# eBIM-GNN : Fast and Scalable energy analysis through BIMs and Graph Neural
Networks
††thanks: Funding Agency: National Geospatial Programme, Department of Science
and Technology, Government of India

Rucha Bhalchandra Joshi
<EMAIL_ADDRESS>National Institute of Science Education and Research,
Bhubaneswar, Odisha 752050
Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094
Annada Prasad Behera
<EMAIL_ADDRESS>National Institute of Science Education and
Research, Bhubaneswar, Odisha 752050
Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094
Subhankar Mishra
<EMAIL_ADDRESS>National Institute of Science Education and Research,
Bhubaneswar, Odisha 752050
Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094

###### Abstract

Building Information Modeling has been used to analyze as well as increase the
energy efficiency of the buildings. It has shown significant promise in
existing buildings by deconstruction and retrofitting. Current cities which
were built without the knowledge of energy savings are now demanding better
ways to become smart in energy utilization. However, the existing methods of
generating BIMs work on building basis. Hence they are slow and expensive when
we scale to a larger community or even entire towns or cities. In this paper,
we propose a method to creation of prototype buildings that enable us to match
and generate statistics very efficiently. Our method suggests better energy
efficient prototypes for the existing buildings. The existing buildings are
identified and located in the 3D point cloud. We perform experiments on
synthetic dataset to demonstrate the working of our approach.

##  1 Introduction

The Building Information Model (BIM)[2] is an instance of a populated data
model of buildings. It constitutes of all the different kinds of data
pertaining to a specific building. These data invariably or uniquely define
the same building. It is not a temporal representation; however a static
projection of the building that defines the building from various different
perspectives across disciplines. BIM has replaced traditional onsite
construction process in being a significant workload percentage of the AEC
industry [7].

In the recent years, BIM has been extended to include specialised tools such
as energy analysis, structural analysis for energy and space management [17].
However, creation of accurate BIMs is time consuming and expensive [1] if it
is done for the existing buildings and the advantages it offers for energy
efficiency calculations [9, 11] are lost. Hence it is imperative that we look
for scalable and low cost approximate BIM modeling for existing infrastructure
in a city or a local community.

In this paper, first we create the notion of the prototype buildings which
serve as the state of the art implementation of energy efficiency techniques
given particular conditions of space, number of residents and energy
consumption are considered. Secondly, we give a method to suggest a better
prototype alternatives for the existing building. We propose using Point-GNN
for detecting the buildings and determining their bounding boxes and then
suggesting the efficient prototypes for the detected buildings.

Contributions of the paper are as follows:

  * •

Introducing the novel prototype buildings for capturing the energy profiles.

  * •

Scalable framework for energy efficiency analysis and recommendation.

  * •

Use Point-GNN for identifying existing buildings and suggesting energy
efficient prototypes for them

  * •

We perform the experiments on completely synthetic data for demonstrating our
proposed approach

The remainder of this paper is organized as follows. We review some of the
important related studies in section 2. We discuss the prototype building
information model (BIM) in detail in section 3. We then discuss out proposed
method in section 4. We then conclude with section 7 and discuss the future
works.

##  2 Related Works

Climate change and impending global energy crisis has moved the scientists
along with Architecture, Engineering and Construction (AEC) industry have
called for wide use of BIM-based energy analysis for building performance
assessment [5]. Although creating BIMs for exisiting building and retrofitting
them to improve the existing building management and maintenance has been done
in practice, it is important to note a few caveats. In [8], the researchers
have tried to update a school building. Even though results are significant,
so is the time as well as expense of creating an accurate BIM.

Similar exercises were also carried out by the authors in [4] for historic
buildings where they added the diagnosis part for automating acquisition and
update of knowledge base for heritage sites. Deconstruction of existing
buildings were also taken up by the research community by building 3D models
through devices such as kinect for generating low accurate and optimized 3D
BIMs in real time [16]. However, the problem of scalability still remained an
open challenge as none of the methods could give us a quick answer for a
larger community such as a city to go through the tedious and time consuming
process of the creating individual BIMs for buildings.

There are three broad approaches that are taken to handle the tasks related to
point clouds using Deep Neural Networks. The first approach takes considers
the point cloud as a set of points. The Multi Layered Perceptron (MLP) is used
to extract the features corresponding to each of the nodes. [20] and [14] are
such networks that aggregate the feature vectors obtained by using neural
networks for the set of points.

The other approach considers the point cloud as a grid. The points in the 3D
grids units called voxels are considered in [21] and applies 3D convolutions
onto it for object detection. Some approaches [6], [19] project the point
cloud into 2D space to apply the 2D convolutions onto it. However, projections
into grids has drawback of regularity that is added due to the regular nature
of the grid, unlike the point cloud.

The third approach is to model the point cloud using an irregular structure of
graphs. The graph neural networks in its every layer updates the vertex/edge
features by aggregating them across their neighborhood[18], [13]. Unlike 2D/3D
convolutions, graph convolutions does not reduce the number of points in every
iterations. [15] first systematically converts the point cloud into a graph
for object detection and also gives the mechanism for determining the bounding
box of the detected objects.

##  3 Prototype Building Information Model (BIM)

###  3.1 3D Model

3D model gives a visual feedback for the geometry the structure[3]. The BIM
model is abstract which contains only information but it is not easy to
visualize BIM. Since the BIM also contains information about the geometry of
the building, they are easily accurately rendered to produce a visual
information BIM. In fact, the 3D model is used to accurately find the geometry
of the various structure in BIM, and not the other way around, since,
obtaining the geometry is difficult humanly without a visual feedback. 3D
model with various levels of details are shown in Figure 1.

Figure 1: 3D view of different level of details. (Top left) Original photo.
(Bottom left) High level of detail. (Top right) Low level of detail. (Bottom
right) Medium level of detail.

###  3.2 Data collection

The first step for the creation of BIM for an entire city requires careful
planning and collection of data. The BIM in itself is a very data intensive
model. A assortment of BIMs for each building in a city increases the amount
of data collected by many folds. Some of which may be redundant while many are
not. But even without taking into factor the huge amount of data to be
collected, there are many other factors that come to play.

First, it is equally challenging to figure out which data can be provided by
whom and at what time. Even though a overall summary of the data maybe made
available, the exact data points are not easy to be maintained and many legacy
systems put in place in modern infrastructure do not even collect the data.
Since, many systems both modern and legacy infrastructure is in place, there
is always a risk of obtaining incomplete data and unbeknownst to the observer,
the data might be scaled down and incorrect information creeps into the model.
Second, in a city, many private parties are involved who do not want to share
data due to privacy and other concerns.

With the above facts taken into consideration, the data to be collected is
then divided into two general category, the geometric and non-geometric data.
The geometric data is concerned mostly with the visual look of the buildings
and the non geometric data with any other information that the BIM is
concerned with.

The geometric data contains the GIS location, structure, space, district/area
around the building. The geometric data can be used for the visualization of
BIMs and as the whole, the BIM integration of the entire city. The BIM of a
given building is different from that of a city, as the later contains more
information than the former. Along with the above information, the BIM of the
city has information, for instance, the type of area the building is located
in industrial, residential or commercial. The geometric data like the roads,
bridges, and other public utilities are also information that not the part any
given building but of the city. This will be helpful when deciding the
prototypes of the building. And the data can be used to estimate the other
non-geometric data as well.

The non-geometric data contains, information like mechanical data water
requirements, electrical data like electric power consumption, HVAC data,
safety or maintenance data. These non-geometric data is specific to the
building’s BIM, the BIMs for the entire city also needs data like the sewage,
traffic, garbage (and waste disposal), drainage system etc, has to be
collected.

Another type of data that is required for creating the BIMs of the entire
city, which is not required by individual building BIM is, source. For
instance these source include, power like the electric grid, the water tanks
that provide the water for the city and the traffic load a road can take. Such
data required for creating the model for the resource constraint.

Once, all the required data that has to be collected is determined, depending
on how many parameters the model has to work on, the data has to be sanitized.
The data sanitization step mostly consists of checking for erroneous data,
missing or incomplete data, statistically improbable data, but there are other
sanitization steps specific to the type of data. The data is also to be
normalized, brought to reasonable range with respect to both time and space,
convert all the different data units to one uniform unit.

The data collection is a continuous process. As the city grows, the new
buildings are created and old buildings are either demolished or renovated,
the model evolves with time as the new data enters the system. Optionally, the
model should be able to make prediction about the BIM instead of only querying
or suggest future development and planning. This adds new data to the system.

###  3.3 Prototype Buildings

There are many buildings in a city and each building is different from the
rest. It is true that no two building are similar in all the features of data
collected. BIMs generally deal with individual buildings and therefore, making
a detailed model is not an issue while, for a city with creating BIMs for all
the building is not only a huge task, it doesn’t provide any additional
benefit to the model. Any perceived advantage is further diminished, since it
becomes a hassle and un-maintainable on a large scale, while not providing any
significant advantage over a statistical measure.

A alternative approach is, based on the data, to create a different prototypes
of buildings and use the prototypes of building to replace all the buildings
with their respective BIMs. The prototypes maybe detailed with respect to a
certain feature, while being very coarse in an different feature. A feature
being coarse means that a prototype can be used to represent many buildings
whose feature in question fall in a certain range. While, a detailed feature
begs for an exact or near exact match of the feature.

Creating a prototype is not a routine part of the model. The prototype is
created when the model is initially created and again when there is a
necessity for it’s creation. Initially, when BIMs for a city is created from
the ground up, a bulk of the work goes into the creation of the prototypes
since many of them has to be created. But when the model has been set up,
there is not a need to create prototypes anymore. But as the model evolves
with more data and new prototype buildings has to be made. But creation of new
prototypes is few and far in between and not a regular part of the life cycle
as illustrated in the Fig. 2.

Figure 2: Life cycle of BIM for cities

The creation of prototypes also depends on the coarse or the detailed feature
of the data. For features that need exact or near exact match of the features
of the data of the buildings with the prototype, a prototype for every such
match is required. Therefore, these features must be prominent features and
must broadly divide the buildings into a small limited category. For instance,
the type of the building, whether residential, commercial or industrial is
such a feature, but the power consumption is not such a feature, since for
every value of power consumption, there has to be a prototype, which defeats
the entire purpose of making prototypes.

Power consumption, however, is coarse feature for a prototype and therefore
it’s prototype must span over a large range of values. The ranges of these
features ideally may be chosen appropriately so that the buildings in the city
may fall as evenly as possible, but this may not be possible as the number of
features increases.

There are many different types of buildings at NISER. The buildings are mostly
categorized based from their top down view as shown in Table 1 . What is most
important for a building is that it should retain an overall coarse structure
for any air or drone traffic. Maintaining the detail structure of the building
is not only adds a lot of data points to the storage but also adds a hit to
the rendering of the buildings and also for the machine learning algorithm.
Making a detailed structure to the building is a “hard” task. But making the
structure of the building too coarse, implies that the building becomes a
cuboid, which is what is to be avoided. A trade-off is needed and hence a top
down view of the building is sufficient enough, while maintaining the overall
structure. Also, for a machine learning algorithm it is easier to create the
top-down view of the buildings.

Table 1: Array of 3D models and their corresponding satellite images |  |  |
---|---|---|---
|  |  |
|  |  |
|  |  |

Many mapping services like Google Maps, Here Maps, OpenStreetMaps, Bing Maps
etc, have top down views of the area where buildings can be easily made out.
Looking at a top down view, the buildings at NISER can be categorised as such.
The entire map with the protoypes at NISER for demonstration purpose is mapped
in Figure 3.

Figure 3: Entire Map of NISER

###  3.4 Storing building prototypes

Building prototypes have to be saved in data structures. There were many
candidates for the data structures that were considered. One way was to have
other data in the building’s geometry while the other was to incorporate the
geometry in the building data structure, shown below or have a separate data
structure for the buildings that includes geometry:

  * •

Building - vertices, edges, faces, power, water, traffic

  * •

Building - Geometry (Vertices, edges, faces) , power, water, traffic

In the long term, the scalability of such a data structure is important. The
data structure must be able to add new data categories and integrate well.
Finally, keeping in mind that the ultimate goal is to have a learning
algorithm to automate many of these tasks, it is imperative that the data
structure should also facilitate modification automatically.

##  4 eBIM-GNN

We are interested in a system that matches the right prototype to the
buildings. Towards this goal, we propose a two-step system to address this
problem, where the first part detects the objects in the scene and classifies
into one of the two classes, building or not a building. Along with this
classification, the bounding boxes of each of the detected objects are
predicted, these are used to localize the buildings. The second part considers
the buildings detected by the first part and further assigns the closest
prototype to it.

(a)

(b)

Figure 4: Predicting prototypes corresponding to the detected objects

###  4.1 Graph Construction

A point cloud is defined as a set $P$ of points, given as
$P=\\{p_{1},...p_{N}\\}$, where $p_{i}=(x_{i},s_{i})$ is a point with
$x_{i}\in\mathbb{R}^{3}$ are the co-ordinates of the points and
$s_{i}\in\mathbb{R}^{m}$ is the feature vector of length $m$ corresponding to
the point $i$. We construct a graph $G=(V,E)$ from the given point cloud. We
consider each point from $P$ as the vertices in the graph. Therefore, $V=P$,
and the edges in the graph $G$ we connect two nodes if they lie within a
distance $r$ from each other. This is given in equation 1.

$E=\\{(p_{i},p_{j})\mid\left\lVert x_{i}-x_{j}\right\rVert_{2}\leq r\\}$ (1)

where $r$ is the radius and $x_{j}$ and $x_{j}$ are the nodes in the graph. We
form a graph from the given point cloud as described previously. The graph
formed using the entire point could might overburden the computation. Hence,
before constructing the graph, the point cloud can be downsampled. Point-GNN
makes use of voxel downsampling.

The initial feature vector $s_{i}$ of node $i$ is obtained by using $Max$
function to aggregate the embeddings generated by a multilayer perceptron
(MLP) that embeds the relative coordinates and other features.

###  4.2 Graph Neural Networks

The Graph Neural Networks (GNNs) address the questions related to systems that
can be modeled as graphs. In each layer $l+1$ of a GNN, the features of the
node’s neighboring nodes are extracted by aggregating them, as shown in the
equation 2.

$\displaystyle v_{i}^{l+1}$
$\displaystyle=g^{l}(\rho(\\{e_{ij}^{l}\mid(i,j)\in E\\}),v_{i}^{l})$ (2)
$\displaystyle e_{ij}^{l}$ $\displaystyle=f^{t}(v_{i}^{l},v_{j}^{l})$ (3)

where $v^{l}$ and $e^{l}$ are feature vectors corresponding to vertex $i$ and
edge $ij$ respectively. The set function $\rho$ aggregates the edge features
for each vertex, function $g^{l}$ updates the next layer vertex features, and
the function $f^{l}$ aggregates the vertex features to update the edge
features.

###  4.3 Point-GNN

####  4.3.1 Overview

The method Point-GNN[15] makes use of GNNs for detecting objects given a point
cloud, it also localizing the bounding box of the object. Poing-GNN uses an
auto-registration mechanism that enables the points to align their coordinates
based on their features. The auto-registration mechanism reduced the
translation variance. The alignment offset is predicted using the structural
features from the previous layers in a GNN. We modify the equation 2 to update
the feature vector $s_{i}$ corresponding to each node $i$, and add auto-
registration to it. The auto-registration mechanism is as follows:

$\displaystyle\Delta x_{i}^{l}$ $\displaystyle=h^{l}(s_{i}^{l})$ (4)
$\displaystyle s_{i}^{l+1}$
$\displaystyle=g^{l}\left(\rho\left(\left\\{f^{l}(x_{j}-x_{i},s_{i}^{l})\mid(i,j)\in
E\right\\}\right),s_{i}^{l}\right)$ (5)

where $\Delta x_{i}^{l}$ is coordination offset, calculated by the function
$h^{l}$ using the center node $i$’s feature vector $s_{i}^{l}$ from previous
layer $l$.

The functions $f^{l},g^{l},$ and $h^{l}$ are modeled using MLPs and $\rho$ is
chosen to be $Max$ function. For each layer of the Point-GNN, the MLPs are
different that are not shared across the layers. As the Point-GNN is used to
handle two tasks, classification and localization, the representations
produced by Point-GNN after $L$ layers is passed to two branches to handle
each of the two tasks.

####  4.3.2 Box Merging and Scoring

In order to improve the localization of the bounding box, Point-GNN uses the
modified Non-maximum suppression(NMS). Multiple bounding boxes can be
predicted as there are many points on the same object. Along with the bounding
box, the corresponding confidence score is calculated. It is done using the
ratio of occupied volumen, called occlusion factor. A box $b_{i}$ has length
$l_{i}$, width $w_{i}$ and height $h_{i}$ and three vectors
$v_{i}^{l},v_{i}^{w},v_{i}^{h}$ are the unit vectors in their directions
respectively. Then the occlusion factor is given as

$o_{i}=\frac{1}{l_{i}w_{i}h_{i}}\prod_{v\in{v_{i}^{l},v_{i}^{w},v_{i}^{h}}}\max_{p_{j}\in
b_{i}}(v^{T}x_{j})-\min_{p_{j}\in b_{i}}(v^{T}x_{j})$ (6)

###  4.4 Loss

The classification branch outputs a multiclass probability distribution
corresponding to each of the vertex. It is given by
$\\{p_{c_{1}},p_{c_{2}},\dots,p_{c_{M}}\\}$ for $M$ classes of objects. The
cross-entropy loss contributed by the first branch of classification, is given
as follows:

$l_{cls}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{M}y_{c_{j}}^{i}log(p_{c_{j}}^{i})$
(7)

where $p_{c}^{i}$ and $y_{c}^{i}$ are the predicted probabilities and one-hot
encoding of the predicted label of $i$th class.

The bounding box is predicted for every detected objects as a 7-tuple
$b=(x,y,z,l,b,w,\theta)$ with 7 degrees of freedom. It consists of the
coordinates of center $x,y,z$, length, width and height $l,h,w$ respectively
and the yaw angle $\theta$ of the bounding box. The bounding box is encoded
with vertex coordinates $(x_{v},y_{v},z_{v})$ as follows:

$\displaystyle\begin{split}\delta_{x}&=\frac{x-x_{v}}{l_{m}},\delta_{y}=\frac{y-y_{v}}{h_{m}},\delta_{z}=\frac{z-z_{v}}{w_{m}}\\\
\delta_{l}&=\log\frac{l}{l_{m}},\delta_{h}=\log\frac{h}{h_{m}},\delta_{w}=\log\frac{w}{w_{m}}\\\
\delta_{\theta}&=\frac{\theta-\theta_{)}}{\theta_{m}}\end{split}$ (8)

The loss from the localization branch that predicts the bounding box, is
computed using Huber loss [12] on ground truth and prediction. The vertices
that need not be localized or lie outside the bounding box contribute nothing
to the loss. The average localization loss is given by:

$l_{loc}=\frac{1}{N}\sum_{i=1}^{N}\mathbb{I}(v_{i}\in
b_{interest})\sum_{\delta\in\delta_{b_{i}}}l_{huber}(\delta-\delta^{gt})$ (9)

##  5 Prototype Matching

We need to match the buildings with the prototypes, so that depending on the
energy efficiency, a better prototype could be suggested for the given
building. We propose two approaches to do this.

###  5.1 Approach 1: Parameterized Aggregation

The bounding box that inscribes the object is predicted for each of the
identified object by point-GNN. We predefine the prototypes based on the prior
knowledge about ideal buildings, their energy usage and energy efficiency. For
each of the building type $b_{i}$, we consider a set of prototypes
$P_{b_{i}}$, where $1\leq i\leq B$ with total $B$ different types of
buildings. For each of these prototypes, there are associated predefined
specifications of each of their features. To $k$th feature $f_{jk}$ of the
prototype $j\in P_{b_{i}}$ we assign a coefficient that denoted the
contribution of the particular feature in deciding the better matching of the
identified objects with prototypes.

We calculate the mean $\mu_{l},\mu_{w},\mu_{h}$ and standard deviation
$\sigma_{l},\sigma_{w},\sigma_{h}$ of the length, width and height
respectively of the identified bounding boxes for each of the classes. The
prototypes are chosen with dimensions
$(\mu_{l},\mu_{w},\mu_{h}),(\mu_{l}\pm\sigma_{l},\mu_{w}\pm\sigma_{w},\mu_{h}\pm\sigma_{h})$
and $(\mu_{l}\pm 2\sigma_{l},\mu_{w}\pm 2\sigma_{w},\mu_{h}\pm 2\sigma_{h})$
of for all different types of buildings.

To match the identified buildings to the prototypes, we consider the closest
values of the dimensions of the prototypes to that of the buildings. The range
for of the five prototypes corresponding to every building type is
$(\mu_{l}-2.5\sigma_{l},\mu_{w}-2.5\sigma_{w},\mu_{h}-2.5\sigma_{h})-(\mu_{l}-1.5\sigma_{l},\mu_{w}-1.5\sigma_{w},\mu_{h}-1.5\sigma_{h})$
,
$(\mu_{l}-1.5\sigma_{l},\mu_{w}-1.5\sigma_{w},\mu_{h}-1.5\sigma_{h})-(\mu_{l}-0.5\sigma_{l},\mu_{w}-0.5\sigma_{w},\mu_{h}-0.5\sigma_{h})$,
$(\mu_{l}-0.5\sigma_{l},\mu_{w}-0.5\sigma_{w},\mu_{h}-0.5\sigma_{h})-(\mu_{l}+0.5\sigma_{l},\mu_{w}+0.5\sigma_{w},\mu_{h}+0.5\sigma_{h})$,
$(\mu_{l}+0.5\sigma_{l},\mu_{w}+0.5\sigma_{w},\mu_{h}+0.5\sigma_{h})-(\mu_{l}+1.5\sigma_{l},\mu_{w}+1.5\sigma_{w},\mu_{h}+1.5\sigma_{h})$,
and
$(\mu_{l}+1.5\sigma_{l},\mu_{w}+1.5\sigma_{w},\mu_{h}+1.5\sigma_{h})-(\mu_{l}+2.5\sigma_{l},\mu_{w}+2.5\sigma_{w},\mu_{h}+2.5\sigma_{h})$
respectively.

We can then match each of the identified building to its closest prototype.
The loss in matching the prototypes of each of the building type is calculated
as follows:

$l_{pro}=\sum_{j=i}^{N}\sum_{i=1}^{F}\phi_{i}(f_{a}^{ij}-f_{mp}^{ij})$ (10)

where, $F$ is the number of features of the buildings, $\phi_{i}$ is the
coefficient corresponding to $i$th features, $f_{a}^{ij}$ is the actual value
of the $i$th feature of $j$th example and $f_{mp}^{ij}$ the value of that
feature in the matched prototype. $\phi_{i}$ can be learnt when the data is
annotated.

###  5.2 Approach 2: Learning with MLP

The second part of the system in figure 4 classifies the detected objects into
classes (prototypes). This network also gives a multi-class probabilities
$\\{p_{p_{1}},p_{p_{2}},\dots,p_{p_{T}}\\}$ corresponding to the $T$
prototypes for each of the detected objects. We compute the cross entropy loss
as follows.

$l_{pro}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{T}z_{p_{j}}^{i}log(p_{p_{j}}^{i})$
(11)

where $z_{p_{j}}$ is the one-hot vector indicating the prototype corresponding
to the example.

The total loss is given by:

$l_{total}=\alpha l_{cls}+\beta l_{loc}+\gamma l_{pro}+\kappa l_{reg}$ (12)

where $\alpha,\beta,\gamma,\kappa$ are the constants to balance losses.

###  5.3 Efficiency

With the use of eBIM-GNN, we get the better and efficient prototypes for the
detected objects. To calculate the gain in efficiency, we compare the total
energy consumption of the current building with the total energy consumption
of the prototype building. The total energy consumption of the detected object
is given by $E_{o}$, and the total energy consumption of suggested prototype
is given by $E_{p}$. We find the percentage gain in total energy efficiency as
given in equation 13.

$gain_{efficiency}=\frac{E_{o}-E_{p}}{E_{o}}\times 100$ (13)

##  6 Evaluation on Synthetic Data

###  6.1 Dataset

We consider the KITTI dataset [10] that consists of point clouds and camera
images. The dataset has 7481 training examples and 7518 testing examples.
Currently due to unavailability of the annotated data consisting of point
clouds of buildings, we demonstrate the results on a similar annotated dataset
with objects beings cars, pedestrians and cyclists instead of buildings. Since
we are considering the KITTI dataset for demonstration, we associate the
energy usage to the passenger capacity of each of the three categories. We
consider the categories as building types for the sake of convenience.
Category ’car’ is considered as building type 1, ’pedestrian’ as building type
2 and ’cyclist’ as building type 3. Once identified, we consider the bounding
boxes along with the identified class for to match it with the predefined
prototypes.

We get five prototypes corresponding to each class. The ranges of dimensions
of bounding boxes for matching the prototypes are as shown in the table 2.
These values are $-0.5\sigma$ to $+0.5\sigma$ to each of the dimensions of
prototypes. We calculate the total efficiency, as well as the efficiency of
each of the building types.

Table 2: Range of dimensions of prototype classes in $(h,w,l)$ format. Prototype |  Building type 1 |  Building type 2 |  Building type 3
---|---|---|---
1 |  (1.6971, 1.7051, 4.3538) - (1.8080, 1.7622, 4.6548) |  (1.8890, 0.7882, 1.0638) - (1.9667, 0.8725, 1.2325) |  (1.8250, 0.6876, 1.9565) - (1.8860, 0.7517, 2.0837)
2 |  (1.5862, 1.6480, 4.0527) - (1.6971, 1.7051, 4.3538) |  (1.8112, 0.7039, 0.8951) - (1.8890, 0.7882, 1.0638) |  (1.7640, 0.6235, 1.8293) - (1.8250, 0.6876, 1.9565)
3 |  (1.4753, 1.5909, 3.7518) - (1.5862, 1.6480, 4.0527) |  (1.7334, 0.6196, 0.7264) - (1.8112, 0.7039, 0.8951) |  (1.7030, 0.5594, 1.7021) - (1.7640, 0.6235, 1.8293)
4 |  (1.3644, 1.5338, 3.4508) - (1.4753, 1.5909, 3.7518) |  (1.6556, 0.5353, 0.5577) - (1.7334, 0.6196, 0.7264) |  (1.6420, 0.4953, 1.5749) - (1.7030, 0.5594, 1.7021)
5 |  (1.2535, 1.4767, 3.1498) - (1.3644, 1.5338, 3.4508) |  (1.5778, 0.4510, 0.3890) - (1.6556, 0.5353, 0.5577) |  (1.5810, 0.4312, 1.4477) - (1.6420, 0.4953, 1.5749)

We define the energy usage for each of the building types randomly for our
experiments. For building type 1 the energy usage is considered as 40 units,
for building type 2 it is 20 units, and for building type 3 it is considered
as 10 unit of energy. However, each of the prototype has different energy
usage values associated with them. These values are as shown in table 3.

Table 3: Predefined energy usage values for each of the building prototypes Prototype | Building type 1 | Building type 2 | Building type 3
---|---|---|---
1 | 40 | 10 | 20
2 | 38 | 8 | 18
3 | 36 | 6 | 16
4 | 34 | 4 | 14
5 | 32 | 2 | 12

###  6.2 Results

Table 4: Counts of buildings in the range of dimensions of respective prototype classes  Prototype | Building type 1 | Building type 2 | Building type 3
---|---|---|---
1 | 76 | 0 | 0
2 | 273 | 269 | 110
3 | 1451 | 2104 | 145
4 | 886 | 109 | 26
5 | 1 | 21 | 1

In order to calculate efficiency after matching with the prototype, we count
the number of examples belonging to each of the prototypes. This is shown in
table 4 By multiplying the energy usage corresponding to each of the
prototypes of every class and summing it up, we get the total energy usage. We
show the energy usage in table 5 along with the reduction in energy usage
corresponding to each of the building type. The total reduction in energy
usage is 16.22%.

Table 5: Percentage reduction in energy usage Prototype | Actual usage | Prototype usage | Red. in usage (%)
---|---|---|---
Building type 1 | 107480 | 95806 | 10.86
Building type 2 | 25030 | 15254 | 39.05
Building type 3 | 5640 | 4676 | 17.09

##  7 Conclusion

eBIM-GNN suggests the energy efficient prototypes for the buildings detected
in a 3D point cloud. The building detection and determination of its bounding
box is done using the method of Point-GNN. After the determination of bounding
box, with the use of other features corresponding to the detected objects, the
second part of system determines the appropriate prototype corresponding to
the object. We demonstrate our approach on synthetic dataset.

In future, we would like to collect real data of the buildings and tune out
the possible scalability issues related with the resolution of the point
clouds and further optimize the matching against the prototypes. Our system
benefits from the number of prototypes it has in the system to more accurately
predict the energy efficiency gains.

## References

  * [1] Shakil Ahmed.  Barriers to implementation of building information modeling (bim) to the construction industry: a review.  Journal of civil engineering and construction, 7(2):107–113, 2018\.
  * [2] Salman Azhar.  Building information modeling (bim): Trends, benefits, risks, and challenges for the aec industry.  Leadership and management in engineering, 11(3):241–252, 2011.
  * [3] Vladimir Bazjanac.  Virtual building environments (vbe)-applying information modeling to buildings. lawrence berkeley national laboratory.  University of California. Berkeley, CA, USA, 2004.
  * [4] Silvana Bruno, Mariella De Fino, and Fabio Fatiguso.  Historic building information modelling: performance assessment for diagnosis-aided information modelling and management.  Automation in Construction, 86:256–276, 2018.
  * [5] Chien-Jung Chen, Shang-yuan Chen, Syuan-hao Li, and Hsiu-ting Chiu.  Green bim-based building energy performance analysis.  Computer-Aided Design and Applications, 14(5):650–660, 2017.
  * [6] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia.  Multi-view 3d object detection network for autonomous driving.  In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 1907–1915, 2017.
  * [7] Amos Darko, Albert PC Chan, Yang Yang, and Mershack O Tetteh.  Building information modeling (bim)-based modular integrated construction risk management–critical survey and future needs.  Computers in Industry, 123:103327, 2020.
  * [8] Giuseppe Martino Di Giuda, Valentina Villa, and Paolo Piantanida.  Bim and energy efficient retrofitting in school buildings.  Energy Procedia, 78:1045–1050, 2015.
  * [9] Stathis Eleftheriadis, Dejan Mumovic, and Paul Greening.  Life cycle energy efficiency in building structures: A review of current developments and future outlooks based on bim capabilities.  Renewable and Sustainable Energy Reviews, 67:811–825, 2017.
  * [10] Andreas Geiger, Philip Lenz, and Raquel Urtasun.  Are we ready for autonomous driving? the kitti vision benchmark suite.  In Conference on Computer Vision and Pattern Recognition (CVPR), 2012.
  * [11] Ali GhaffarianHoseini, Tongrui Zhang, Okechukwu Nwadigo, Amirhosein GhaffarianHoseini, Nicola Naismith, John Tookey, and Kaamran Raahemifar.  Application of nd bim integrated knowledge-based building management system (bim-ikbms) for inspecting post-construction energy efficiency.  Renewable and Sustainable Energy Reviews, 72:935–949, 2017.
  * [12] Peter J Huber.  Robust estimation of a location parameter.  The Annals of Mathematical Statistics, pages 73–101, 1964.
  * [13] Rucha Bhalchandra Joshi and Subhankar Mishra.  Learning Graph Representations, pages 209–228.  Springer Singapore, Singapore, 2022.
  * [14] Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas.  Pointnet: Deep learning on point sets for 3d classification and segmentation.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 652–660, 2017.
  * [15] Weijing Shi and Ragunathan (Raj) Rajkumar.  Point-gnn: Graph neural network for 3d object detection in a point cloud.  In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
  * [16] Rebekka Volk, Thu Huong Luu, Johannes Sebastian Mueller-Roemer, Neyir Sevilmis, and Frank Schultmann.  Deconstruction project planning of existing buildings based on automated acquisition and reconstruction of building information.  Automation in construction, 91:226–245, 2018.
  * [17] Rebekka Volk, Julian Stengel, and Frank Schultmann.  Building information modeling (bim) for existing buildings—literature review and future needs.  Automation in construction, 38:109–127, 2014.
  * [18] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and S Yu Philip.  A comprehensive survey on graph neural networks.  IEEE transactions on neural networks and learning systems, 32(1):4–24, 2020.
  * [19] Bin Yang, Wenjie Luo, and Raquel Urtasun.  Pixor: Real-time 3d object detection from point clouds.  In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 7652–7660, 2018.
  * [20] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola.  Deep sets.  Advances in Neural Information Processing Systems, 30, 2017.
  * [21] Yin Zhou and Oncel Tuzel.  Voxelnet: End-to-end learning for point cloud based 3d object detection.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4490–4499, 2018.


### Extension: Anticipated lockdown duration

In Lemma 3 we solve, using the maximum principle, the infinite horizon
maximization problem (56), with $t\in[T,\infty)$. Then in Lemma 4, we solve
problem (55) as a finite horizon optimization problem where the term
$v(T,b(T),h(T))$ is the terminal cost. In the following the same assumptions
of Section 5 hold and to avoid cumbersome notation, we write

$v(0,b,h)=:v(b,h).$

######  Lemma 3.

Suppose that both sectors are active with production
$y_{1,NL}=(\xi\bar{l})^{\alpha}$, and $y_{2,NL}=[(1-\xi)\bar{l}]^{\alpha}$.
Then, a given state-control quadruple
$((c_{1})_{TS,AL},(c_{2})_{TS,AL},h_{TS,AL},b_{TS,AL})$ is optimal for the
infinite horizon maximization problem after lockdown (56) if and only if it is
a solution of the following system

$\displaystyle u_{c_{1_{TS,AL}}}+\mu_{TS,AL}\phi-\lambda_{TS,AL}=0$ (112)
$\displaystyle u_{c_{2_{TS,AL}}}-p\lambda_{TS,AL}=0$ (113)
$\displaystyle\dot{\mu}_{TS,AL}=(\phi+\rho)\mu_{TS,AL}-u_{h_{TS,AL}},\quad
t\in[T,\infty)$ (114) $\displaystyle\dot{\lambda}_{TS,AL}=0,\quad
t\in[T,\infty)$ (115)
$\displaystyle\lim_{t\to+\infty}b_{TS,AL}\cdot\lambda_{TS,AL}e^{-\rho t}=0$
(116) $\displaystyle\lim_{t\to+\infty}h_{TS,AL}\cdot\mu_{TS,AL}e^{-\rho t}=0$
(117) $\displaystyle\dot{h}_{TS,AL}=\phi(c_{1_{TS,AL}}-h_{TS,AL}),\quad
t\in(T,\infty)$ (118)
$\displaystyle\dot{b}_{TS,AL}+c_{1_{TS,AL}}=rb_{TS,AL}+y_{1},\quad
t\in(T,\infty)$ (119) $\displaystyle c_{2_{TS,AL}}=y_{2}.$ (120)
$\displaystyle h_{TS,AL}(T)=h_{TS,L}(T),$ (121) $\displaystyle
b_{TS,AL}(T)=b_{TS,L}(T).$ (122)

Note that $(b_{TS,L}(T),h_{TS,L}(T))$ are taken as exogenously given constants
at this stage of the analysis.

######  Lemma 4.

Consider now an economy with only sector 1 being active and producing
$y_{1,L}=A\\{[\xi+a(1-\xi)]\bar{\ell}\\}^{\alpha}$ with $a\in(0,1).$ Then, a
given state-control quadruple $(c_{1_{TS,L}},0,h_{1_{TS,L}},b_{1_{TS,L}})$ is
optimal for the problem (55) if and only if it is solution of the following
system

$\displaystyle u_{c_{1_{TS,L}}}+\mu_{TS,L}\phi-\lambda_{TS,L}=0$ (123)
$\displaystyle\dot{\mu}_{TS,L}=(\phi+\rho)\mu-u_{h_{TS,L}}\,\,t\in[0,T)$ (124)
$\displaystyle\lambda_{TS,L}=v_{b}(b_{TS,L}(T),h_{TS,L}(T))$ (125)
$\displaystyle\mu_{TS,L}=v_{h}(b_{TS,L}(T),h_{TS,L}(T))$ (126)
$\displaystyle\dot{h}_{TS,L}=\phi(c_{1_{TS,L}}-h_{TS,L})\,\,t\in(0,T]$ (127)
$\displaystyle\dot{b}_{TS,L}+c_{1_{TS,L}}=rb_{TS,L}+y_{1,L}\,\,t\in(0,T]$
(128) $\displaystyle b_{TS,L}(0)=b_{0}$ (129) $\displaystyle
h_{TS,L}(0)=h_{0},$ (130)

where $v_{b}(b_{TS,L}(T),h_{TS,L}(T))$,$v_{h}(b_{TS,L}(T),h_{TS,L}(T))$ are
treated as exogenously given constants at this stage of the analysis.

######  Proposition 10.

Let $\psi_{1},\psi_{2}$ be defined in (85). Denote
$a=r+\phi(1-a_{c_{1}h}),b=1-a_{c_{1}h}^{2},d=\phi(a_{c_{1}h}-1),c=\phi^{2}$.
The optimal state-control quadruple $({c_{1}}_{TS,L},0,h_{TS,L},b_{TS,L})$ for
the infinite horizon maximization problem after lockdown, solution of
(123)-(130), satisfies

$\displaystyle
c_{1_{TS,L}}(t)=a_{c_{1}}+a_{c_{1}h}[h^{*}_{TS,L}+(h_{0}-h^{*}_{TS,L}-D)e^{\psi_{1}t}+De^{\psi_{2}t}]+$
$\displaystyle\hskip
113.81102pt+\phi[\mu^{*}_{TS,L}-(h_{0}-h^{*}_{TS,L}-D)\frac{b}{a-\psi_{1}}e^{\psi_{1}t}-D\frac{b}{a-\psi_{2}}e^{\psi_{2}t}]-v_{b}(b(\tilde{t}),h(\tilde{t}))$
$\displaystyle
h_{TS,L}(t)=h^{*}_{TS,L}+(h_{0}-h^{*}_{TS,L}-D)e^{\psi_{1}t}+De^{\psi_{2}t}$
(131) $\displaystyle
b_{TS,L}(t)=e^{rt}b_{0}+A(e^{rt}-1)+B(e^{\psi_{1}t}-e^{rt})+C(e^{\psi_{2}t}-e^{rt})$
(132)

where

$h^{*}_{TS,L}=\frac{\phi\lambda_{TS,L}-\phi a_{c_{1}}-c\mu^{*}_{TS,L}}{d},$

$\lambda_{TS,L}$ is given by (125) and

$\mu^{*}_{TS,L}=\frac{a_{h}d+a_{c_{1}h}a_{c_{1}}d-a_{c_{1}h}\lambda_{TS,L}d-b\phi\lambda+b\phi
a_{c_{1}}}{ad-bc},$
$D=\left(v_{h}(b_{TS,L}(\tilde{t}),h_{TS,L}(\tilde{t}))-\mu^{*}_{TS,L}+(h_{0}-h^{*}_{TS,L})\frac{b}{a-\psi_{1}}e^{\psi_{1}\tilde{t}}\right)\frac{(a-\psi_{1})(a-\psi_{2})}{be^{\psi_{1}\tilde{t}}(a-\psi_{2})-be^{\psi_{2}\tilde{t}}(a-\psi_{1})},$
$A:=\frac{y_{1,L}-a_{c_{1}}-a_{c_{1}h}h^{*}_{TS,L}-\phi\mu^{*}_{L,TS}+v_{b}(b_{TS,L}(\tilde{t}),h_{TS,L}(\tilde{t}))}{r},$
$B:=\frac{\phi(h_{0}-h^{*}_{TS,L}-D)\frac{b}{a-\psi_{1}}-a_{c_{1}h}(h_{0}-h^{*}_{TS,L}-D)}{\psi_{1}-r},$
$C:=\frac{\phi D\frac{b}{a-\psi_{2}}-a_{c_{1}h}D}{\psi_{2}-r},$

For $t>T$, the optimal state-control quadruple
$({c_{1}}_{TS,AL},{c_{2}}_{TS,AL},h_{TS,AL},b_{TS,AL})$, solution to
(112)-(122), satisfies

$\displaystyle
c_{1_{TS,AL}}(t)=h^{*}_{TS,AL}+\frac{\phi+\psi_{1}}{\phi}(h_{TS,L}(T)-h^{*}_{TS,AL})e^{\psi_{1}(t-T)}$
$\displaystyle
h_{TS,AL}(t)=h^{*}_{TS,AL}+(h_{TS,L}(T)-h^{*}_{TS,AL})e^{\psi_{1}(t-T)}$
$\displaystyle
b_{TS,AL}(t)=b^{*}_{TS,AL}+\frac{\psi+\psi_{1}}{\phi(r-\psi_{1})}[h_{TS,AL}(t)-h^{*}_{TS,AL}]$
$\displaystyle c_{2_{TS,AL}}(t)\equiv y_{2}$

where

$h^{*}_{TS,AL}=\frac{\phi(\psi_{1}-r)}{(\phi+r)\psi_{1}}\left[rb_{TS,L}(T)+y_{1}+\frac{r(\phi+\psi_{1})}{\phi(\psi_{1}-r)}h_{TS,L}(T)\right]$

and $b^{*}_{TS,AL}$ is defined in (34).

It is possible to find the following explicit formula for the value function
by some computations done in MATLAB.

######  Proposition 11.

The value function is given by the following formula:

$\displaystyle
v(b,h)=(A_{0}+A_{b}b+A_{h}h+A_{2,b}b^{2}+A_{2,h}h^{2}+A_{b,h}bh)+$
$\displaystyle+\frac{1}{\rho-2\psi_{1}}(\tilde{A}_{0}+\tilde{A}_{b}b+\tilde{A}_{h}h+\tilde{A}_{2,b}b^{2}+\tilde{A}_{2,h}h^{2}+\tilde{A}_{b,h}bh)$
(133)
$\displaystyle+\frac{1}{\rho-\psi_{1}}(A^{*}_{0}+A^{*}_{b}b+A^{*}_{h}h+A^{*}_{2,b}b^{2}+A^{*}_{2,h}h^{2}+A^{*}_{b,h}bh)$
(134)

where the coefficients
$A_{0},A_{b},A_{h},A_{2,b},A_{2,h},A_{b,h},\tilde{A}_{0},\tilde{A}_{b},\tilde{A}_{h},\tilde{A}_{2,b},\tilde{A}_{2,h},\tilde{A}_{b,h},A^{*}_{0},A^{*}_{b},A^{*}_{h},A^{*}_{2,b}$
$A^{*}_{2,h},A^{*}_{b,h}$ are calculated using MATLAB.

Finally, note that in Proposition 10 the constants
$h^{*}_{TS,L},\mu^{*}_{TS,L},A,B,C,D$ and $h^{*}_{TS,AL}$ depend on the
“parameters” $v_{b}(h_{TS,L}(T),b_{TS,L}(T))$ $v_{h}(h_{TS,L}(T),b_{TS,L}(T))$
and on $(b_{TS,L}(T),h_{TS,L}(T))$, respectively. To solve the problem, we
need to find their explicit values. The procedure one could use is the
following. The values of $v_{h}(b_{TS,L}(T),h_{TS,L}(T))$, and
$v_{b}(b_{TS,L}(T),h_{TS,L}(T))$ are given in Proposition 11 in terms of
$b_{TS,L}(T),h_{TS,L}(T)$. Then, $(b_{TS,L}(T),h_{TS,L}(T))$ can be found by
plugging $t=T$ in (132), coupling it with the equation (131) and plugging in
the expressions for the derivatives of the value function. The result is a
linear system in the unknown $(b(T),h(T))$, easily solvable in MATLAB.

## SUPPLEMENTARY MATERIAL (For Online Pubblication)

The purpose of this section is to show that a lockdown leads to no economic
consequences after its end if the economy is the same as the one described in
the main part of the paper but without habits.

Under this assumption the first order conditions are

$\displaystyle u_{c_{1}}(c_{1},c_{2})=\lambda$ (135) $\displaystyle
u_{c_{2}}(c_{1},c_{2})=p\lambda$ (136) $\displaystyle\lambda=\ constant\qquad
as\ we\ assume\ r=\rho$ (137) $\displaystyle\lim_{t\rightarrow\infty}b\lambda
e^{-\rho t}=0$ (138)

Moreover, at the equilibrium we have that

$c_{2}=y_{2}\qquad and\qquad\dot{b}+rb=y_{1}-c_{1}$

Assuming that the marginal utility with respect to $c_{1}$ is an invertible
function, it follows immediately that in equilibrium the consumption of good 1
is constant:

$c_{1}=u^{-1}_{c_{1}}(\lambda;y_{2})=\ constant.$

Therefore, the good 1 market equilibrium condition is a linear ODE in the
variable $b$ whose solution is

$b_{t}=\left(b_{0}+\frac{y_{1}-c_{1}}{r}\right)e^{rt}-\frac{y_{1}-c_{1}}{r}$

Using the TVC it follows immediately that the equilibrium path of $b$ is

$b_{t}=-\frac{y_{1}-c_{1}}{r}\qquad for\ all\ t\geq 0$

Based on this finding, we have that good 1 consumption before, i.e. $t=0$, and
during the lockdown, i.e. $t\in(0,\tilde{t}]$, are respectively

$\displaystyle c^{BL}_{1}=rb_{0}+y^{BL}_{1}$ $\displaystyle
c^{L}_{1}=rb_{0}+y^{L}_{1}$

Observe that since $b_{0}$ is the same in both expressions (this is because it
is a predetermined variable), then the increase in good 1 consumption during
the lockdown is fully driven by an expansion of good 1 production due to the
change in labor composition. As a consequence, the economy will immediately
return after the lockdown to its pre-lockdown levels since labor readjusts to
its original levels, i.e.

$y_{1}^{AL}=y_{1}^{BL}\qquad\Rightarrow\qquad c_{1}^{AL}=c_{1}^{BL},\ and\ \
\lambda^{AL}=\lambda^{BL}.$

Therefore, there will not be no change in the relative price either.

We conclude that all the effects found in the paper depends on the habits and
that their introduction induces also transitional dynamics in the economy.


left multiplication on both sides of the equation (226) by $\sigma$ and taking
the trace.

######  Lemma X.1.

Consider the trace $\mathrm{Tr}$ over a vector space $C_{N}^{\otimes
2n}=C_{N}^{\otimes n}\otimes C_{N}^{\otimes n}$. Let $\mathrm{P_{2}Tr}$ be the
partial trace over the second $C_{N}^{\otimes n}$ vector space, and
$\alpha=(1,1+n)(2,2+n)\cdots(n,2n)$ be the permutation that switches the first
and second $C_{N}^{\otimes n}$ vector spaces. For a given $n$,

$\sum_{\gamma\in[2^{n}]}\mathrm{P_{2}Tr}(\gamma\,\alpha)=\sum_{\rho\in
S_{n}}\mathrm{Tr}(\rho)\,\rho\;\big{(}1+{\mathcal{O}}(N^{-2})\big{)}\;.$ (226)

Before we begin our proof, let us see an example of (226), checking it for
$n=2$. The permutations in $[2^{2}]$ are shown in Fig. 17. For each
permutation $\sigma$ in $[2^{2}]$, $\mathrm{P_{2}Tr}(\sigma\,\alpha)$ is shown
in Fig. 18. This way, we indeed check

$\sum_{\gamma\in[2^{2}]}\mathrm{P_{2}Tr}(\gamma\;\alpha)=[1+N^{2}](1)(2)+N(1\,2)=[N^{2}(1)(2)+N(1\,2)][1+{\mathcal{O}}(N^{-2})]\;.$
(227)

(a) $(1\,2)(3\,4)$

(b) $(1\,3)(2\,4)$

(c) $(1\,4)(2\,3)$

Figure 17: Permutations in $[2^{2}]$.

(a) $(1)(2)$

(b) $N^{2}(1)(2)$

(c) $N(1\,2)$

Figure 18: Partial trace of permutations in $[2^{2}]$.

###### Proof.

Let $\gamma\in[2^{n}]$. Defining the interval of integers
$[a,b]=\\{x|x\in\mathbb{Z},a\leq x\leq b\\}$ and the sets
$p_{f}=\\{x|x\in[1,n],\gamma x\in[1,n]\\}$ and $p_{i}=\\{\alpha
x|x\in[n+1,2n]|\gamma x\in[n+1,2n]\\}$, where
$\alpha=(1,1+n)(2,2+n)\cdots(n,2n)$. Remark $\alpha=\alpha^{-1}$ and
$\gamma=\gamma^{-1}$. Noticing the cardinalities of $p_{f}$ and $p_{i}$ are
equal, we call $s=|p_{f}|=|p_{i}|$. Define the permutation
$\nu_{f}\in[2^{\frac{s}{2}}]$ by

$\nu_{f}x=\begin{cases}\gamma x&\text{ if }x\in p_{f}\\\ x&\text{ if
}x\in[1,n]\setminus p_{f}\end{cases}\;.$ (228)

Define the permutation $\nu_{i}\in[2^{\frac{s}{2}}]$ by

$\nu_{i}x=\begin{cases}\alpha\gamma\alpha x&\text{ if }x\in p_{i}\\\ x&\text{
if }x\in[1,n]\setminus p_{i}\end{cases}\;.$ (229)

Define also $\mu\in S_{n}$ by

$\mu x=\begin{cases}\gamma\alpha x&\text{ if }x\in[1,n]\setminus p_{i}\\\
(\gamma\alpha)^{-k(x)}x&\text{ if }x\in p_{i}\end{cases}\,,$ (230)

where $k(x)=\mathrm{min}\\{m|m\geq 0,(\gamma\alpha)^{-m}x\in p_{f}\\}$.
Checking each case we see that

$\gamma x=\begin{cases}\nu_{f}x&\text{ if }x\in p_{f}\\\ \alpha\nu_{i}\alpha
x&\text{ if }x\in\alpha p_{i}\\\ \mu\alpha x&\text{ if
}x\in[n+1,2n]\setminus\alpha p_{i}\\\ (\mu\alpha)^{-1}x&\text{ if
}x\in[1,n]\setminus p_{f}\end{cases}\,,$ (231)

and applying a multiplication by $\alpha$,

$\gamma\alpha x=\begin{cases}\nu_{f}\alpha x&\text{ if }x\in\alpha p_{f}\\\
\alpha\nu_{i}x&\text{ if }x\in p_{i}\\\ \mu x&\text{ if }x\in[1,n]\setminus
p_{i}\\\ \alpha\mu^{-1}\alpha x&\text{ if }x\in[n+1,2n]\setminus\alpha
p_{f}\end{cases}\;.$ (232)

Let us review the definition of partial trace of an operator
$\eta\in\mathrm{End}(\mathbb{C}_{N}^{2n})$. The tensor coefficients of $\eta$
are defined by

$\eta_{i_{1},...i_{2n}}^{j_{1},...,j_{2n}}=\langle
e_{j_{1}},...,e_{j_{2n}}|\,\eta\,|e_{i_{1}},...,e_{i_{2n}}\rangle\,,$ (233)

where $|e_{i_{1}},...,e_{i_{2n}}\rangle$, for $1\leq i_{k}\leq N$ with $1\leq
k\leq 2n$, is a basis of ${\mathbb{C}}_{N}^{2n}$. In the case $\eta$ is a
permutation, the coefficients (233) can be evaluated as

$\eta_{i_{1},...,i_{2n}}^{j_{1},...,j_{2n}}=\delta_{i_{1}}^{j_{\eta(1)}}...\delta_{i_{2n}}^{j_{\eta(2n)}}=\delta_{l_{1}}^{j_{1}}...\delta_{l_{2n}}^{j_{2n}}\delta_{i_{1}}^{l_{\eta(1)}}...\delta_{i_{2n}}^{l_{\eta(2n)}}\;.$
(234)

We define the partial trace of $\eta$ as a trace over its last $n$ indices,

$\mathrm{P_{2}Tr}(\eta)_{i_{1},...,i_{n}}^{j_{1},...,j_{n}}=\sum_{k_{n+1},...,k_{2n}}\eta_{i_{1},...,i_{n},k_{n+1},...,k_{2n}}^{j_{1},...,j_{n},k_{n+1},...,k_{2n}}\;.$
(235)

If $\eta$ is a permutation in $S_{n}$ represented in
$\mathrm{End}(\mathbb{C}_{N}^{2n})$, its partial trace is equal to

$\mathrm{P_{2}Tr}(\eta)_{i_{1},...,i_{n}}^{j_{1},...,j_{n}}=\sum_{\begin{subarray}{c}k_{n+1},...,k_{2n}\\\
l_{1},...,l_{2n}\end{subarray}}\delta_{l_{1}}^{j_{1}}...\delta_{l_{n}}^{j_{n}}\delta_{l_{n+1}}^{k_{n+1}}...\delta_{l_{2n}}^{k_{2n}}\delta_{i_{1}}^{l_{\eta(1)}}...\delta_{i_{n}}^{l_{\eta(n)}}\delta_{k_{n+1}}^{l_{\eta(n+1)}}...\delta_{k_{2n}}^{l_{\eta(2n)}}\,,$
(236)

Let $\mathrm{cy}(\mu)$ be the number of cycles in $\mu$ (note that indeed
$\mathrm{cy}(\mu)=\mathrm{cy}(\mu^{-1})$). Checking case by case, we then see
that for $\eta=\gamma\alpha$,

$\mathrm{P_{2}Tr}(\gamma\alpha)x=\begin{cases}N^{\mathrm{cy}(\mu^{mu})-s}\,\mu\,x&\text{
if }x\in[1,n]\setminus p_{i}\\\
N^{\mathrm{cy}(\mu^{-1})-s}\,\nu_{f}\,\mu^{-k(x)}\,\nu_{i}\,x&\text{ if }x\in
p_{i}\end{cases}\;,$ (237)

where the power of N, which is $\mathrm{cy}(\mu)-s$, is the number of cycles
of $\alpha\gamma$ which contains only elements from $n+1$ to $2n$. This power
also counts the number of closed loops in the graphic representation, see Fig.
18. By noticing that the two cases are both equal to $\nu_{f}\mu\nu_{i}$
restricted to their conditions, simply we can write

$\mathrm{P_{2}Tr}(\gamma\alpha)=N^{\mathrm{cy}(\mu)-s}\,\nu_{f}\,\mu\,\nu_{i}\;.$
(238)

If $s=0$, then $\nu_{f}=\nu_{i}=id$ and $\gamma\alpha=\mu\otimes\mu^{-1}$,
therefore

$\mathrm{P_{2}Tr}(\alpha\gamma)=N^{{\mathrm{cy}}(\mu)}\mu\;.$ (239)

If $s>0$, the coefficient of $\nu_{f}\mu\nu_{i}$ is of lower order than from
the from $(\nu_{f}\,\mu\,\nu_{i})\otimes(\nu_{f}\,\mu\,\nu_{i})^{-1}$,
therefore

$\sum_{\gamma\in[2^{n}]}\mathrm{P_{2}Tr}(\gamma\alpha)=\sum_{\mu\in
S_{n}}\mathrm{P_{2}Tr}(\mu\otimes\mu^{-1})\;\left(1+{\mathcal{O}}(N^{-2})\right)=\sum_{\mu\in
S_{n}}N^{\mathrm{cy}(\mu)}\mu\;\left(1+{\mathcal{O}}(N^{-2})\right)\,,$ (240)

where the order is ${\mathcal{O}}(N^{-2})$ since $s$ is even.

∎

As an illustration of the proof above, we consider a permutation $\gamma$ in
$[2^{10}]$ given by

$\gamma=(1\;5)(2\;17)(3\;18)(4\;15)(6\;12)(7\;10)(8\;13)(9\;11)(14\;16)(19\;20)$
(241)

and is represented graphically in Fig. 19. For all the lines connecting the
bottom to the top, call $x$ and $y$ the values at the bottom and top of the
line, respectively. Each pair $x$ and $y$ falls into one of four distinct
cases: $1\leq x\leq 10$ and $1\leq y\leq 10$; $11\leq x\leq 20$ and $11\leq
y\leq 20$; $11\leq x\leq 20$ and $1\leq y\leq 10$; $1\leq x\leq 10$ and
$11\leq y\leq 20$. In Fig. 19 we highlight the first two cases in red, the
third case in green and the fourth case in pink.

Figure 19: An example of a permutation in $[2^{10}]$.

The sets $p_{i}$ and $p_{f}$ used in the lemma X.1 are $p_{i}=\\{4,6,9,10\\}$
and $p_{f}=\\{1,5,7,10\\}$. The permutation $\nu_{f}$ defined in (228) is the
permutation associated with the case $1\leq x\leq 10$ and $1\leq y\leq 10$,
shown in red on the left side of Fig. 19, and given by

$\nu_{f}=(1\;5)(7\;10)(2)(3)(4)(6)(8)(9)\,.$ (242)

The permutation $\nu_{i}$ defined in (229) is the permutation associated with
the case $11\leq x\leq 20$ and $11\leq y\leq 20$, shown in red on the right
side of Fig. 19, and given by

$\nu_{i}=(4\;6)(9\;10)(1)(2)(3)(5)(7)(8)\,.$ (243)

The permutation $\mu$ defined in (230) is the permutation associated with the
case $11\leq x\leq 20$ and $1\leq y\leq 10$, shown in green in Fig. 19, and
given by

$\mu=(1\;9)(5\;4)(7\;2\;6)(3\;8)(10)\,.$ (244)

The permutation $\gamma\alpha$ is

$\gamma\alpha=(1\;9\;20\;7\;2\;6\;14\;15)(5\;4\;16\;12\;17\;10\;19\;11)(3\;8)(18\;13)\;,$
(245)

and according to (238), its partial trace is

$\mathrm{P_{2}Tr}(\gamma\alpha)=N\;(1\;9\;7\;2\;6)(3\;8)(4\;10\;5)\,.$ (246)

## References

  * ’t Hooft and Veltman [1974] G. ’t Hooft and M. J. G. Veltman, One loop divergencies in the theory of gravitation, Ann. Inst. H. Poincare Phys. Theor. A 20, 69 (1974).
  * Christensen and Duff [1980] S. M. Christensen and M. J. Duff, Quantizing Gravity with a Cosmological Constant, Nucl. Phys. B 170, 480 (1980).
  * Goroff and Sagnotti [1986] M. H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266, 709 (1986).
  * Frohlich [1982] J. Frohlich, On the Triviality of Lambda (phi**4) in D-Dimensions Theories and the Approach to the Critical Point in D $>$= Four-Dimensions, Nucl. Phys. B 200, 281 (1982).
  * Gell-Mann and Low [1954] M. Gell-Mann and F. E. Low, Quantum electrodynamics at small distances, Phys. Rev. 95, 1300 (1954).
  * Gockeler  _et al._ [1998] M. Gockeler, R. Horsley, V. Linke, P. E. L. Rakow, G. Schierholz, and H. Stuben, Is there a Landau pole problem in QED?, Phys. Rev. Lett. 80, 4119 (1998), arXiv:hep-th/9712244 .
  * Callaway [1988] D. J. E. Callaway, Triviality Pursuit: Can Elementary Scalar Particles Exist?, Phys. Rept. 167, 241 (1988).
  * Donoghue [1994] J. F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72, 2996 (1994), arXiv:gr-qc/9310024 .
  * Donoghue [1995] J. F. Donoghue, Introduction to the effective field theory description of gravity, in  _Advanced School on Effective Theories_ (1995) arXiv:gr-qc/9512024 .
  * Burgess [2004] C. P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel. 7, 5 (2004), arXiv:gr-qc/0311082 .
  * Politzer [1973] H. D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30, 1346 (1973).
  * Gross and Wilczek [1973] D. J. Gross and F. Wilczek, Ultraviolet Behavior of Nonabelian Gauge Theories, Phys. Rev. Lett. 30, 1343 (1973).
  * Hawking and Israel [1979] S. W. Hawking and W. Israel,  _General Relativity: An Einstein Centenary Survey_ (Univ. Pr., Cambridge, UK, 1979).
  * Kawai and Ninomiya [1990] H. Kawai and M. Ninomiya, Renormalization Group and Quantum Gravity, Nucl. Phys. B 336, 115 (1990).
  * Kawai  _et al._ [1993a] H. Kawai, Y. Kitazawa, and M. Ninomiya, Scaling exponents in quantum gravity near two-dimensions, Nucl. Phys. B 393, 280 (1993a), arXiv:hep-th/9206081 .
  * Kawai  _et al._ [1993b] H. Kawai, Y. Kitazawa, and M. Ninomiya, Ultraviolet stable fixed point and scaling relations in (2+epsilon)-dimensional quantum gravity, Nucl. Phys. B 404, 684 (1993b), arXiv:hep-th/9303123 .
  * Kawai  _et al._ [1996] H. Kawai, Y. Kitazawa, and M. Ninomiya, Renormalizability of quantum gravity near two-dimensions, Nucl. Phys. B 467, 313 (1996), arXiv:hep-th/9511217 .
  * Reuter [1998] M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57, 971 (1998), arXiv:hep-th/9605030 .
  * Reuter and Saueressig [2012] M. Reuter and F. Saueressig, Quantum Einstein Gravity, New J. Phys. 14, 055022 (2012), arXiv:1202.2274 [hep-th] .
  * Eichhorn [2019] A. Eichhorn, An asymptotically safe guide to quantum gravity and matter, Front. Astron. Space Sci. 5, 47 (2019), arXiv:1810.07615 [hep-th] .
  * Bonanno  _et al._ [2020] A. Bonanno, A. Eichhorn, H. Gies, J. M. Pawlowski, R. Percacci, M. Reuter, F. Saueressig, and G. P. Vacca, Critical reflections on asymptotically safe gravity, Front. in Phys. 8, 269 (2020), arXiv:2004.06810 [gr-qc] .
  * Pawlowski and Reichert [2021] J. M. Pawlowski and M. Reichert, Quantum Gravity: A Fluctuating Point of View, Front. in Phys. 8, 551848 (2021), arXiv:2007.10353 [hep-th] .
  * Eichhorn and Schiffer [2022] A. Eichhorn and M. Schiffer, Asymptotic safety of gravity with matter, (2022), arXiv:2212.07456 [hep-th] .
  * Saueressig [2023] F. Saueressig, The Functional Renormalization Group in Quantum Gravity, (2023), arXiv:2302.14152 [hep-th] .
  * Ambjorn and Loll [1998] J. Ambjorn and R. Loll, Nonperturbative Lorentzian quantum gravity, causality and topology change, Nucl. Phys. B 536, 407 (1998), arXiv:hep-th/9805108 .
  * Ambjorn  _et al._ [2001] J. Ambjorn, J. Jurkiewicz, and R. Loll, Dynamically triangulating Lorentzian quantum gravity, Nucl. Phys. B 610, 347 (2001), arXiv:hep-th/0105267 .
  * Ambjorn  _et al._ [2012] J. Ambjorn, A. Goerlich, J. Jurkiewicz, and R. Loll, Nonperturbative Quantum Gravity, Phys. Rept. 519, 127 (2012), arXiv:1203.3591 [hep-th] .
  * Loll [2020] R. Loll, Quantum Gravity from Causal Dynamical Triangulations: A Review, Class. Quant. Grav. 37, 013002 (2020), arXiv:1905.08669 [hep-th] .
  * Manrique  _et al._ [2011] E. Manrique, S. Rechenberger, and F. Saueressig, Asymptotically Safe Lorentzian Gravity, Phys. Rev. Lett. 106, 251302 (2011), arXiv:1102.5012 [hep-th] .
  * Biemans  _et al._ [2017] J. Biemans, A. Platania, and F. Saueressig, Quantum gravity on foliated spacetimes: Asymptotically safe and sound, Phys. Rev. D 95, 086013 (2017), arXiv:1609.04813 [hep-th] .
  * Bonanno  _et al._ [2022] A. Bonanno, T. Denz, J. M. Pawlowski, and M. Reichert, Reconstructing the graviton, SciPost Phys. 12, 001 (2022), arXiv:2102.02217 [hep-th] .
  * Fehre  _et al._ [2023] J. Fehre, D. F. Litim, J. M. Pawlowski, and M. Reichert, Lorentzian Quantum Gravity and the Graviton Spectral Function, Phys. Rev. Lett. 130, 081501 (2023), arXiv:2111.13232 [hep-th] .
  * Saueressig and Wang [2023] F. Saueressig and J. Wang, Foliated asymptotically safe gravity in the fluctuation approach, JHEP 09, 064, arXiv:2306.10408 [hep-th] .
  * Di Francesco  _et al._ [1995] P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, 2-D Gravity and random matrices, Phys. Rept. 254, 1 (1995), arXiv:hep-th/9306153 .
  * Nakayama [2004] Y. Nakayama, Liouville field theory: A Decade after the revolution, Int. J. Mod. Phys. A 19, 2771 (2004), arXiv:hep-th/0402009 .
  * David [1985a] F. David, Planar Diagrams, Two-Dimensional Lattice Gravity and Surface Models, Nucl. Phys. B 257, 45 (1985a).
  * David [1985b] F. David, A model of random surfaces with non-trivial critical behaviour, Nuclear Physics B 257, 543 (1985b).
  * Kazakov  _et al._ [1985a] V. A. Kazakov, A. A. Migdal, and I. K. Kostov, Critical Properties of Randomly Triangulated Planar Random Surfaces, Phys. Lett. B 157, 295 (1985a).
  * Douglas and Shenker [1990] M. R. Douglas and S. H. Shenker, Strings in Less Than One-Dimension, Nucl. Phys. B 335, 635 (1990).
  * Brezin and Kazakov [1990] E. Brezin and V. A. Kazakov, Exactly Solvable Field Theories of Closed Strings, Phys. Lett. B 236, 144 (1990).
  * Gross and Migdal [1990a] D. J. Gross and A. A. Migdal, Nonperturbative Two-Dimensional Quantum Gravity, Phys. Rev. Lett. 64, 127 (1990a).
  * Gross and Migdal [1990b] D. J. Gross and A. A. Migdal, A Nonperturbative Treatment of Two-dimensional Quantum Gravity, Nucl. Phys. B 340, 333 (1990b).
  * Ginsparg and Moore [1993] P. H. Ginsparg and G. W. Moore, Lectures on 2-D gravity and 2-D string theory, in  _Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles_ (1993) pp. 277–469, arXiv:hep-th/9304011 .
  * Gurau [2011a] R. Gurau, Colored Group Field Theory, Commun. Math. Phys. 304, 69 (2011a), arXiv:0907.2582 [hep-th] .
  * Gurau [2011b] R. Gurau, The 1/N expansion of colored tensor models, Annales Henri Poincare 12, 829 (2011b), arXiv:1011.2726 [gr-qc] .
  * Gurau and Rivasseau [2011] R. Gurau and V. Rivasseau, The 1/N expansion of colored tensor models in arbitrary dimension, EPL 95, 50004 (2011), arXiv:1101.4182 [gr-qc] .
  * Gurau and Ryan [2012] R. Gurau and J. P. Ryan, Colored Tensor Models - a review, SIGMA 8, 020 (2012), arXiv:1109.4812 [hep-th] .
  * Gurau and Ryan [2014] R. Gurau and J. P. Ryan, Melons are branched polymers, Annales Henri Poincare 15, 2085 (2014), arXiv:1302.4386 [math-ph] .
  * Loll [1998] R. Loll, Discrete approaches to quantum gravity in four-dimensions, Living Rev. Rel. 1, 13 (1998), arXiv:gr-qc/9805049 .
  * Ambjorn and Jurkiewicz [1995] J. Ambjorn and J. Jurkiewicz, Scaling in four-dimensional quantum gravity, Nucl. Phys. B 451, 643 (1995), arXiv:hep-th/9503006 .
  * Bialas  _et al._ [1996] P. Bialas, Z. Burda, A. Krzywicki, and B. Petersson, Focusing on the fixed point of 4-D simplicial gravity, Nucl. Phys. B 472, 293 (1996), arXiv:hep-lat/9601024 .
  * de Bakker [1996] B. V. de Bakker, Further evidence that the transition of 4-D dynamical triangulation is first order, Phys. Lett. B 389, 238 (1996), arXiv:hep-lat/9603024 .
  * Laiho and Coumbe [2011] J. Laiho and D. Coumbe, Evidence for Asymptotic Safety from Lattice Quantum Gravity, Phys. Rev. Lett. 107, 161301 (2011), arXiv:1104.5505 [hep-lat] .
  * Laiho  _et al._ [2017] J. Laiho, S. Bassler, D. Coumbe, D. Du, and J. T. Neelakanta, Lattice Quantum Gravity and Asymptotic Safety, Phys. Rev. D 96, 064015 (2017), arXiv:1604.02745 [hep-th] .
  * Dai  _et al._ [2021] M. Dai, J. Laiho, M. Schiffer, and J. Unmuth-Yockey, Newtonian binding from lattice quantum gravity, Phys. Rev. D 103, 114511 (2021), arXiv:2102.04492 [hep-lat] .
  * Bassler  _et al._ [2021] S. Bassler, J. Laiho, M. Schiffer, and J. Unmuth-Yockey, The de Sitter Instanton from Euclidean Dynamical Triangulations, Phys. Rev. D 103, 114504 (2021), arXiv:2103.06973 [hep-lat] .
  * Dai  _et al._ [2023] M. Dai, W. Freeman, J. Laiho, M. Schiffer, and J. Unmuth-Yockey, An improved algorithm for dynamical triangulations and simulations of finer lattices, (2023), arXiv:2309.12257 [hep-lat] .
  * Asaduzzaman and Catterall [2023] M. Asaduzzaman and S. Catterall, Euclidean dynamical triangulations revisited, Phys. Rev. D 107, 074505 (2023), arXiv:2207.12642 [hep-lat] .
  * Hamber [2009] H. W. Hamber, Quantum Gravity on the Lattice, Gen. Rel. Grav. 41, 817 (2009), arXiv:0901.0964 [gr-qc] .
  * Hamber  _et al._ [2013] H. W. Hamber, R. Toriumi, and R. M. Williams, Wheeler-DeWitt equation in 3+1 dimensions, Phys. Rev. D 88, 084012 (2013), arXiv:1212.3492 [hep-th] .
  * Hamber [2015] H. W. Hamber, Scaling Exponents for Lattice Quantum Gravity in Four Dimensions, Phys. Rev. D 92, 064017 (2015), arXiv:1506.07795 [hep-th] .
  * Eichhorn and Koslowski [2018] A. Eichhorn and T. Koslowski, Flowing to the continuum limit in tensor models for quantum gravity, Ann. Inst. H. Poincare D Comb. Phys. Interact. 5, 173 (2018), arXiv:1701.03029 [gr-qc] .
  * Eichhorn  _et al._ [2019a] A. Eichhorn, T. Koslowski, J. Lumma, and A. D. Pereira, Towards background independent quantum gravity with tensor models, Class. Quant. Grav. 36, 155007 (2019a), arXiv:1811.00814 [gr-qc] .
  * Eichhorn  _et al._ [2019b] A. Eichhorn, T. Koslowski, and A. D. Pereira, Status of background-independent coarse-graining in tensor models for quantum gravity, Universe 5, 53 (2019b), arXiv:1811.12909 [gr-qc] .
  * Eichhorn  _et al._ [2020a] A. Eichhorn, J. Lumma, A. D. Pereira, and A. Sikandar, Universal critical behavior in tensor models for four-dimensional quantum gravity, JHEP 02, 110, arXiv:1912.05314 [gr-qc] .
  * Ambjorn  _et al._ [2010] J. Ambjorn, A. Gorlich, J. Jurkiewicz, and R. Loll, CDT - an Entropic Theory of Quantum Gravity, in  _International Workshop on Continuum and Lattice Approaches to Quantum Gravity_ (2010) arXiv:1007.2560 [hep-th] .
  * Ambjorn  _et al._ [2000] J. Ambjorn, J. Jurkiewicz, and R. Loll, A Nonperturbative Lorentzian path integral for gravity, Phys. Rev. Lett. 85, 924 (2000), arXiv:hep-th/0002050 .
  * Ambjorn  _et al._ [2004] J. Ambjorn, J. Jurkiewicz, and R. Loll, Emergence of a 4-D world from causal quantum gravity, Phys. Rev. Lett. 93, 131301 (2004), arXiv:hep-th/0404156 .
  * Ambjorn  _et al._ [2005] J. Ambjorn, J. Jurkiewicz, and R. Loll, Reconstructing the universe, Phys. Rev. D 72, 064014 (2005), arXiv:hep-th/0505154 .
  * Ambjorn  _et al._ [2008] J. Ambjorn, A. Gorlich, J. Jurkiewicz, and R. Loll, Planckian Birth of the Quantum de Sitter Universe, Phys. Rev. Lett. 100, 091304 (2008), arXiv:0712.2485 [hep-th] .
  * Ambjorn  _et al._ [2011a] J. Ambjorn, A. Gorlich, J. Jurkiewicz, R. Loll, J. Gizbert-Studnicki, and T. Trzesniewski, The Semiclassical Limit of Causal Dynamical Triangulations, Nucl. Phys. B 849, 144 (2011a), arXiv:1102.3929 [hep-th] .
  * Ambjorn  _et al._ [2011b] J. Ambjorn, S. Jordan, J. Jurkiewicz, and R. Loll, A Second-order phase transition in CDT, Phys. Rev. Lett. 107, 211303 (2011b), arXiv:1108.3932 [hep-th] .
  * Jordan and Loll [2013] S. Jordan and R. Loll, De Sitter Universe from Causal Dynamical Triangulations without Preferred Foliation, Phys. Rev. D 88, 044055 (2013), arXiv:1307.5469 [hep-th] .
  * Ambjorn  _et al._ [2020] J. Ambjorn, J. Gizbert-Studnicki, A. Görlich, J. Jurkiewicz, and R. Loll, Renormalization in quantum theories of geometry, Front. in Phys. 8, 247 (2020), arXiv:2002.01693 [hep-th] .
  * Benedetti [2023] D. Benedetti, Landau Theory of Causal Dynamical Triangulations (2023) arXiv:2212.11043 [hep-th] .
  * Benedetti and Henson [2009] D. Benedetti and J. Henson, Imposing causality on a matrix model, Phys. Lett. B 678, 222 (2009), arXiv:0812.4261 [hep-th] .
  * Di Francesco and Itzykson [1993] P. Di Francesco and C. Itzykson, A Generating function for fatgraphs, Ann. Inst. H. Poincare Phys. Theor. 59, 117 (1993), arXiv:hep-th/9212108 .
  * Kazakov  _et al._ [1996a] V. A. Kazakov, M. Staudacher, and T. Wynter, Character expansion methods for matrix models of dually weighted graphs, Communications in Mathematical Physics 177, 451 (1996a).
  * Kazakov  _et al._ [1996b] V. A. Kazakov, M. Staudacher, and T. Wynter, Almost flat planar diagrams, Commun. Math. Phys. 179, 235 (1996b), arXiv:hep-th/9506174 .
  * Kazakov  _et al._ [1996c] V. A. Kazakov, M. Staudacher, and T. Wynter, Exact solution of discrete two-dimensional R**2 gravity, Nucl. Phys. B 471, 309 (1996c), arXiv:hep-th/9601069 .
  * Oriti [2011] D. Oriti, The microscopic dynamics of quantum space as a group field theory, in  _Foundations of Space and Time: Reflections on Quantum Gravity_ (2011) pp. 257–320, arXiv:1110.5606 [hep-th] .
  * Jercher  _et al._ [2022] A. F. Jercher, D. Oriti, and A. G. A. Pithis, Complete Barrett-Crane model and its causal structure, Phys. Rev. D 106, 066019 (2022), arXiv:2206.15442 [gr-qc] .
  * Marchetti  _et al._ [2023a] L. Marchetti, D. Oriti, A. G. A. Pithis, and J. Thürigen, Phase transitions in TGFT: a Landau-Ginzburg analysis of Lorentzian quantum geometric models, JHEP 02, 074, arXiv:2209.04297 [gr-qc] .
  * Marchetti  _et al._ [2023b] L. Marchetti, D. Oriti, A. G. A. Pithis, and J. Thürigen, Mean-Field Phase Transitions in Tensorial Group Field Theory Quantum Gravity, Phys. Rev. Lett. 130, 141501 (2023b), arXiv:2211.12768 [gr-qc] .
  * Knorr  _et al._ [2022] B. Knorr, C. Ripken, and F. Saueressig, Form Factors in Asymptotically Safe Quantum Gravity, (2022), arXiv:2210.16072 [hep-th] .
  * Pawlowski and Reichert [2023] J. M. Pawlowski and M. Reichert, Quantum Gravity from dynamical metric fluctuations, (2023), arXiv:2309.10785 [hep-th] .
  * Ambjorn [2022] J. Ambjorn, Lattice Quantum Gravity: EDT and CDT (2022) arXiv:2209.06555 [hep-lat] .
  * Kazakov [1986a] V. A. Kazakov, Exact Solution of the Ising Model on a Random Two-dimensional Lattice, JETP Lett. 44, 133 (1986a).
  * Boulatov and Kazakov [1987] D. V. Boulatov and V. A. Kazakov, The Ising Model on Random Planar Lattice: The Structure of Phase Transition and the Exact Critical Exponents, Phys. Lett. B 186, 379 (1987).
  * Kazakov [1986b] V. A. Kazakov, Ising model on a dynamical planar random lattice: Exact solution, Phys. Lett. A 119, 140 (1986b).
  * Brezin  _et al._ [1990] E. Brezin, M. R. Douglas, V. Kazakov, and S. H. Shenker, The Ising Model Coupled to 2-$D$ Gravity: A Nonperturbative Analysis, Phys. Lett. B 237, 43 (1990).
  * Bonzom  _et al._ [2012] V. Bonzom, R. Gurau, and V. Rivasseau, The Ising Model on Random Lattices in Arbitrary Dimensions, Phys. Lett. B 711, 88 (2012), arXiv:1108.6269 [hep-th] .
  * Sasakura and Sato [2014] N. Sasakura and Y. Sato, Ising model on random networks and the canonical tensor model, PTEP 2014, 053B03 (2014), arXiv:1401.7806 [hep-th] .
  * Lahoche  _et al._ [2020] V. Lahoche, D. Ousmane Samary, and A. D. Pereira, Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices, Phys. Rev. D 101, 064014 (2020), arXiv:1911.05173 [hep-th] .
  * Ambjorn  _et al._ [1999] J. Ambjorn, K. N. Anagnostopoulos, and R. Loll, A New perspective on matter coupling in 2-D quantum gravity, Phys. Rev. D 60, 104035 (1999), arXiv:hep-th/9904012 .
  * Benedetti and Loll [2007a] D. Benedetti and R. Loll, Unexpected spin-off from quantum gravity, Physica A 377, 373 (2007a), arXiv:hep-lat/0603013 .
  * Benedetti and Loll [2007b] D. Benedetti and R. Loll, Quantum gravity and matter: Counting graphs on causal dynamical triangulations, Gen. Rel. Grav. 39, 863 (2007b), arXiv:gr-qc/0611075 .
  * Napolitano and Turova [2016] G. M. Napolitano and T. Turova, The Ising model on the random planar causal triangulation: bounds on the critical line and magnetization properties, J. Statist. Phys. 162, 739 (2016), arXiv:1504.03828 [cond-mat.stat-mech] .
  * Sato and Tanaka [2018] Y. Sato and T. Tanaka, Criticality at absolute zero from Ising model on two-dimensional dynamical triangulations, Phys. Rev. D 98, 026026 (2018), arXiv:1710.03402 [hep-th] .
  * Cerda-Hernández [2017] J. Cerda-Hernández, Critical region for an Ising model coupled to causal triangulations, J. Stat. Mech. 1702, 023209 (2017).
  * Ambjørn  _et al._ [2020] J. Ambjørn, Y. Sato, and T. Tanaka, Towards elucidation of zero-temperature criticality of the Ising model on 2D dynamical triangulations, Phys. Rev. D 101, 106019 (2020), arXiv:2003.08524 [hep-th] .
  * Eichhorn and Koslowski [2013] A. Eichhorn and T. Koslowski, Continuum limit in matrix models for quantum gravity from the Functional Renormalization Group, Phys. Rev. D 88, 084016 (2013), arXiv:1309.1690 [gr-qc] .
  * Eichhorn and Koslowski [2014] A. Eichhorn and T. Koslowski, Towards phase transitions between discrete and continuum quantum spacetime from the Renormalization Group, Phys. Rev. D 90, 104039 (2014), arXiv:1408.4127 [gr-qc] .
  * Lahoche and Ousmane Samary [2020] V. Lahoche and D. Ousmane Samary, Revisited functional renormalization group approach for random matrices in the large-$N$ limit, Phys. Rev. D 101, 106015 (2020), arXiv:1909.03327 [hep-th] .
  * Perez-Sanchez [2021] C. I. Perez-Sanchez, On Multimatrix Models Motivated by Random Noncommutative Geometry I: The Functional Renormalization Group as a Flow in the Free Algebra, Annales Henri Poincare 22, 3095 (2021), arXiv:2007.10914 [math-ph] .
  * Eichhorn  _et al._ [2020b] A. Eichhorn, A. D. Pereira, and A. G. A. Pithis, The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization, JHEP 12, 131, arXiv:2009.05111 [gr-qc] .
  * Castro and Koslowski [2021] A. Castro and T. Koslowski, Renormalization Group Approach to the Continuum Limit of Matrix Models of Quantum Gravity with Preferred Foliation, Front. in Phys. 9, 114 (2021), arXiv:2008.10090 [gr-qc] .
  * Ambjørn and Loll [1998] J. Ambjørn and R. Loll, Non-perturbative lorentzian quantum gravity, causality and topology change, Nuclear Physics B 536, 407 (1998).
  * Itzykson and Zuber [1980] C. Itzykson and J. B. Zuber, The Planar Approximation. 2., J. Math. Phys. 21, 411 (1980).
  * Mehta and Mahoux [1990] M. L. Mehta and G. Mahoux, A Method of integration over matrix variables: 3., (1990).
  * Eynard [1997] B. Eynard, Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices, Nuclear Physics B 506, 633 (1997).
  * Eynard [2003] B. Eynard, Large N expansion of the 2 matrix model, JHEP 01, 051, arXiv:hep-th/0210047 .
  * Bertola and Eynard [2003] M. Bertola and B. Eynard, Mixed correlation functions of the two matrix model, J. Phys. A 36, 7733 (2003), arXiv:hep-th/0303161 .
  * Kazakov [1986c] V. Kazakov, Ising model on a dynamical planar random lattice: Exact solution, Physics Letters A 119, 140 (1986c).
  * Mehta [2004] M. Mehta,  _Random Matrices_ , ISSN (Elsevier Science, 2004).
  * Walker [2003] P. Walker, The zeros of the partial sums of the exponential series, The American Mathematical Monthly 110, 337 (2003).
  * Zemyan [2005] S. M. Zemyan, On the zeroes of the nth partial sum of the exponential series, The American Mathematical Monthly 112, 891 (2005).
  * Collins  _et al._ [2021] B. Collins, S. Matsumoto, and J. Novak, The weingarten calculus (2021), arXiv:2109.14890 [math-ph] .
  * Francesco and Itzykson [1992] P. D. Francesco and C. Itzykson, A generating function for fatgraphs (1992), arXiv:hep-th/9212108 [hep-th] .
  * Eynard  _et al._ [2015] B. Eynard, T. Kimura, and S. Ribault, Random matrices, (2015), arXiv:1510.04430 [math-ph] .
  * Anninos and Mühlmann [2020] D. Anninos and B. Mühlmann, Notes on matrix models (matrix musings), J. Stat. Mech. 2008, 083109 (2020), arXiv:2004.01171 [hep-th] .
  * Zhang [2017] L. Zhang, Volumes of orthogonal groups and unitary groups (2017), arXiv:1509.00537 [math-ph] .
  * de Bruijn [1955] N. G. de Bruijn, On some multiple integrals involving determinants, Journal of the Indian Mathematical Society. New Series 19, 133 (1955).
  * Macdonald [1998] I. Macdonald,  _Symmetric Functions and Hall Polynomials_ , Oxford classic texts in the physical sciences (Clarendon Press, 1998).
  * Schilling [2017] R. Schilling,  _Measures, Integrals and Martingales_ , Measures, Integrals and Martingales (Cambridge University Press, 2017).
  * Bleistein and Handelsman [1986] N. Bleistein and R. Handelsman,  _Asymptotic Expansions of Integrals_ , Dover Books on Mathematics Series (Dover Publications, 1986).
  * Hamber  _et al._ [2012] H. W. Hamber, R. Toriumi, and R. M. Williams, Wheeler-DeWitt Equation in 2 + 1 Dimensions, Phys. Rev. D 86, 084010 (2012), arXiv:1207.3759 [hep-th] .
  * Baldazzi  _et al._ [2019] A. Baldazzi, R. Percacci, and V. Skrinjar, Wicked metrics, Class. Quant. Grav. 36, 105008 (2019), arXiv:1811.03369 [gr-qc] .
  * Visser [2022] M. Visser, Feynman’s i$\epsilon$ prescription, almost real spacetimes, and acceptable complex spacetimes, JHEP 08, 129, arXiv:2111.14016 [gr-qc] .
  * Gérard  _et al._ [2022] C. Gérard, S. Murro, and M. Wrochna, Quantization of linearized gravity by Wick rotation in Gaussian time, (2022), arXiv:2204.01094 [math-ph] .
  * Kontsevich and Segal [2021] M. Kontsevich and G. Segal, Wick Rotation and the Positivity of Energy in Quantum Field Theory, Quart. J. Math. Oxford Ser. 72, 673 (2021), arXiv:2105.10161 [hep-th] .
  * ’t Hooft [1974] G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72, 461 (1974).
  * Francesco  _et al._ [1995] P. Francesco, P. Ginsparg, and J. Zinn-Justin, 2d gravity and random matrices, Physics Reports 254, 1 (1995).
  * Brezin  _et al._ [1978] E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Planar Diagrams, Commun. Math. Phys. 59, 35 (1978).
  * Kazakov  _et al._ [1985b] V. Kazakov, I. Kostov, and A. Migdal, Critical properties of randomly triangulated planar random surfaces, Physics Letters B 157, 295 (1985b).
  * Brézin and Kazakov [1990] E. Brézin and V. Kazakov, Exactly solvable field theories of closed strings, Physics Letters B 236, 144 (1990).
  * Shenker [1991] S. H. Shenker, The strength of nonperturbative effects in string theory, in  _Random Surfaces and Quantum Gravity_ , edited by O. Alvarez, E. Marinari, and P. Windey (Springer US, Boston, MA, 1991) pp. 191–200.
  * Eynard and Orantin [2007] B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Num. Theor. Phys. 1, 347 (2007), arXiv:math-ph/0702045 .
  * Miller and Sheffield [2021] J. Miller and S. Sheffield, Liouville quantum gravity and the brownian map ii: geodesics and continuity of the embedding (2021), arXiv:1605.03563 [math.PR] .
  * Miller and Sheffield [2016] J. Miller and S. Sheffield, Liouville quantum gravity and the Brownian map III: the conformal structure is determined, (2016), arXiv:1608.05391 [math.PR] .
  * Marckert and Mokkadem [2006] J.-F. Marckert and A. Mokkadem, Limit of normalized quadrangulations: The brownian map, The Annals of Probability 34, 10.1214/009117906000000557 (2006).
  * Gall and Paulin [2006] J. F. L. Gall and F. Paulin, Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere (2006), arXiv:math/0612315 [math.PR] .
  * Gall [2007] J.-F. L. Gall, The topological structure of scaling limits of large planar maps, Inventiones mathematicae 169, 621 (2007).
  * Gall and Miermont [2012] J.-F. L. Gall and G. Miermont, Scaling limits of random trees and planar maps (2012), arXiv:1101.4856 [math.PR] .
  * Bonzom  _et al._ [2011] V. Bonzom, R. Gurau, A. Riello, and V. Rivasseau, Critical behavior of colored tensor models in the large N limit, Nucl. Phys. B 853, 174 (2011), arXiv:1105.3122 [hep-th] .
  * Rivasseau [2012a] V. Rivasseau, Quantum Gravity and Renormalization: The Tensor Track, AIP Conf. Proc. 1444, 18 (2012a), arXiv:1112.5104 [hep-th] .
  * Rivasseau [2012b] V. Rivasseau, The Tensor Track: an Update, in  _29th International Colloquium on Group-Theoretical Methods in Physics_ (2012) arXiv:1209.5284 [hep-th] .
  * Rivasseau [2014] V. Rivasseau, The Tensor Track, III, Fortsch. Phys. 62, 81 (2014), arXiv:1311.1461 [hep-th] .
  * Rivasseau  _et al._ [2016] V. Rivasseau, , and and, Random tensors and quantum gravity, Symmetry, Integrability and Geometry: Methods and Applications 10.3842/sigma.2016.069 (2016).
  * Rivasseau [2016] V. Rivasseau, The Tensor Track, IV, PoS CORFU2015, 106 (2016), arXiv:1604.07860 [hep-th] .
  * Collins  _et al._ [2022] B. Collins, R. Gurau, and L. Lionni, The tensor harish-chandra-itzykson-zuber integral i: Weingarten calculus and a generalization of monotone hurwitz numbers (2022), arXiv:2010.13661 [math.CO] .
  * Kazakov and Zinn-Justin [1999] V. A. Kazakov and P. Zinn-Justin, Two-matrix model with ABAB interaction, Nuclear Physics B 546, 647 (1999).
  * Douglas and Kazakov [1993] M. R. Douglas and V. A. Kazakov, Large n phase transition in continuum QCD2, Physics Letters B 319, 219 (1993).
  * Bulgakova [2020] D. V. Bulgakova,  _Some Aspects Of Representation Theory Of Walled Brauer Algebras_ , Theses, Aix Marseille Université (2020).