parameters, we find a significant improvement in the precision on the growth
rate.

##  1 Introduction

Einstein’s theory of General Relativity (GR) and modified gravity theories
(see e.g. [1, 2, 3, 4]) prescribe the relation between peculiar velocities and
the growth of large-scale structure. Peculiar velocities generate redshift-
space distortions (RSD) in the power spectrum, which consequently provide a
powerful probe for testing theories of gravity via the linear growth rate
$f=-\mathrm{d}\ln\ln D/\mathrm{d}\ln(1+z)$, where
$D(z)=\delta(z,\bm{k})/\delta(0,\bm{k})$ and $\delta$ is the matter density
contrast. Here we assume that the growth rate is scale-independent on linear
scales. We confine our analysis to scales where linear perturbation theory is
accurate, using a conservative $k_{\rm max}$. Although this leads to a
significant loss in signal, it has the advantage that we can avoid the
theoretical complexities and uncertainties involved in the modelling of small-
scale RSD.

Precision measurements of RSD require the redshift accuracy of spectroscopic
surveys. Currently, one of the best constraints on the growth index is from
the extended Baryon Oscillation Spectroscopic Survey (eBOSS) survey, Data
Release 14 Quasar [5]:

$\displaystyle\gamma\equiv\frac{\ln f(z)}{\ln\Omega_{m}(z)}=0.580\pm 0.082\,.$
(1.1)

This is consistent with the standard value $\gamma=0.55$, which is predicted
by GR in the $\Lambda$CDM model. This value of $\gamma$ is also a good
approximation for simple models of evolving dark energy, whose clustering is
negligible [6]. Statistically significant deviations from $\gamma=0.55$ could
indicate either non-standard dark energy in GR or a breakdown in GR itself.
The next generation of multi-wavelength spectroscopic surveys (e.g. [7, 8, 9,
10, 11, 12, 13, 14]) promises to deliver high-precision measurements of RSD,
using complementary types of dark matter tracers.

The effect of linear RSD on the power spectrum is degenerate with the
amplitude of the matter power spectrum and with the linear clustering bias.
This degeneracy can be broken by using information in the multipoles of the
Fourier power spectrum (see e.g. [15, 16, 17, 5, 18]), or by using the angular
power spectrum and including cross-bin correlations [19]. By combining
information from different tracers, the multi-tracer technique [20] can
significantly improve constraints on the growth rate [21, 22, 23, 24, 25, 26].

We use Fourier power spectra in the flat-sky approximation. We perform a
simple Fisher forecast on pairs of next-generation spectroscopic surveys at
low and at higher redshifts. The low-$z$ samples are similar to the Dark
Energy Spectroscopic Instrument (DESI) Bright Galaxy Sample (BGS) [11, 27, 28]
and the Square Kilometer Array Observatory (SKAO) HI galaxies sample or the HI
intensity mapping (IM) Band 2 sample [13, 29]. For the higher-$z$ samples, we
use samples similar to the DESI Emission Line Galaxies (ELG) and SKAO Band 1
IM samples.

##  2 Multi-tracer power spectra

In redshift space, the positions of observed sources are made up of two parts.
The first is due to the background expansion of the universe, and the second
is due to the peculiar velocities of the sources. Peculiar velocities are the
result of the gravitational effect of local large-scale structure, and they
induce shifts in the redshift-space positions of the sources. On large scales,
linear RSD produce an increase in clustering. For a given tracer $A$ of the
dark matter distribution, the observed density contrast at linear order is

$\Delta_{A}(z,\bm{\hat{n}})=\underbrace{b_{A}(z)\,\delta(z,\bm{\hat{n}})}_{\mbox{density}}-\underbrace{\frac{(1+z)}{H(z)}\,\hat{\bm{n}}\cdot\bm{\nabla}\big{[}\hat{\bm{n}}\cdot\bm{v}(z,\bm{\hat{n}})\big{]}}_{\mbox{linear
RSD}}\;,$ (2.1)

where $b_{A}$ is the linear bias, $\bm{v}$ is the peculiar velocity, and
$\bm{\hat{n}}$ is the unit vector in the line of sight direction of the
observer. In the flat-sky approximation (fixed $\bm{\hat{n}}$), the Fourier
transform of (2.1) gives

$\Delta_{A}(z,\bm{k})=\Big{[}b_{A}(z)+f(z)\mu^{2}\Big{]}\delta(z,\bm{k})\quad\mbox{where}\quad\mu=\bm{\hat{k}}\cdot\bm{\hat{n}}\;.$
(2.2)

Here we used the first-order continuity equation

$\bm{\nabla}\cdot\bm{v}=-\frac{H}{(1+z)}\,f\,\delta\;.$ (2.3)

The tree-level Fourier power spectra are then defined by

$\big{\langle}\Delta_{A}(z,\bm{k})\,\Delta_{B}(z,\bm{k}^{\prime})\big{\rangle}=(2\pi)^{3}\,P_{AB}(z,\bm{k})\,\delta^{\mathrm{D}}(\bm{k}+\bm{k}^{\prime})\;.$
(2.4)

By (2.2),

$P_{AB}(z,\bm{k})=P_{AB}(z,k,\mu)=\Big{[}b_{A}(z)+f(z)\mu^{2}\Big{]}\Big{[}b_{B}(z)+f(z)\mu^{2}\Big{]}P(z,k)\;,$
(2.5)

where $P$ is the linear matter power spectrum (computed from CLASS [30]). We
can split it into a shape function $\mathcal{P}$ and an amplitude parameter
$\sigma_{8,0}$ as:

$P(z,\bm{k})=\sigma_{8,0}^{2}\,\mathcal{P}(z,\bm{k})\;.$ (2.6)

Note that in general there is a scale-dependent cross-correlation coefficient,
$0<r\leq 1$, that multiplies the $P_{AB}$ in (2.5) [31, 32]. On the large,
linear scales that we consider, it is expected that $r$ can be taken to be 1
(e.g. [33]).

###  2.1 Sample specifications

We consider mock samples similar to the following spectroscopic samples:

  * •

galaxies: DESI BGS and ELG [11, 27] and SKAO Band 2 HI galaxies [13].

  * •

intensity mapping: SKAO HI IM Band 1,2 [13].

Table 1, based on [11, 13], shows the sky and redshift coverage of the
individual and overlapping samples, together with the survey time for the HI
samples. For the overlap sky areas, we assume nominal values.

Table 1: Sky area and redshift range of samples, and survey time for HI samples. Survey | Sample |  $\Omega_{\mathrm{sky}}$ |  $t_{\mathrm{tot}}$ |  redshift
---|---|---|---|---
|  | $\big{[}10^{3}\,\mathrm{deg}^{2}\big{]}$ | $\big{[}10^{3}\,\mathrm{hr}\big{]}$ |  range
$g$ (DESI-like) | ELG | 14 | - | 0.60–1.70
| BGS | 14 | - | 0.00–0.50
$g$ (SKAO-like) | HI gal. | 5 | 10 | 0.00–0.50
$H$ (SKAO-like) | IM Band 1 | 20 | 10 | 0.35–3.05
| IM Band 2 | 20 | 10 | 0.10–0.58
$g_{1}\times g_{2}$ | BGS $\times$ HI gal. | 5 | 5 | 0.00–0.50
$g\times H$ | ELG $\times$ IM Band 1 | 10 | 5 | 0.60-1.70
$g\times H$ | BGS $\times$ IM Band 2 | 10 | 5 | 0.10–0.50

For the linear clustering biases $b_{A}$, we use one-parameter models where
the redshift evolution is assumed known, as suggested by [34]. For the DESI-
like samples we use [35]:

$b_{g}(z)=\frac{b_{g0}}{D(z)}\quad\text{with fiducial
value}~{}b_{g0}=1.34~{}\mbox{(BGS)}\quad\mbox{and}\quad
0.84~{}\mbox{(ELG)}\,.$ (2.7)

For the SKAO-like HI galaxy sample, we use [13]:

$b_{g}(z)={b_{g0}}\big{(}1+0.880\,z-0.739\,z^{2}\big{)}\quad\text{with
fiducial value}~{}b_{g0}=0.625\,.$ (2.8)

For HI IM, we use a fit based on simulations [36]:

$b_{H}(z)=b_{H0}\big{(}1+0.693\,z-0.046\,z^{2}\big{)}\quad\text{with fiducial
value}~{}b_{H0}=0.842\,.$ (2.9)

The background brightness temperature of HI IM is modelled via the fit given
in [12]:

$\bar{T}_{H}(z)=0.0559+0.2324\,z-0.0241\,z^{2}~{}~{}\mathrm{mK}\,.$ (2.10)

###  2.2 Noise

For galaxy surveys, the noise that affects the auto-power spectrum measurement
is the shot noise (assumed to be Poissonian):

$P^{\rm shot}_{gg}(z)=\frac{1}{\bar{n}_{g}(z)}\;,$ (2.11)

where $\bar{n}_{g}$ is the comoving background number density. The total
signal for the galaxy auto-power spectra is

$\tilde{P}_{gg}(z,k,\mu)=P_{gg}(z,k,\mu)+P^{\rm shot}_{gg}(z)\;.$ (2.12)

Figure 1 shows the fiducial clustering biases and number density and
brightness temperature for all the samples.

Figure 1: _Left:_ Fiducial clustering bias for galaxy (red) and intensity
mapping (blue) samples. _Right:_ Comoving number density for galaxy surveys
(red, left $y$-axis) and brightness temperature for IM surveys (blue, right
$y$-axis)

There is shot noise in HI IM surveys – but on the linear scales that we
consider, this shot noise is much smaller than the thermal noise (see below)
and can be safely neglected [37, 36].

For the cross-power spectra, the cross shot-noise may be neglected if the
overlap of halos hosting the two samples is negligible. This is shown to be
the case for BGS $\times$ IM in [26] (see also [38]). We assume that it is a
reasonable approximation in the cases ELG $\times$ IM and BGS $\times$ HI
galaxies, so that $P_{gH}^{\rm shot}\approx 0\approx P_{gg^{\prime}}^{\rm
shot}$. (Note that we do not consider the multi-tracer case HI galaxies
$\times$ HI IM.)

The thermal noise in HI IM depends on the sky temperature in the radio band,
the survey specifications and the array configuration (single-dish or
interferometer). For the single-dish mode of SKAO-like IM surveys, the thermal
noise power spectrum is [39, 40, 41]:

$P_{H\\!H}^{\mathrm{therm}}(z)=\frac{\Omega_{\mathrm{sky}}}{2\nu_{21}t_{\mathrm{tot}}}\,\frac{(1+z)^{2}r(z)^{2}}{H(z)}\,\left[\frac{T_{\mathrm{sys}}(z)}{\bar{T}_{H}(z)}\right]^{2}\,\frac{1}{N_{\rm
d}}\,,$ (2.13)

where $\nu_{21}=1420\,\mathrm{MHz}$ is the rest-frame frequency of the 21 cm
emission, $t_{\mathrm{tot}}$ is the total observing time, and the number of
dishes is $N_{\mathrm{d}}=197$ (with dish diameter $D_{\mathrm{d}}=15\,$m).
The system temperature is modelled as [42]:

$T_{\mathrm{sys}}(z)=T_{\rm d}(z)+T_{\rm sky}(z)=T_{\rm
d}(z)+2.7+25\bigg{[}\frac{400\,\mathrm{MHz}}{\nu_{21}}(1+z)\bigg{]}^{2.75}~{}\mathrm{K},$
(2.14)

where $T_{\rm d}$ is the dish receiver temperature given in [26]. The total
signal is then

$\tilde{P}_{H\\!H}(z,k,\mu)=P_{H\\!H}(z,k,\mu)+P_{H\\!H}^{\mathrm{therm}}(z)\,.$
(2.15)

###  2.3 Intensity mapping beam and foregrounds

HI IM surveys in single-dish mode have poor angular resolution, which results
in power loss on small transverse scales, i.e. for large
$k_{\perp}=(1-\mu^{2})^{1/2}k$. This effect is typically modelled by a
Gaussian beam factor [39]:

${\cal D}_{\rm
beam}(z,k,\mu)=\exp\left[-\frac{(1-\mu^{2})k^{2}r(z)^{2}\theta_{\rm
b}(z)^{2}}{16\ln 2}\right]\quad\mbox{with}\quad\theta_{\rm
b}(z)=1.22\,\frac{\lambda_{21}(1+z)}{D_{\rm d}}\,.$ (2.16)

HI IM surveys are also contaminated by foregrounds much larger than the HI
signal. Since these foregrounds are spectrally smooth, they can be separated
from the non-smooth signal on small to medium scales. However, on very large
radial scales, i.e. for small $k_{\|}=\mu k$, the signal becomes smoother and
therefore, the separation fails. A comprehensive treatment requires
simulations of foreground cleaning of the HI signal (e.g. [43, 44]). For a
simplified Fisher forecast, we can instead use a foreground avoidance approach
by excising the regions of Fourier space where the foregrounds are
significant. This means removing large radial scales, which can be modelled by
the foreground-avoidance factor:

$\mathcal{D}_{\mathrm{fg}}(k,\mu)=\Theta\left(\left|k_{\|}\right|-k_{\mathrm{fg}}\right)=\begin{cases}1,&\left|k_{\|}\right|>k_{\mathrm{fg}}\\\
0,&\left|k_{\|}\right|\leq k_{\mathrm{fg}}\end{cases}$ (2.17)

where $\Theta$ is the Heaviside step function. We assume the cut is made at a
minimum value of

$k_{\mathrm{fg}}=0.01\,h\,\mathrm{Mpc}^{-1}\,.$ (2.18)

In summary, the HI IM density contrast is modified by beam and foreground
effects as follows:

$\displaystyle\Delta_{H}(z,k,\mu)~{}~{}\to~{}~{}\mathcal{D}_{\rm
beam}(z,k,\mu)\,\mathcal{D}_{\mathrm{fg}}(k,\mu)\,\Delta_{H}(z,k,\mu)\,.$
(2.19)

##  3 Multi-tracer Fisher Analysis

The Fisher matrix in each redshift bin for the combination of two dark matter
tracers is [45, 46]

$F_{\alpha\beta}^{\bm{P}}=\sum_{\mu=-1}^{+1}\,\sum_{k=k_{\mathrm{min}}}^{k_{\mathrm{max}}}\,\partial_{\alpha}\,\bm{P}\cdot\mathrm{Cov}(\bm{P},\bm{P})^{-1}\cdot\partial_{\beta}\,\bm{P}^{\mathrm{T}}\;,$
(3.1)

where $\partial_{\alpha}=\partial\,/\,\partial\vartheta_{\alpha}$, with
$\vartheta_{\alpha}$ the parameters, and $\bm{P}$ is the data vector of the
power spectra:

$\bm{P}=\big{(}P_{gg}\,,\,{P}_{gH}\,,\,{P}_{H\\!H}\big{)}\quad\mbox{or}~{}~{}\big{(}P_{gg}\,,\,{P}_{gg^{\prime}}\,,\,{P}_{g^{\prime}g^{\prime}}\big{)}\;.$
(3.2)

Note that the sum over $\mu$ incorporates the foreground avoidance via the
Heaviside factor (2.17) in $P_{HA}$. Also note that $\bm{P}$ contains no noise
terms – these appear in the covariance below. The reason is that noise does
not depend on the cosmological parameters. Although the thermal noise in HI IM
depends on $H$, this arises from mapping the Gaussian pixel noise term to
Fourier space.

The multi-tracer covariance includes the shot and thermal noises, and is given
by [22, 24, 45, 46]:

$\mathrm{Cov}(\bm{P},\bm{P})=\frac{k_{\rm f}^{3}}{2\pi k^{2}\Delta
k}\,\frac{2}{\Delta\mu}\,\begin{pmatrix}\tilde{P}_{gg}^{2}&&\tilde{P}_{gg}\tilde{P}_{gH}&&\tilde{P}_{gH}^{2}\\\
&&&&\\\
\tilde{P}_{gg}\tilde{P}_{gH}&&\frac{1}{2}\big{[}\tilde{P}_{gg}\tilde{P}_{H\\!H}+\tilde{P}_{gH}^{2}\big{]}&&\tilde{P}_{H\\!H}\tilde{P}_{gH}\\\
&&&&\\\
\tilde{P}_{gH}^{2}&&\tilde{P}_{H\\!H}\tilde{P}_{gH}&&\tilde{P}_{H\\!H}^{2}\end{pmatrix}\;,$
(3.3)

and similarly for the case $g\times g^{\prime}$. Here $\Delta k$ and
$\Delta\mu$ are bin-widths and the fundamental mode $k_{\mathrm{f}}$,
corresponding to the longest wavelength, is determined by the comoving survey
volume of the redshift bin centred at $z$:

$V(z)=\frac{\Omega_{\mathrm{sky}}}{3}\bigg{[}r\Big{(}z+\frac{\Delta
z}{2}\Big{)}^{3}-r\Big{(}z-\frac{\Delta
z}{2}\Big{)}^{3}\bigg{]}=\bigg{[}\frac{2\pi}{k_{\mathrm{f}}(z)}\bigg{]}^{3}\;.$
(3.4)

We choose the bin-widths following [47, 48, 49]:

$\Delta z=0.1,~{}~{}\Delta\mu=0.04,~{}~{}\Delta k=k_{\mathrm{f}}\;.$ (3.5)

In order to exclude the small length scales that are beyond the validity of
linear perturbation theory, we impose a conservative maximum wavenumber of
$.08h/$Mpc at $z=0$, with a redshift evolution as proposed in [50]:

$k_{\mathrm{max}}(z)=0.08\,(1+z)^{2/(2+n_{s})}\;h\,\mathrm{Mpc}^{-1}\;.$ (3.6)

The largest length scale that can be measured in galaxy surveys corresponds to
the smallest wavenumber, given by

$k_{\mathrm{min}}=k_{\mathrm{f}}\,.$ (3.7)

For HI IM surveys,
$k_{\mathrm{min}}=\text{max}\big{(}k_{\mathrm{f}},k_{\mathrm{fg}}\big{)}$.

The multi-tracer Fisher matrix applies for a perfectly overlapping region in
both redshift range and sky area for the two tracers. If the samples differ in
redshift and sky area, then we can add the independent non-overlapping Fisher
matrix information of the individual surveys. The full Fisher matrix, denoted
by $g\otimes H$, is [26, 51]

$F_{\alpha\beta}^{g\otimes
H}=F_{\alpha\beta}^{\bm{P}}(\text{overlap})+F_{\alpha\beta}^{g}(\text{non-
overlap})+F_{\alpha\beta}^{H}(\text{non-overlap})\;,$ (3.8)

and similarly for the case $g\otimes g^{\prime}$.

For the cosmological parameters, we choose $\sigma_{8,0}$, $\Omega_{b,0}$,
$\Omega_{c,0}$, $n_{s}$ and $h$, since we are focusing on constraining the
growth rate index $\gamma$, which should be minimally affected by the
remaining $\Lambda$CDM parameters on linear scales. We fix these remaining
cosmological parameters to their Planck 2018 best-fit values [52]. We
therefore consider the following set of cosmological parameters together with
the two nuisance bias parameters:

$\vartheta_{\alpha}=\big{(}\gamma,\,\sigma_{8,0},\,n_{s},\,h,\,\Omega_{b0},\,\Omega_{c0}\,;\,b_{A0}\big{)}\quad\mbox{where}~{}~{}A=\mbox{BGS,
ELG, HIg, IM1, IM2}\;.$ (3.9)

The marginalised errors are then obtained as

$\sigma\big{(}\vartheta_{\alpha}\big{)}=\Big{[}\big{(}F^{-1}\big{)}_{\alpha\alpha}\Big{]}^{1/2}\;.$
(3.10)

We compute numerically the Fisher derivatives with respect to $n_{s}$, $h$,
$\Omega_{b0}$ and $\Omega_{c0}$, using the 5-point stencil approximation with
selected step sizes, shown in Table 2. The derivatives are stable for
$0.0003\leq\mbox{step size}\leq 0.1$ [27]. The derivatives with respect to
$\gamma$, $\sigma_{8,0}$ and $b_{A0}$ are computed analytically, with for
example

$\displaystyle\frac{1}{P_{AA}}\,\partial_{\gamma}P_{AA}=\frac{2\mu^{2}\,\Omega_{m}^{\gamma}\,\ln\Omega_{m}}{b_{A}+\mu^{2}\,\Omega_{m}^{\gamma}}\,.$
(3.11)

Table 2:  Numerical step sizes for Fisher derivatives. Parameters | $n_{s}$ | $h$ | $\Omega_{b0}$ | $\Omega_{c0}$
---|---|---|---|---
Optimal step size | 0.1 | 0.01 | 0.01 | 0.02

##  4 Results

Figure 2–Figure 4 show the 1$\sigma$ error contours for the parameter $\gamma$
and the cosmological parameters, after marginalising over the 2 bias nuisance
parameters in (3.9). There are significant degeneracies, which the multi-
tracer partly alleviates, allowing for improved precision on the cosmological
parameters. The improvement is small, unlike the case of the growth index,
which shows significant improvement. This is not surprising, since the multi-
tracer removes cosmic variance from the effective bias, i.e. the clustering
bias plus the RSD contribution, as shown in [21]. All multi-tracer pairs show
a significant reduction in errors on $\gamma$ compared to the best single
tracer. The improvements are shown in the fractional errors listed in Table 3.
We note that these multi-tracer errors are obtained using only linear scales.

For the BGS and HI galaxy combination in Figure 2, the multi-tracer fractional
error on $\gamma$ is less than half of the BGS value. We note that our
constraint on $\gamma$ for the single-tracer BGS is weaker than that in [26].
The reason is that [26] uses the angular power spectra with a large number of
very thin redshift bins (width 0.01) and considers all possible cross-bin
correlations. By contrast, our standard Fourier analysis only uses 5 redshift
bins of width 0.1, and does not include cross-correlations between different
redshift bins. The single-tracer HI galaxy delivers the weakest constraints,
mainly due to its smaller sky area and number density.

Table 3: Fractional errors $\sigma(\vartheta_{\alpha})/\vartheta_{\alpha}$. Sample |  $\gamma$ |  $\sigma_{8,0}$ |  $n_{s}$ |  $h$ |  $\Omega_{b0}$ | $\Omega_{c0}$
---|---|---|---|---|---|---
BGS | 0.467 | 0.169 | 0.319 | 0.465 | 0.523 | 0.105
ELG | 0.119 | 0.018 | 0.066 | 0.102 | 0.107 | 0.020
HI gal. | 0.664 | 0.25 | 0.921 | 1.445 | 1.586 | 0.239
IM Band 2. | 0.217 | 0.067 | 0.225 | 0.353 | 0.391 | 0.059
IM Band 1. | 0.061 | 0.013 | 0.050 | 0.077 | 0.084 | 0.018
BGS $\otimes$ HI gal. | 0.228 | 0.096 | 0.295 | 0.465 | 0.520 | 0.089
BGS $\otimes$ IM Band 2 | 0.139 | 0.048 | 0.168 | 0.267 | 0.278 | 0.052
ELG $\otimes$ IM Band 1 | 0.047 | 0.008 | 0.037 | 0.057 | 0.062 | 0.013

When BGS is combined with HI IM Band 2 (Figure 3) the situation changes. HI IM
Band 2 gives much better constraints than BGS, with an error on $\gamma$ less
than half of the BGS error. This arises despite the effects of foreground
noise because HI IM Band 2 covers a much larger area of the sky, which results
in more Fourier modes that contribute to the Fisher analysis. In addition,
foreground noise affects the largest scales where the $\gamma$ is not strong.
When IM surveys are combined with spectroscopic galaxy surveys, the impact of
foreground noise on the multi-tracer constraints is further mitigated. Table 3
shows that the multi-tracer error on $\gamma$ is reduced by $\sim 40\%$
compared to the best single-tracer error from HI IM Band 2.

The best $\gamma$ precision is delivered at high redshifts by ELG $\otimes$ IM
Band 1 (Figure 4). The IM Band 1 error on $\gamma$ is $\sim 6\%$ while ELG
produces about double this error. The multi-tracer reduces the error to $\sim
5\%$.

Table 4: Fractional errors on bias parameters. Survey | Sample |  ${b_{A0}}$ |  ${b_{B0}}$
---|---|---|---
$g$ (DESI-like) | BGS | 0.1719 | -
| ELG | 0.0169 | -
$g$ (SKAO-like) | HI gal. | 0.2691 | -
$H$ (SKAO-like) | IM Band 2 | 0.0696 | -
| IM Band 1 | 0.0138 | -
$g\otimes g^{\prime}$ | BGS $\otimes$ HI gal. | 0.0997 | 0.0989
$g\otimes H$ | BGS $\otimes$ IM Band 2 | 0.0501 | 0.0496
| ELG $\otimes$ IM Band 1 | 0.0083 | 0.0085
Figure 2: Marginal 1$\sigma$ contours for $\gamma$ and cosmological
parameters, from the BGS and HI galaxy combination.

Figure 3: As in Figure 2, from the BGS and HI IM Band 2 combination.

Figure 4: As in Figure 2, from the ELG and HI IM Band 1 combination.

Table 4 displays the fractional errors for the 2 bias nuisance parameters for
the 3 survey combinations. The multi-tracer constraints on bias nuisance
parameters are much tighter than those obtained from the individual single-
tracers (compare [19, 26]).

All of our constraints are obtained from scales $k<k_{\rm max}$ where linear
perturbations are accurate. In Figure 5 we investigate the effect on the
marginalised fractional error for $\gamma$ of changing $k_{\rm max,0}$ from
its value given in (3.6). The plots confirm that constraints are sensitive to
$k_{\mathrm{max,0}}$. We would unnecessarily lose information by reducing our
$k_{\rm max,0}$ value. On the other hand, increasing it leads to higher
precision – but at the risk of moving into the regime of nonlinear effects –
especially in RSD – which requires much more effort to model. The multi-tracer
has the advantage of allowing us to avoid these difficulties while at the same
time delivering constraints that would not be possible with single tracers in
the linear regime.

Figure 5: Change in fractional errors on $\gamma$ as a function of $k_{\rm
max,0}$, for the BGS and HI galaxy (top left), BGS and HI IM Band 2 (top
right) and ELG and HI IM Band 1 (bottom) samples. Our choice is $k_{\rm
max,0}=0.08\,h$/Mpc.

##  5 Conclusion

Using a simplified Fisher analysis we have estimated the multi-tracer
constraints on the growth rate of large-scale structure, for pairs of tracer
samples that are similar to those expected from the specifications of SKAO and
DESI surveys. The multi-tracer is known to be more effective, the more
different are the pairs of surveys – and this motivates our choice of DESI-
like and SKAO-like samples.

We applied a foreground-avoidance filter to the HI intensity mapping samples
and included the effects of the radio telescope beam, but we have not dealt
with the many other systematics. Our aim is not a realistic forecast for
specific surveys, but rather a proof of principle analysis to answer the
question: what is the potential of the multi-tracer to improve constraints
using only linear scales?

By confining the signal to linear scales, we avoid the highly complex
modelling, especially for RSD, that is required to access nonlinear scales.
The cross-power spectra represent an additional complexity in the nonlinear
regime [24]. Our Fisher analysis suggests that significant improvements in
precision on the growth rate could be achieved by multi-tracing next-
generation radio-optical pairs of samples. The details are summarised
qualitatively in Figure 2, Figure 3 and Figure 4, and quantitatively in Table
3. The biggest improvements, $\sim 40-60\%$, are for the low-redshift pairs.

This indicates that it is worthwhile to perform a more realistic analysis and
derive more realistic forecasts. We leave this for further work. Finally, we
note that multi-tracer improvements can be delivered without requiring
additional observational resources.

Acknowledgements We are supported by the South African Radio Astronomy
Observatory (SARAO) and the National Research Foundation (Grant No. 75415).

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