# Personalization of Dataset Retrieval Results using a Metadata-based Data
Valuation Method

Malick Ebiele<EMAIL_ADDRESS>0000-0001-5019-6839 University
College DublinDublinIreland ,  Malika Bendechache
<EMAIL_ADDRESS>University of GalwayGalwayIreland ,
Eamonn Clinton<EMAIL_ADDRESS>Tailte ÉireannDublinIreland and  Rob
Brennan<EMAIL_ADDRESS>University College DublinDublinIreland

(2024)

###### Abstract.

In this paper, we propose a novel data valuation method for a Dataset
Retrieval (DR) use case in Ireland’s National mapping agency. To the best of
our knowledge, data valuation has not yet been applied to Dataset Retrieval.
By leveraging metadata and a user’s preferences, we estimate the personal
value of each dataset to facilitate dataset retrieval and filtering. We then
validated the data value-based ranking against the stakeholders’ ranking of
the datasets. The proposed data valuation method and use case demonstrated
that data valuation is promising for dataset retrieval. For instance, the
outperforming dataset retrieval based on our approach obtained 0.8207 in terms
of NDCG@5 (the truncated Normalized Discounted Cumulative Gain at 5). This
study is unique in its exploration of a data valuation-based approach to
dataset retrieval and stands out because, unlike most existing methods, our
approach is validated using the stakeholders’ ranking of the datasets.

Data valuation, Data value, Personalized data value, Dataset retrieval,
Quantitative data valuation

††copyright: acmlicensed††journalyear: 2024††doi: XXXXXXX.XXXXXXX††conference:
Make sure to enter the correct conference title from your rights confirmation
email; October 21–25, 2024; Boise, Idaho, USA††ccs: Information systems
Evaluation of retrieval results††ccs: Information systems Relevance assessment

##  1\. Introduction

Given rapidly rising data volumes, knowing which data to keep and which to
discard has become an essential task. Data valuation has emerged as a
promising approach to tackle this problem (Even and Shankaranarayanan, 2005).
The primary focus of data valuation research is the development of
methodologies for determining the value of data (Wang et al., 2020, 2021;
Laney, 2017; Turczyk et al., 2007; Khokhlov and Reznik, 2020; Qiu et al.,
2017).

Data valuation methods have been applied to data management, machine learning,
system security, and energy (Wang et al., 2020; Turczyk et al., 2007; Khokhlov
and Reznik, 2020; Wang et al., 2021), to name a few. No previous attempts to
apply data valuation to dataset retrieval. Dataset retrieval is a
specialization of information retrieval with the difference that instead of
retrieving relevant documents the engine returns a list of relevant datasets
(Kunze and Auer, 2013). Dataset retrieval systems will return relevant
datasets according to a given query. However, they do not take into account
the preferences of the user in terms of metadata. Some dataset retrieval
softwares allow users to sort the results by each metadata separately like
creation date, usage, and last update. But none of them allow the users to
sort the results by a combination of those metadata. In this paper, we
proposed a metadata-based data valuation method that will allow users to sort
dataset retrieval results using a combination of metadata.

Many of the existing data valuation approaches are subjective due to the
subjective nature of some dimensions that characterize data value (Attard and
Brennan, 2018) or the subjective weighting techniques (in the case of weighted
averaging or summing) (Odu, 2019; Deng et al., 2023). Subjective metrics of
data value dimensions or weighting techniques can only be defined by
individual users or experts based on their personal views, experiences, and
backgrounds. These are opposed to objective metrics that can be determined
precisely on the basis of a detailed analysis of the data or extracted from
the data infrastructure (Bodendorf et al., 2022). This makes it challenging to
develop a fully objective data valuation model. We believe that instead of
generalizing subjective metrics and weighting techniques, it would be better
to attempt to develop personalized data valuation models.

Choosing a suitable weighting technique is an additional challenge for
weighted approaches. For instance, usage-over-time is one of the first data
valuation methods and developed a weighting technique based on recency (Chen,
2005). The recency-based weighting technique is per objective. However, the
choice of using recency-based as a weighting technique is subjective or the
choice of assigning high or low weights to the more recency metadata is also
subjective. In our case, the desired weighting technique should be subjective
and straightforward for the stakeholders to interact with.

The research question is: To what extent can metadata-based data valuation
methods improve the results of dataset retrieval systems in terms of ranking
performance?

To answer the research question above, we designed and implemented a metadata-
based data valuation method and applied it to a dataset retrieval use case for
a National Mapping Agency. The goal is to make retrieval in a data catalog
more effective by correctly identifying the datasets that are more valuable to
customers by taking into account the customers’ dataset preferences.

The contributions of this paper are as follows:

  * •

First attempt to apply a metadata-based data valuation method to dataset
retrieval.

  * •

Proposed a personalized and interactive data valuation method. Extant methods
are mainly subjective approaches.

The remainder of this paper is structured as follows. Section 2 gives a
description of the use case. Section 3 describes the related work. Our
proposed metadata-based data valuation method is explained in Section 4.
Section 5 explains our experimental design. In Section 6, the experimental
results are shown and discussed. Finally, the conclusion and future work are
presented in Section 7.

##  2\. Related Work

This section covers different approaches to metadata-based data valuation
methods. It highlights their challenges especially challenges related to
weighted average approaches because the approach proposed in this paper falls
into that category.

Overview of data valuation methods.Wijnhoven et al. ((2014)) (Wijnhoven et
al., 2014) used file age and last modification time metadata to train machine
learning models to classify files as waste (or nonwaste). Random Forest
achieved a performance of 84% accuracy using 300 data points (20% for
training). The main problem with this approach is the scarcity of labeled
data. To achieve better performance, more data will be needed. The main reason
Random Forest achieved 84% accuracy is because the classification problem is
binary classification with only two dependent variables which is quite
straightforward. In many cases, there are more than two dependent variables
and classes. Another approach is Fuzzy expert system. Hanratty et al. ((2013))
(Hanratty et al., 2013) used Fuzzy Associative Memory (FAM) with three inputs:
source reliability, information content, and timeliness to calculate the value
of information [data] in a military application context. There are two options
here: either subjectively choosing the membership functions of the Fuzzy
system or fine-tuning using data to objectively select the membership
functions. This is not the direction we would like to follow in this paper
because our first choice will be fine-tuning the Fuzzy system using data which
will require a lot more data than we can currently collect. Probability and
Statistics and Matrix multiplication are other approaches to metadata-based
data valuation. Probability and Statistics are applied to estimate the
probability of further use of a dataset (Turczyk et al., 2007). As to matrix
multiplication, it is chosen in the context of two-level indicators data
valuation (Chen et al., 2011; Fang, 2016). These approaches are not relevant
because our proposed approach is not a two-level data valuation method and we
are also not trying to predict the probability of future use of the datasets.

Weighted average data valuation methods. There were also previous attempts to
calculate the data value using weighted averaging (Chen, 2005; Ma and Zhang,
2019; Qiu et al., 2017). For instance, usage-over-time is one of the first
data valuation methods used and it estimates data value with a weighted
averaging approach (Chen, 2005). It consists of splitting the usage data into
a series of time slots, assigning a weight to each time slot, and then
computing the data value using the weighted average. The weights are the
normalized recency weights. The more recent time slots are assigned the higher
weights (Chen, 2005). Ma and Zhang ((2019)) (Ma and Zhang, 2019) extended the
usage-over-time model by adding the age and size dimensions. Their Multi-
Factors Data Valuation Method ($MDV$) is a trade-off between dynamic and
static data value. The dynamic data value is the usage-over-time model of
Chen. The static data value is the weighted average of the normalized age and
size. The weights of the age and size are assigned subjectively by experts.

Qiu et al. ((2017)) (Qiu et al., 2017) used the Analytical Hierarchy Process
(AHP) which is a different weighting approach. AHP requires a subjective
rating of the input dimensions in pairs. These pairwise comparisons are then
arranged in a matrix, from which a final weighting of the dimensions will be
calculated. AHP is technically straightforward to implement and more
importantly allows to assess the transitivity consistency of the pairwise
comparisons matrix by assigning a consistency score to it. However, experts
are still needed for the pairwise rating of the input dimensions. Qiu et al.
((2017)) (Qiu et al., 2017) use the measure of 6 dimensions in their model.
For more details on the dimensions used, please refer to (Qiu et al., 2017).

The challenge of weighted approaches is the weighting technique. In our case,
the desired weighting technique should be straightforward for the stakeholders
to interact with. The weighting approach used in this paper is detailed in
Section 4.1.

To the best of our knowledge, the data valuation approach proposed in this
study and the application of data value to dataset retrieval is unique. Also,
none of the studies described above validated the outputs of their data
valuation approaches. Our approach is validated using data collected from
stakeholders.

##  3\. Proposed Data Valuation Method

###  3.1. Weight Determination

Analytic Hierarchy Process (AHP) was our first choice because of its sound
mathematical basis (Saaty, 1987). However, it was challenging to interact
with, as instead of assigning a weight to each metadata or dimension, a
pairwise comparison of the dimensions is needed (Saaty, 1987). E.g. usage is
twice as important as creation date, usage is 5 times more important than
number of spatial objects, or usage is twice less important than utility. This
exercise was difficult for the stakeholders who participated in the
experiments. They confessed to being more comfortable with a rating-like
weighting approach e.g. 1 to 5 websites or products rating mechanism. Also,
AHP assumes that preferences are transitive and has a transitivity consistency
test. Saaty ((1987)) (Saaty, 1987) advise to discard the current weights
deduced from the pairwise comparisons if the consistency ratio is greater than
0.1. There are previous studies showing that preferences are not always
transitive (Alós-Ferrer et al., 2023; Alós-Ferrer and Garagnani, 2021; Gendin,
1996; Fishburn, 1991). Alós-Ferrer et al. ((2023)) (Alós-Ferrer et al., 2023)
shows using two preference datasets that no matter the initial assumptions,
even when the preferences are supposed to be transitive, a maximum of 27.45%
of individual preferences are non-transitive. We believe that assuming that
all preferences are transitive implies ignoring some individual preferences.
Therefore, we used a slider from 0 to 10 (with a step of 1) as the weights
determination technique; the presence of a zero rating allows the individual
to discard a particular metadata as irrelevant to the use case under
consideration. This approach seems to be straightforward and inclusive after
testing it during the interviews with the stakeholders. The only constraint in
our weighting approach is that at least one of the provided weights should be
non-zero.

###  3.2. Data Value Calculation

As the collected metadata values have different scales, they must be
normalized. The weights also must be normalized. For usage and number of
objects metadata, the values must be divided by the maximum value. Utility
values have been normalized by dividing the values by 100 (the maximum
possible utility value). As to the creation date, we applied the probabilistic
approach of calculating data currency with a decline rate of 20%. This
approach was proposed by Heinrich and Klier ((2011)) (Heinrich and Klier,
2011) and the data currency $Q_{Curr.}(\omega,A)$ formula is shown in the
Equation 1 below.

(1) $Q_{Curr.}(\omega,A):=exp(-decline(A)\cdot age(\omega,A))$

For the weights, the weight of each metadata has been divided by the sum of
the weights of all four metadata per stakeholder. The data value is then the
weighted average of the metadata values using the Equation 2 below.

(2)
$V(d_{i})=w_{\text{U}}\times\text{U}_{i}+w_{\text{S}}\times\text{S}_{i}+w_{\text{C}}\times\text{C}_{i}+w_{\text{O}}\times\text{O}_{i}$

Where $w_{\\{\text{U, S, C, O}\\}}$ in [0,1] are the weights and $V(d_{i})$ in
]0,1] the data value. U, S, C, and O stand for Utility, uSage, Creation date,
and number of Objects, respectively.

##  4\. Experimental Design

Figure 1 below shows the flowchart of our experimental design. The experiments
consist of three main steps: Data collection, Data value calculation, and
Analysis and conclusions.

Step 1: Data collection. It has two main substeps: metadata collection and
interview of stakeholders. The stakeholders included in this study are
managers within the mapping agency with data management responsibilities for
at least 3 years each.

  * •

Step 1.1: Metadata collection. This consists of extracting metadata from the
dataset repository system. For this use case, only three metadata types have
been extracted: creation date, number of spatial objects (a domain-relevant
measure of data volume and information content), and usage. We also collected,
using spreadsheets, an estimation of the utility of the datasets from four
stakeholders who were domain experts.

  * •

Step 1.2: Stakeholder interview. The main goal of each interview (15-20
minutes) was to get the stakeholders to assign weights to each of the metadata
and their ideal dataset ranking based on a data retrieval scenario. A slider
from 0 to 10 (with a step of 1) is used to assign the weight to each value
metadata. Three stakeholders participated in the interview with only one of
them who also provided an estimation of the utility of each dataset.

Step 2: Data value calculation. Four metadata have been collected on 15
datasets and three sets of weights from three stakeholders. The data valuation
process described in Section 4.2 is used to estimate the value of each dataset
for each stakeholder.

Step 3: Analysis and conclusions. The data value-based dataset ranking is
compared with the dataset ranking provided by the stakeholders. The main
objective is to check to what extent both rankings match. The findings are
discussed in the following Section.

Figure 1. Experimental design for personalized metadata-based data valuation.

##  5\. Experimental Results

Table 1 shows the weights provided by each stakeholder. Stakeholder 2 (SH2)
provided an invalid set of weights (all of the weights are zero) because all
of the metadata selected for this case study was invalid for them. This shows
the practicality of giving the possibility of users or experts to assign zero
weights.

The personal data value was calculated for each dataset using the valid
weights provided by stakeholders SH1 and SH3 and eqt. 2. The datasets were
then ranked by data value. The resulting calculated personalized ranking was
then compared to the ideal ranking provided by the stakeholder. The ideal
ranking of each stakeholder has also been compared to special cases where the
datasets are ranked using each metadata independently (univariate ranking).

The Normalized Discounted Cumulative Gain (NDCG) is chosen to evaluate the
performance of the data value-based rankings. There are two main reasons for
choosing NDCG. Firstly, NDCG is widely used and involves a discount function
over the rank while many other measures uniformly weight all positions. This
feature is important for our use case but also for search engines as users
care about top-ranked documents much more than others; the truncated NDCG also
known as NDCG@k, which only focuses on the top k elements, is preferred for
our use case. Secondly, NDCG is proven to be a good ranking measure from a
learning-to-rank point of view (Wang et al., 2013). NDCG ranges between 0 and
1, with 1 being the optimal performance. We used the scikit-learn
implementation of NDCG with the default parameters111https://scikit-
learn.org/stable/modules/generated/sklearn.metrics.ndcg_score.html.

###  5.1. Evaluation Results

Table 2 presents the evaluation results. The performance of the different data
value-based rankings is expressed in terms of NDCG and NDCG@5.

NDCG and NDCG@5 are calculated per stakeholder and data value calculation
technique (weighted average, simple average and univariate). Simple average
and univariate data valuation are special cases of weighted approaches where
all the weights are equal and where only one weight is non-zero, respectively.

We highlighted the best NDCG and NDCG@5 per stakeholder and data value
calculation technique.

As Usage metadata has been collected monthly from January 2017 to January
2023, we calculated the total and average usage and used them as independent
metadata. For Utility, only experimental stakeholder 1 provided a Utility
estimate and weights of the metadata. Therefore, we included their Utility
estimate as metadata. We also computed the average of the Utility metadata and
used it as another metadata.

###  5.2. Discussion

NDCG and the univariate rankings help us to identify which metadata is the
best determinant in each of the three stakeholders’ ideal ranking of the
datasets. The most determinant metadata for stakeholder 1 are Creation date
and Usage, respectively. For stakeholders 2 and 3, the most determinant
metadata are Utility and Creation date, on one hand, and Utility and Number of
spatial objects, on the other, respectively.

NDCG@5 measures the dataset retrieval performance of the weighted average data
valuation method and its variants. For stakeholder 1, the outperforming
ranking method is the Creation date-based univariate and the second best is
the weighted average using their provided Utility estimate. For stakeholder 2,
the outperforming ranking method is the simple average using average Usage and
the second best is the Creation date-based univariate. For stakeholder 3, the
outperforming ranking methods, with a tie, are the average Utility-based and
stakeholder 1 provided Utility-based univariate and the third best is the
Creation date-based univariate.

Table 1. Dataset value dimension (metadata field) weights provided by stakeholders | Stakeholders
---
(SH)
Utility | Creation Date |  | #Spatial Objects
---
Usage
SH1 | 8 | 10 | 8 | 5
SH2 | 0 | 0 | 0 | 0
SH3 | 7 | 9 | 9 | 4
Table 2. Evaluation results |  |  | NDCG | NDCG@5
---|---|---|---|---
Stakeholder 1 | Weighted Average Raking | Total Usage | 0.7522 | 0.3988
Average Usage | 0.7481 | 0.3868
Provided Utility | 0.9121 | 0.7469
Simple Average Raking | Total Usage | 0.8057 | 0.5853
Average Usage | 0.7766 | 0.4662
Provided Utility | 0.7771 | 0.4564
Univariate Raking | Stakeholder 1 Provided Utility | 0.8062 | 0.6288
Average Utility | 0.8047 | 0.6288
Number of Spatial Objects | 0.8002 | 0.5331
Creation Date | 0.8984 | 0.8984
Total Usage | 0.8881 | 0.6869
Average Usage | 0.8870 | 0.6869
Stakeholder 2 | Simple Average Raking | Total Usage | 0.9159 | 0.7557
Average Usage | 0.9303 | 0.8207
Univariate Raking | Stakeholder 1 Provided Utility | 0.9298 | 0.7694
Average Utility | 0.9195 | 0.7694
Number of Spatial Objects | 0.7797 | 0.5095
Creation Date | 0.8150 | 0.8150
Total Usage | 0.7190 | 0.3652
Average Usage | 0.7241 | 0.3652
Stakeholder 3 | Weighted Average Raking | Total Usage | 0.8155 | 0.4756
Average Usage | 0.8155 | 0.4756
Simple Average Raking | Total Usage | 0.8450 | 0.6153
Average Usage | 0.8148 | 0.4853
Univariate Raking | Stakeholder 1 Provided Utility | 0.9331 | 0.9187
Average Utility | 0.9423 | 0.9187
Number of Spatial Objects | 0.8577 | 0.5680
Creation Date | 0.8505 | 0.8505
Total Usage | 0.8116 | 0.5650
Average Usage | 0.8053 | 0.5650

##  6\. Conclusion

This paper presents a weighted average data valuation method. The proposed
method is then applied to a dataset retrieval use case. The best two (weighted
or simple) averaging performances in terms of NDCG@5 are stakeholder 2 simple
average ranking using average Usage at 0.8207 and stakeholder 1 weighted
average ranking using their own provided Utility at 0.7469. This shows that
the proposed approach could successfully be applied to dataset retrieval.
However, more work is needed to improve the performance of the proposed
method.

There are many directions that this research can follow for further
explorations. For instance, stakeholders, users, or experts can be asked to
provide an ideal ranking of the datasets given a set of metadata. Then, use
NDCG and the univariate rankings to statistically analyse the significance of
the weights of each metadata. This could be a new objective weighting
technique. Another direction could be to compute data value using all the
possible weight distributions (weighted and simple average, univariate,
bivariate, trivariate, and so on). A third direction could be combining the
previous two.

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# Loop Vertex
Representation for Cumulants

V. Rivasseau
Université Paris-Saclay, CNRS/IN2P3
IJCLab, 91405 Orsay, France

###### Abstract

In this paper we construct cumulants for stable random matrix models with
single trace interactions of arbitrarily high even order. We obtain explicit
and convergent expansions for the cumulants. We show that any cumulant is an
analytic function inside a cardioid domain in the complex plane and we prove
their Borel-LeRoy summability at the origin of the coupling constant. Our
proof is uniform in the external variables.

keywords Random Matrix; Cumulants; Constructive Field Theory

Mathematics Subject Classification 81T08

##  1 Introduction

Random matrix theory [1, 2] studies probability laws for matrices. Application
of random matrices to 2d quantum gravity [3] relies on their associated
_combinatorial maps_ , which depend on (at least) two parameters: a coupling
constant $\lambda$ and the size of the matrix, $N$. A _formal_ expansion in
the parameter $\lambda$ yields generating functions for maps of arbitrary
genus. The coupling constant $\lambda$ roughly measures the size of the map
while the parameter $1/N$ turns out to measure the genus of the map [4].

We are interested in the loop vertex representation (LVR) [5]. This is a
improvement of the loop vertex expansion (LVE) [6]. This LVE was introduced
itself as a tool for constructive field theory to deal with random matrix
fields. A common main feature of the LVR and LVE is that it is written in
terms of trees which are exponentially bounded. It means that the outcome of
the LVR-LVE is convergent and is _the Borel-LeRoy sum in $\lambda$_, whereas
the usual perturbative quantum field theory _diverges_ at the point
$\lambda=0$. The essential components of LVE are the Hubbard-Stratonovich
intermediate field representation [7, 8], the replica method [9] and the BKAR
formula [10, 11]. The added ingredients of the LVR are combinatorial, based on
the selective Gaussian integration [5], and the Fuss-Catalan numbers and their
generating function [12]. We think that the LVR has _more power_ than the LVE,
since the LVR can treat more models, with higher polynomial interactions.

For a general exposition of constructive field theory, see [13, 14, 15]; for
an early application to the generating function of connected Schwinger
functions - which in this paper are denoted cumulants \- see [16]; for the
actual mechanism of replacing Feynman graphs, which are not exponentially
bounded, by trees, see [17]; for a review of the LVE, we suggest consulting
[18]; and for the LVE applied to cumulants, we refer to [19]. Together with
[20]-[21] this is our main source of inspiration for this paper.

Figure 1: In blue the cardioid domain considered in [22], in red the cardioid
domain considered in [23].

The authors of [20] join this formalism to Cauchy holomorphic matrix calculus
and have been applied to the simplest complex matrix model with stable
monomial interaction. In [21] the same authors have extended it to the case of
_Hermitian_ or _real symmetric_ matrices, in a manner both _simpler and more
powerful_. The basic formalism is still the LVR, but while [5, 20] used
contour integral parameters attached to every _vertex_ of the loop
representation, [21] introduces more contour integrals, one for each _loop
vertex corner_. This results in simpler bounds for the norm of the corner
operators.

But we should remember that the LVE is older and their authors have more time
to fine-tune their models. They construct their models with the coupling
constant in a cardioid-shaped domain (see Figure 1) which has opening angle
arbitrarily close to $2\pi$ [22] or even exceeding $2\pi$ [23]. In this case
the LVE is capable to compute some typically non-perturbative effects like
instantons by resuming perturbative field theory. In [24], Sazonov combined
the LVE with ideas of the variational perturbation theory.

Figure 2: The pacman domain with parameters $(\eta,\epsilon)$, defined by
$P(\epsilon,\eta):=\\{0<|\lambda|<\eta,|\arg\lambda|<\frac{\pi}{2}+\frac{\pi}{p-1}-\epsilon\\}$,
in the case $p=3$ which corresponds to a sextic matrix interaction.

The results of this article are expressed mainly by our section 4 which
concerns the analyticity and Borel summability for cumulants. One additional
remark is that in [20, 21] we only prove analyticity and Borel-LeRoy
summability inside a pacman domain like Figure 2 (see [25, 26]). For this
article, we extend to the more up-to-date cardioid domain of Nevanlinna-Sokal
[27, 28, 29].

Acknowledgement We would like to thank T. Krajewski, L. Ferdinand, R. Gurău,
P. Radpay and V. Sazonov for comments on the present work when we were in some
preliminary stage and simply for expressing some interest and motivating us to
pursue. We thank Fabien Vignes-Tourneret to have corrected mistakes in Eq. (7)
and Eq. (13) of Section 2 of the arXiv 2305.08399. We also acknowledge the
support of the CEA-LIST through the chair “Artificial Intelligence and
Complexity”.

##  2 The model

In this paper, ${\cal H}$ is the Hilbert space ${\cal H}=C^{N}$,
$\operatorname{Tr}$ always means _the trace on ${\mathbf{C}}^{N}$_,
$\operatorname{Tr}_{\otimes}$ always means _the trace on
${\mathbf{C}}^{N\times N}$_, and ${\bf 1}_{\otimes}$ always means the
$N^{2}\times N^{2}$ matrix whose all eigenvalues are $1$.

Consider a complex square matrix model with stable interaction of order $2p$,
where $p\geq 2$ is an integer which is fixed through all this paper. We assume
the reader is reasonably familiar with the notations of [19, 20, 21] and with
Appendix B of the book [34]. Let us recall some basics of our LVR in the
scalar and $d=0$ case [5]. One of the key elements of the LVR construction is
the Fuss-Catalan numbers of order $p$, which we denote by $C_{n}^{(p)}$, and
their generating function $T_{p}$ [12]. This generating function $T_{p}$ is
defined by

$T_{p}(z)=\sum_{n=0}^{\infty}C_{n}^{(p)}z^{n}.$ (1)

It is analytic at the origin and obeys the algebraic equation

$zT_{p}^{p}(z)-T_{p}(z)+1=0.$ (2)

In the case $p=3$ the LVR is somewhat simplified; the Fuss-Catalan equation is

$zT_{3}^{3}(z)-T_{3}(z)+1=0,$ (3)

which is soluble by radicals. We give in [5], section VI.2, the details
derived from Cardano’s solution.

We shall only present our main result for _complex square matrices_ in a
perturbation $(MM^{\dagger})^{p}$. In a simplification with respect to [20],
we consider only square matrices. The generalisation to cumulants of other
cases, for instance rectangular complex matrices, or Hermitian matrices, or
real symmetric matrices, is not too difficult for someone who is familiar of
[20, 21]. 111For practical applications such as data analysis, the case $p=3$
seems to be the main one and it is interesting to treat the case of _real
symmetric_ matrices and _rectangular matrices_.

To motivate the introduction of function $T_{p}$, let us first briefly recall
how the loop vertex representation (LVR) works in the simple scalar case $N=1$
[5]. In this case, the partition function is simply

$Z(\lambda,1)=\int dzd\bar{z}e^{-z\bar{z}-\lambda(z\bar{z})^{p}}.$ (4)

The LVR in this case simply changes variable such that the original action
becomes Gaussian; hence, $z\bar{z}+\lambda(z\bar{z})^{p}=w\bar{w}$. This can
be done by choosing $\bar{w}=\bar{z},w=z+\lambda(z\bar{z})^{p-1}z$. This of
course will cost us a Jacobian:

$\frac{\partial(w,\bar{w})}{\partial(z,\bar{z})}=1+p\lambda(z\bar{z})^{p-1}.$
(5)

Using $z\bar{z}=w\bar{w}T_{p}[-\lambda(w\bar{w})^{p-1}]$, we rewrite the
partition function as

$Z(\lambda,1)=\int
dwd\bar{w}e^{-w\bar{w}-\log\big{[}1+p\lambda(w\bar{w}T_{p}(-\lambda(w\bar{w})^{p-1}))^{p-1}\big{]}}.$
(6)

The derivatives of the log are uniformly bounded in $w,\bar{w}$ because

$1-z[T_{p}(z)]^{p-1}$ (7)

has only one cut in the complex plane, which one can avoid by tweaking the
phase of $\lambda$. This allows to control the expansion for $\log Z$.

The case without sources, of the partition function and its logarithm, has
been treated in [20]. Therefore in this paper we are dealing with the case
with sources fields $J$. The source $J$ is itself a $N\times N$ complex matrix
and $J^{\dagger}$ is its adjoint.

Therefore we define the matrix model with perturbation

$\frac{\lambda}{N^{p-1}}(MM^{\dagger})^{p}.$ (8)

Since $p$ is fixed we omit the subscript $p$ from $T,Z,S$… when no confusion
is possible. The stable case corresponds to
$\text{Re}\,\frac{\lambda}{N^{p-1}}>0$.

######  Definition 1.

The measure $dM$ and the action $S(M)$ are defined by

$\displaystyle dM$ $\displaystyle:=$ $\displaystyle\pi^{-N^{2}}\prod_{1\leq
i,j\leq N}d\text{Re}(M_{ij})d\text{Im}(M_{ij})\,,$ (9) $\displaystyle S(M)$
$\displaystyle:=$
$\displaystyle\operatorname{Tr}\\{MM^{\dagger}+\frac{\lambda}{N^{p-1}}(MM^{\dagger})^{p}\\}.$
(10)

The model that we consider in this paper but _without sources_ has partition
function

$\displaystyle Z(\lambda,N)$ $\displaystyle:=$
$\displaystyle\int\,dM\,e^{-S(M,M^{\dagger})}.$ (11)

The same matrix model _but with sources_ is defined by:

$\displaystyle Z(\lambda,N,J)$ $\displaystyle=$ $\displaystyle\frac{\int
dMe^{-S(M)+\sqrt{N}\operatorname{Tr}(JM^{\dagger})+\sqrt{N}\operatorname{Tr}(MJ^{\dagger})}}{\int\,dM\,e^{-S(M)}}$
(12)

where $dM$ and $S(M)$ are defined in (9)-(10). 222 The normalization in $N$
has been chosen in a same way as in [19], so that the amplitude of a ribbon
graph is $N^{\chi(G)}$, with $\chi(G)$ the Euler characteristic of the graph.
Recall that [19] is dealing with the LVE, thus with a $p=2$ (quartic)
perturbation.

For any $N$ by $N$ square matrix $X$ we define the matrix-valued function

$\displaystyle A(\lambda,X)$ $\displaystyle:=$ $\displaystyle XT_{p}(-\lambda
X^{p-1})\,,$ (13)

so that from (2)

$X=A(\lambda,X)+\lambda A^{p}(\lambda,X)\,.$ (14)

We often write $A(X)$ for $A(\lambda,X)$, or even simply $A$, when no
confusion is possible. Next we define an $N$ by $N$ square matrix $X_{l}$ and
an $N$ by $N$ square matrix $X_{r}$ through

$\displaystyle X_{l}$ $\displaystyle:=$ $\displaystyle MM^{\dagger},\quad
X_{r}:=M^{\dagger}M.$ (15)

Crucially these two matrices have the same trace, therefore we simply call
them $\operatorname{Tr}X$:

$\displaystyle\operatorname{Tr}X_{l}=\operatorname{Tr}X_{r}=\operatorname{Tr}X.$
(16)

For the partition function and its logarithm without sources the following
proposition holds:

######  Proposition 1.

In the sense of _formal power series in_ $\lambda$

$\displaystyle Z(\lambda,N)$ $\displaystyle=$
$\displaystyle\int\,dM\,\exp\\{-N\operatorname{Tr}X+{\cal S}\\},$ (17)

where ${\cal S}$, the _loop vertex action_ , is

$\displaystyle{\cal S}$ $\displaystyle=$
$\displaystyle-\operatorname{Tr}_{\otimes}\log\Big{[}{\bf
1}_{\otimes}+\lambda\sum_{{\mathfrak{k}}=0}^{p-1}A^{\mathfrak{k}}(X_{l})\otimes
A^{p-1-{\mathfrak{k}}}(X_{r})\Big{]}$ (18) $\displaystyle=$
$\displaystyle-\operatorname{Tr}_{\otimes}\log\big{[}{\bf
1}_{\otimes}+\Sigma(\lambda,X)\big{]},$ (19)

where
$\Sigma(\lambda,X):=\lambda\sum_{{\mathfrak{k}}=0}^{p-1}A^{\mathfrak{k}}(X_{l})\otimes
A^{p-1-{\mathfrak{k}}}(X_{r})$. In (18) the matrix $A^{\mathfrak{k}}(X_{l})$
acts on the left index of ${\cal H}\otimes{\cal H}$ and the matrix
$A^{p-1-{\mathfrak{k}}}(X_{r})$ acts on the right index of ${\cal
H}\otimes{\cal H}$.

Proof This proposition is proved in [20], Section 2, under the name of Theorem
2.1. But we recall and summarize the two proofs of [20] in the notations of
this paper, because Proposition 1 is of fundamental importance. One proof in
by performing a change of variables $M\to P$. $P$ is again an $N$ by $N$
square matrix. In a manner similar to [20] we write $Y:=PP^{\dagger}$, and
define $P(M)$ through the implicit function formal power series equation
$X:=A(Y)$.

$\displaystyle S(M)=\operatorname{Tr}(X+\lambda
X^{p})=\operatorname{Tr}[A(Y)+\lambda A^{p}(Y)]=\operatorname{Tr}Y,$ (20)

hence it becomes the ordinary Gaussian measure on $P,P^{\dagger}$. The new
interaction lies therefore entirely in the _Jacobian_ of the $M\to P$
transformation. This transformation can be written more explicitly as

$\displaystyle M$ $\displaystyle:=$ $\displaystyle
PP^{\dagger}T_{p}(-\lambda(PP^{\dagger})^{p-1})(P^{\dagger})^{-1}=A(PP^{\dagger})(P^{\dagger})^{-1},$
(21) $\displaystyle M^{\dagger}$ $\displaystyle:=$ $\displaystyle
P^{\dagger}\,.$ (22)

Taken into account [20] Section 2, Proposition 1, together with equations
(17)-(19), hold.

The second proof is in [20], Appendix A. It uses the method of selective
Gaussian integration [5]. It consists essentially by representing integration
as differentiation:

$\displaystyle
Z(\lambda,N)=e^{\operatorname{Tr}[\frac{1}{N}\partial_{M}^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\dagger}}\partial_{M^{\dagger}}]}e^{N\operatorname{Tr}[-\lambda(MM^{\dagger})^{p-1}MM^{\dagger}]}\Big{|}_{M^{\dagger}=0,M=0}$
$\displaystyle=e^{\operatorname{Tr}\big{[}\frac{1}{N}\partial_{M}\partial_{A}\big{]}}\Big{(}e^{\frac{1}{N}\operatorname{Tr}[\partial_{M}\partial_{B}]}e^{N\operatorname{Tr}[-\lambda(MA)^{p-1}MB]}\Big{|}_{B=0}\Big{)}\Big{|}_{A=0,M=0}\,,$
(23)

where
$\operatorname{Tr}[\partial_{M}\partial_{M^{\dagger}}]=\sum_{a,b}\frac{\partial}{\partial
M_{ab}^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\dagger}}}\frac{\partial}{\partial
M_{ba}^{\dagger}}$. Renaming matrix $A$ back as $M^{\dagger}$, we obtain

$\displaystyle
Z(\lambda,N)=\int\,dM\,\exp\\{-N\operatorname{Tr}[MM^{\dagger}]+{\cal
S}(M)\\}$ (24)

with the effective action given by

$\displaystyle{\cal
S}=\log\Big{(}e^{\frac{1}{N}\operatorname{Tr}[\partial_{M}\partial_{B}]}e^{N\operatorname{Tr}[-\lambda(MM^{\dagger})^{p-1}MB]}\Big{|}_{B=0}\Big{)}\,.$
(25)

Within the framework of the perturbation theory, the exponential of ${\cal S}$
can be transformed as

$\displaystyle e^{{\cal S}}$ $\displaystyle=$
$\displaystyle\int\,dC\,dB\,e^{-N\operatorname{Tr}[CB]+N\operatorname{Tr}[-\lambda((M+C)M^{\dagger})^{p-1}(M+C)B]}\,.$

Then, the matrix $B$ can be interpreted as a Lagrange multiplier and ${\cal
S}$ write

$\displaystyle{\cal
S}=-\operatorname{Tr}_{\otimes}[\log\big{|}\mathbf{1}_{\otimes}+\lambda\sum_{{\mathfrak{k}}=0}^{p-1}[(C_{0}+M)M^{\dagger}]^{\mathfrak{k}}\otimes[(C_{0}+M)M^{\dagger}]^{p-1-{\mathfrak{k}}}\big{|}].$
(26)

The matrix $C_{0}$ in (26) is a solution of the equation

$C_{0}+\lambda((M+C_{0})M^{\dagger})^{p-1}(M+C_{0})=0$ (27)

and it is given by

$C_{0}=MM^{\dagger}T(-\lambda(MM^{\dagger})^{p-1})(M^{\dagger})^{-1}-M\,.$
(28)

Combining (28) and (26) we arrive at (18) and at Proposition 1. ∎

In terms of $X$ and $M$, equation (12) (after the change of variables $P\to M$
in (21)) write

$Z(\lambda,N,J):=\frac{\int\,dMe^{-N\operatorname{Tr}X+{\cal
S}+\sqrt{N}\operatorname{Tr}(JM^{\dagger})+\sqrt{N}\operatorname{Tr}(A(MM^{\dagger})(M^{\dagger})^{-1}J^{\dagger})}}{\int\,dMe^{-N\operatorname{Tr}X+{\cal
S}}}.$ (29)

The quantity $Z(\lambda,N,J)$ may be written in terms of a Gaussian measure
whose covariance is $N^{-1}$:

$\displaystyle d\mu(M)$ $\displaystyle=$ $\displaystyle dMe^{-N{\rm
Tr}M^{\dagger}M},$ (30) $\displaystyle Z(\lambda,N,J)$ $\displaystyle=$
$\displaystyle\frac{\int d\mu(M)e^{{\cal S}(J,M)+{\cal S}}}{\int
d\mu(M)e^{{\cal S}}}.$ (31)

${\cal S}$ does not depend on the sources, ${\cal S}(J,M)$ depends on the
sources and is

$\displaystyle{\cal S}(J,M)$ $\displaystyle=$
$\displaystyle\sqrt{N}\operatorname{Tr}(JM^{\dagger})+\sqrt{N}\operatorname{Tr}(T_{p}(-\frac{\lambda}{N^{p-1}},MM^{\dagger})(M^{\dagger})^{-1}J^{\dagger})\;.$
(32)

where $T_{p}(x,u)$ is the generating function of rooted $p$-ary trees with a
factor $x$ per internal vertex and $u$ per leaf.

First we shall define a mathematical expression for the scalar cumulants.

######  Definition 2.

The scalar cumulant of order $2k$ is:

$\displaystyle\hskip-17.07182pt\mathfrak{K}^{k}(\lambda,N)\\!$
$\displaystyle:=$
$\displaystyle\\!\Big{[}\frac{\partial^{2}}{J^{\ast}_{a_{1}b_{1}}J_{c_{1}d_{1}}}\cdots\frac{\partial^{2}}{J^{\ast}_{a_{k}b_{k}}J_{c_{k}d_{k}}}\log
Z(\lambda,N,J)\Big{]}_{J=0}.$ (33)

$J_{ab}^{\ast}$ is the complex conjugate of $J_{ab}$, so that
$(J^{\dagger})_{ab}=J^{\ast}_{ba}$.

Note that all the derivatives of $\log Z$ which are not of this form vanish.
In the case $p=2$, which is treated by [19], the order 2 cumulant is
proportional to $\langle M_{ab}M^{\ast}_{cd}\rangle-\langle
M_{ab}\rangle\langle M^{\ast}_{cd}\rangle$ whereas, if we transpose, it is not
the same thing: $\langle M_{ba}M^{\ast}_{cd}\rangle-\langle
M_{ba}\rangle\langle M^{\ast}_{cd}\rangle$ is subtly different from $\langle
M_{ab}M^{\ast}_{cd}\rangle-\langle M_{ab}\rangle\langle M^{\ast}_{cd}\rangle$.

Any quantity $F$ in quantum field theory which is an integral over a Gaussian
measure $\int_{C}d\mu(\phi)\ F(\phi)$ can be combinatorially represented as a
sum over the set ${\mathfrak{F}}$ of _oriented forests_ 333Oriented forests
simply distinguish edges $(i,j)$ and $(j,i)$ so have edges with arrows. It
allows to distinguish below between operators $\frac{\partial}{\partial
M^{\dagger}_{i}}\frac{\partial}{\partial
M_{j}^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\dagger}}}$
and $\frac{\partial}{\partial M^{\dagger}_{j}}\frac{\partial}{\partial
M_{i}^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\dagger}}}$.
by applying the BKAR formula [10, 11]. For readers who want to look further
into BKAR formula and oriented forests, ordered or not, see [17, 31, 19].

In this context, where the Gaussian measure is $d\mu(M)$ and the covariance is
$C=\frac{1}{N}$, we start by replacing the covariance by
$C_{ij}(x)=\frac{x_{ij}+x_{ji}}{2}\frac{1}{N}$ evaluated at $x_{ij}=1$ for
$i\neq j$ and $C_{ii}(x)=\frac{1}{N}\;\forall i$. Then the Taylor BKAR formula
for oriented forests ${\mathfrak{F}}_{n}$ on $n$ labeled vertices yields

$\displaystyle\hskip-105.2751ptF(M)=\sum_{n=0}^{\infty}\frac{1}{n!}\;\;\sum_{{\cal
F}\in{\mathfrak{F}}_{n}}\;\int dw_{\cal F}\ \partial_{\cal
F}\;\int\,d\mu_{C\\{x_{ij}^{\cal
F}\\}}(M)\,F_{n}(M)\;\Big{|}_{x_{ij}=x_{ij}^{\cal F}(w)},$ (34)
$\displaystyle\text{where }\int dw_{\cal F}$ $\displaystyle:=$
$\displaystyle\prod_{(i,j)\in{\cal
F}}\int_{0}^{1}dw_{ij}\;,\quad\partial_{\cal F}:=\prod_{(i,j)\in{\cal
F}}\frac{\partial}{\partial x_{ij}}\;,$ (35) $\displaystyle x_{ij}^{\cal
F}(w)$ $\displaystyle:=$
$\displaystyle\left\\{\begin{array}[]{c}\text{inf}_{(k,l)\in P^{\cal
F}_{i\leftrightarrow j}}w_{kl}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ P^{\cal F}_{i\leftrightarrow
j}\leavevmode\nobreak\ \text{exists}\,,\\\ 0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ P^{\cal F}_{i\leftrightarrow
j}\leavevmode\nobreak\ \text{does not exist}\,.\end{array}\right.$ (38)

In this formula $w_{ij}$ is the weakening parameter of the edge $(i,j)$ of the
forest, and $P^{\cal F}_{i\leftrightarrow j}$ is the unique path in ${\cal F}$
joining $i$ and $j$ when it exists.

Remember that a main property of the forest formula is that the symmetric $n$
by $n$ matrix $C\\{x^{\cal F}_{ij}\\}=\frac{x^{\cal F}_{ij}(w)+x^{\cal
F}_{ji}(w)}{2}\frac{1}{N}$ is positive for any value of $w_{kl}$, hence the
Gaussian measure $d\mu_{C\\{x^{\cal F}_{ij}\\}}(M)$ is well-defined. Since the
fields, the measure and the integrand are now factorized over the connected
components of ${\cal F}$, its _logarithm_ is easily computed as exactly the
same sum but restricted to the spanning trees.

##  3 LVR Amplitudes of Combinatorial Maps

444This section is essentially a condensed version of the corresponding
section of [19].

For the task of computing $\mathfrak{K}^{k}(\lambda,N)$, we have to introduce
combinatorial maps, a refinement of the usual Feynman graphs . A combinatorial
map is a graph with a distinguished cyclic ordering of the half edges incident
at each vertex. Combinatorial maps are conveniently represented as _ribbon
graphs_ whose vertices are disks and whose edges are ribbons (allowing one to
encode graphically the ordering of the half edges incident at a vertex). When
applied to cumulants, it is based on combinatorial maps with cilia. A _cilium_
is a half edge hooked to a vertex. We denote $k(G)$, $v(G)$, $e(G)$, $f(G)$
and $c(G)$ the cilia, vertices, edges, faces and _corners_ of $G$
555Attention: in a simplification to the notations of [19], we shall no longer
use in this paper the distinction between $k(G),v(G),e(G),f(G),c(G)$ and their
_numbers_ $|k(G)|,|v(G)|,|e(G)|,|f(G)|,|c(G)|$. A corner of $G$ is a pair of
consecutive half edges attached to the same vertex. The faces of $G$ are
partitioned between the faces which do not contain any cilium (which we
sometimes call internal faces) and the ones which contain at least a cilium,
which we call _broken faces_. We denote $b(G)$ the set of broken faces of $G$.
Each broken face corresponds to a puncture in the Riemann surface in which $G$
is embedded. The Euler characteristic of the graph $G$ is:

$\chi(G)=v(G)-e(G)+f(G)-b(G)=2-2g(G)-b(G).$ (39)

where $g(G)$ is the genus of the graph $G$.

######  Definition 3 (LVR graphs and trees).

A LVR graph $(G,T)$ is a connected ribbon graph $G$ with labels on its
vertices having furthermore:

  * •

a distinguished spanning tree $T\subset G$,

  * •

a labeling of the edges of $G$ not in $T$,

  * •

at most one cilium per vertex.

A LVR tree is a graph such that the set $l(G,T):=e(G)-e(T)$ is empty, so
$(G,T)=(T,T)$.

We associate to every LVR graph $(G,T)$ _its amplitude_ ${\cal
A}_{(G,T)}(\lambda,N,J)$. We emphasize here that the following definition of
amplitudes is almost the same that in [19], but is subtly _different for the
LVE and for the LVR_ ; we have to replace the intermediate field (denoted by
$A$ in [19]) by a family of functions $X_{l},X_{r}$ depending only on the
fields $M_{i},M_{i}^{\dagger}$ by eq. (15) but having the same trace, see
(16). This point is subtle and we want to explain in more detail. First we
want to make the following definition :

######  Definition 4.

$\displaystyle\Sigma(\lambda,M)$ $\displaystyle:=$
$\displaystyle\lambda\sum_{{\mathfrak{k}}=0}^{p-1}A^{\mathfrak{k}}(MM^{\dagger})\otimes
A^{p-1-{\mathfrak{k}}}(M^{\dagger}M).$ (40)

where $A$ is defined by (13).

######  Definition 5.

We define the _amplitude_ of LVR graph $(G,T)$, ${\cal A}_{(G,T)}(\lambda,N)$,
by

$\displaystyle{\cal A}_{(G,T)}(\lambda,N,J)$ $\displaystyle=$
$\displaystyle\frac{(-\lambda)^{e(G)}N^{v(G)-e(G)}}{v(G)!}\int
dw_{T}\partial_{T}\int d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)$ (41)
$\displaystyle\prod_{f\in
f(G)}\operatorname{Tr}\bigg{\\{}\mathop{\prod}\limits_{c\in\partial
f}^{\longrightarrow}\big{[}{\bf
1}_{\otimes}+\Sigma(\lambda,M)\big{]}^{-1}(JJ^{\dagger})^{\eta_{c}}\bigg{\\}}\;,$

where:

  * •

${\displaystyle\mathop{\prod}^{\longrightarrow}}$ is the oriented product
around the corners $c$ on the boundary $\partial f$ of the face $f$,

  * •

$\eta_{c}=1,0$ depending on whether $c$ is followed by a cilium or no cilia,

  * •

the Gaussian measure $\int d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)$ can also be
written as the differential operator:

$\displaystyle\hskip-51.21504pt\int
d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)F\\{M_{i},M_{i}^{\dagger}\\}=\left[e^{\frac{x_{ij}^{T}+x_{ji}^{T}}{2}\frac{\partial}{\partial
M_{i}}\frac{\partial}{\partial
M^{\dagger}_{j}}}\;F\Big{\\{}M_{i},M^{\dagger}_{i}\Big{\\}}\right]_{\\{M_{i}\\}=0},$
(42)

  * •

and $\Sigma(\lambda,M)$ is defined by (40).

If the graph $(G,T)$ has $k$ cilia we use for its amplitude the notation
${\cal A}^{k}_{G,T}(\lambda,N,J)$; if the graph $(G,T)$ is reduced to a tree
we use the shorthand notation ${\cal A}^{k}_{T}(\lambda,N,J)$ instead of
${\cal A}^{k}_{(T,T)}(\lambda,N,J)$. The amplitude simplifies drastically in
this case: one trace is obtained (as trees have only one face). Hence

$\displaystyle{\cal A}^{k}_{T}(\lambda,N,J)$ $\displaystyle=$
$\displaystyle\frac{(-\lambda)^{e(T)}N^{v(T)-e(T)}}{v(T)!}$ $\displaystyle\int
d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)\operatorname{Tr}\bigg{\\{}\mathop{\prod}\limits_{c\in\partial
f}^{\longrightarrow}\big{[}{\bf
1}_{\otimes}+\Sigma(\lambda,M)\big{]}^{-1}(JJ^{\dagger})^{\eta_{c}}\bigg{\\}}\;.$

Figure 3: Cardioid domain ${\cal C}$ in the complex $\lambda$ plane in the
case $p=3$.

Let ${\cal C}$ be the cardioid domain

${\cal
C}=\bigg{\\{}\lambda\in{\mathbb{C}}\quad\text{with}\quad|\lambda|<\frac{1}{2(p-1)}\cos^{p-1}\Big{(}{\frac{\arg\lambda}{p-1}}\Big{)}\bigg{\\}}\;,$
(44)

where we choose the determination
$-\frac{\pi}{2}<\frac{\arg\lambda}{p-1}<\frac{\pi}{2}$ of the argument. This
domain is also defined in the Appendix of this article by ${\cal D}_{R}$,
simply change $q\to p-1$, $z\to\lambda$, $R\to\frac{1}{2(p-1)}$. For the case
$p=3$, ${\cal C}$ is pictured in Figure 3.

Our first proposition is a convergent expansion of $\log Z(\lambda,N,J)$ as a
sum over LVR trees.

######  Proposition 2.

For any $\lambda\in{\cal C}$, there exists $\epsilon_{\lambda}>0$ depending on
$\lambda$ such that for $\|JJ^{\dagger}\|<\epsilon_{\lambda}$ the logarithm of
$Z(\lambda,N,J)$ is given by the following absolutely convergent expansion:

$\log Z(\lambda,N,J)=\sum_{T\,\text{LVR tree}}{\cal A}_{T}(\lambda,N,J)\;.$
(45)

Proof This proposition was proved in [19]. ∎

In order to compare the tree expansion of Proposition 2 with the conventional
perturbative expansion, it is necessary to further expand some of the loop
edges. The following proposition is obtained by recursively adding loop edges
to the LVR trees.

######  Proposition 3.

For any $\lambda\in{\cal C}$, there exists $\epsilon_{\lambda}>0$ depending on
$\lambda$ such that for $\|JJ^{\dagger}\|<\epsilon_{\lambda}$:

$\displaystyle\hskip-28.45274pt\log Z(\lambda,N,J)=\sum_{{G\text{ ciliated
labeled}\atop\text{ ribbon graph}}\atop e(G)\leq
n}\frac{(-\lambda)^{e(G)}N^{\chi(G)}}{v(G)}\prod_{f\in
b(G)}\operatorname{Tr}(JJ^{\dagger})^{c(f)}+{\cal R}_{n}(\lambda,N,J)\;,$

where $c(f)$ is the number of cilia in the broken face $f$ and the
perturbative remainder at order $n$ is a convergent sum over LVR graphs with
at least $n+1$ edges and at most $n+1$ loop edges

${\cal R}_{n}(\lambda,N,J)=\sum_{(G,T)\text{ LVR graph}\atop e(G)=n+1}{\cal
A}_{(G,T)}(\lambda,N,J)+\sum_{T\text{ LVR tree}\atop e(T)\geq n+2}{\cal
A}_{T}(\lambda,N,J)\;.$ (46)

Proof This proposition was proved in [19]. ∎

Note that the first term in (3) involves a sum over ribbon graphs (not LVR
graphs) and reproduces the perturbative expansion over maps.

Next we introduce Weingarten functions who were defined in [32], [33] and also
in [19]. As the authors of [19] remark, scalar cumulants arise when
integrating over unitary matrices $\text{U}(N)$ with the invariant normalized
Haar measure. Denoting $U_{ab}^{*}$ the complex conjugate of $U_{ab}$ we have
[32, 33]:

$\displaystyle\int dU\;U_{a_{1}b_{1}}\dots
U_{a_{k}b_{k}}U^{*}_{c_{1}d_{1}}\dots U^{*}_{c_{l}d_{l}}=$ (47)
$\displaystyle\hskip
56.9055pt\delta_{kl}\sum_{\sigma,\tau\in\mathfrak{S}_{k}}\delta_{a_{\tau(1)c_{1}}}\dots\delta_{a_{\tau(k)}c_{k}}\delta_{b_{\sigma(1)}d_{1}}\dots\delta_{b_{\sigma(k)}d_{k}}{\cal{W}}\hskip-1.70709pt_{g}(\tau\sigma^{-1},N)\;.$

The functions ${\cal{W}}\hskip-1.70709pt_{g}(\zeta=\tau\sigma^{-1},N)$ only
depend on the cycle structure of $\zeta$. Here is a few examples of Weingarten
functions:

$\displaystyle{\cal{W}}\hskip-1.70709pt_{g}\big{(}(1),N\big{)}$
$\displaystyle=\frac{1}{N}$
$\displaystyle{\cal{W}}\hskip-1.70709pt_{g}\big{(}(1,1,1),N\big{)}$
$\displaystyle=\frac{N^{2}-2}{N(N^{2}-1)(N^{2}-4)}$
$\displaystyle{\cal{W}}\hskip-1.70709pt_{g}\big{(}(1,1),N\big{)}$
$\displaystyle=\frac{-1}{N^{2}-1}$
$\displaystyle{\cal{W}}\hskip-1.70709pt_{g}\big{(}(1,2),N\big{)}$
$\displaystyle=\frac{-1}{(N^{2}-1)(N^{2}-4)}$
$\displaystyle{\cal{W}}\hskip-1.70709pt_{g}\big{(}(2),N\big{)}$
$\displaystyle=\frac{-1}{N(N^{2}-1)}$
$\displaystyle{\cal{W}}\hskip-1.70709pt_{g}\big{(}(3),N\big{)}$
$\displaystyle=\frac{2}{N(N^{2}-1)(N^{2}-4)}\;.$

For any permutation of $k$ elements $\zeta\in{\mathfrak{S}}_{{k}}$, let us
write $C(\zeta)$ the integer partition of associated to the cycle
decomposition of $\zeta$ and $|C(\zeta)|$ the number of cycles it contains.
Let us also denote by $\Pi_{{k}}$ the set of integer partitions of $k$ (recall
that a partition $\pi\in\Pi_{{k}}$ is an increasing sequence of $|\pi|$
integers $0<{k}_{1}\leq\cdots\leq{k}_{|\pi|}$ such that
${k}_{1}+\cdots+{k}_{|\pi|}={k}$). To any integer partition of ${k}$ we
associate a _trace invariant_ :

$\operatorname{Tr}_{\pi}({\mathfrak{X}})=\operatorname{Tr}({\mathfrak{X}}^{{k}_{1}})\cdots\operatorname{Tr}({\mathfrak{X}}^{{k}_{p}})\;.$
(48)

Let us chose a permutation $\zeta\in{\mathfrak{S}}_{{k}}$ whose cycle
decomposition reproduces the contribution of the _broken faces_ to the
amplitude of a graph. Specifically, if there are $b=b(G)$ broken faces with
${k}_{1},\dots,{k}_{b}$ cilia, we choose $\zeta$ to have a cycle decomposition
of the form:

$\zeta=(i_{1}^{1}\dots i_{{k}_{1}}^{1})\cdots(i_{1}^{b}\dots
i_{{k}_{b}}^{b})\;.$ (49)

This permutation defines a labeling of the cilia in such a way that the
product of traces over the broken faces can be expressed as:

$\displaystyle\prod_{1\leq m\leq
b}\operatorname{Tr}\Big{[}JJ^{\dagger}\mathop{\prod}\limits_{1\leq
r\leq{k}_{m}}^{\longrightarrow}{\mathfrak{X}}^{i^{m}_{r}}\Big{]}=\sum_{1\leq
p_{1},q_{1}\dots\leq N}(JJ^{\dagger})_{p_{l}q_{l}}\prod_{1\leq
l\leq{k}}{\mathfrak{X}}^{l}_{q_{l}p_{\zeta(l)}}$ (50)

where ${\mathfrak{X}}^{l}$ is the product of the resolvents

$\displaystyle\big{[}{\bf 1}_{\otimes}+\Sigma(\lambda,M)\big{]}^{-1}$
$\displaystyle=$ $\displaystyle\big{[}{\bf
1}_{\otimes}+\lambda\sum_{{\mathfrak{k}}=0}^{p-1}A^{\mathfrak{k}}(MM^{\dagger})\otimes
A^{p-1-{\mathfrak{k}}}(M^{\dagger}M)\big{]}^{-1}$ (51)

located on the corners separating the cilia labeled $l$ and $\zeta(l)$.

Similarly, for the $f(G)-b(G)$ unbroken faces we denote by
${\mathfrak{X}}^{m}$ the product of the resolvents

$\displaystyle\big{[}{\bf 1}_{\otimes}+\Sigma(\lambda,M)\big{]}^{-1}$
$\displaystyle=$ $\displaystyle\big{[}{\bf
1}_{\otimes}+\lambda\sum_{{\mathfrak{k}}=0}^{p-1}A^{\mathfrak{k}}(MM^{\dagger})\otimes
A^{p-1-{\mathfrak{k}}}(M^{\dagger}M)\big{]}^{-1}$ (52)

around the unbroken face labeled $m$.

The amplitude of a LVR graph with $k$ cilia expands in trace invariants as:

${\cal
A}_{(G,T)}^{k}(\lambda,N,J)=\sum_{\pi\in\Pi_{k}}{A}_{(G,T)}^{\pi}(\lambda,N)\;\operatorname{Tr}_{\pi}\\{J,J^{\dagger}\\}\;,$
(53)

with
$Tr_{\pi}\\{J,J^{\dagger}\\}=\operatorname{Tr}[JJ^{\dagger}]^{k_{1}}\cdots\operatorname{Tr}[JJ^{\dagger}]^{{k}_{p}}$
is a. trace invariant defined by an increasing sequence of $|\pi|$ integers
$0<{k}_{1}\leq\cdots\leq{k}_{|\pi|}$ such that
${k}_{1}+\cdots+{k}_{|\pi|}={k}$, and the definition of
${A}_{(G,T)}^{\pi}(\lambda,N)$ is

######  Definition 6 (Weingarten definition).

$\displaystyle{A}_{(G,T)}^{\pi}(\lambda,N)=\frac{(-\lambda)^{e(G)}N^{v(G)-e(G)}}{v(G)!}\int
dw_{T}\partial_{T}\int d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)$ (54)
$\displaystyle\sum_{\tau,\sigma\in{\mathfrak{S}}_{k}\atop
C(\sigma)=\pi}\sum_{1\leq p_{1},\dots,p_{k}\leq
N}\text{Wg}(\tau\sigma^{-1},N)\prod_{1\leq m\leq
f(G)-b(G)}\operatorname{Tr}\Big{[}{\mathfrak{X}}^{m}\Big{]}\prod_{1\leq l\leq
k}{\mathfrak{X}}^{l}_{p_{\tau(l)}p_{\zeta(l)}}\;.$

where ${\mathfrak{X}}$ is defined by (51) and (52).

If the LVR graph $(G,T)$ is reduced to a tree we use the shorthand notation
${A}_{T}^{\pi}(\lambda,N)$ instead of ${A}_{(T,T)}^{\pi}(\lambda,N)$.

######  Proposition 4.

The _scalar cumulants_ of order $2k$ can be written as a sum over partitions
of $k$ and over two permutations of $k$ elements:

${\mathfrak{K}}^{k}(\lambda,N)\\!=\\!\sum_{\pi\in\Pi_{k}}{\mathfrak{K}}^{k}_{\pi}(\lambda,N)\sum_{\rho,\sigma\in\mathfrak{S}_{k}}\prod_{1\leq
l\leq
k}\delta_{d_{l},a_{\rho\tau_{\pi}\sigma^{-1}(l)}}\delta_{c_{l},b_{\rho\xi_{\pi}\sigma^{-1}(l)}},$
(55)

where $\tau_{\pi}$ and $\xi_{\pi}$ are arbitrary permutations such that
$\tau_{\pi}(\xi_{\pi})^{-1}$ has a cycle structure corresponding to the
partition $\pi$. Then the scalar cumulants
${\mathfrak{K}}^{k}_{\pi}(\lambda,N)$ are given by the expansion:

${\mathfrak{K}}^{k}_{\pi}(\lambda,N)=\sum_{T\text{ LVR tree with $k$
cilia}}{\cal A}^{\pi}_{T}(\lambda,N)\;.$ (56)

Proof This proposition was proved in [19]. ∎

Now we want to explicit further the dependency on the indices of $J$ by the
following expanded definition :

$\displaystyle{\mathfrak{K}}_{a_{1}b_{1}c_{1}d_{1},\dots,a_{k}b_{k}c_{k}d_{k}}(\lambda,N,J)$
$\displaystyle:=$
$\displaystyle\mathfrak{K}^{k}(\lambda,N,J):=\mathfrak{K}^{k}(\lambda,N).$
(57)

Choosing any other pair of permutations $\tau_{\pi}$ and $\xi_{\pi}$ leads to
an identical result, after reorganizing the sum over $\rho$ and $\sigma$.
Therefore it explains why we call it _scalar cumulant_ :
${\mathfrak{K}}^{k}_{\pi}(\lambda,N)$ only depends on the partition $\pi$ and
not on the index structure of
${\mathfrak{K}}_{a_{1}b_{1}c_{1}d_{1},\dots,a_{k}b_{k}c_{k}d_{k}}(\lambda,N,J)$.

##  4 Results

We state an analyticity result and a Borel summability for the constructive
expansion of the cumulant of order $2k$ or J-cumulants.

######  Theorem 1 (Constructive expansion for the J-cumulant).

Let $k\leq k_{\max}$. There exists $\epsilon_{\lambda}>0$ depending on
$\lambda$ such that $\mathfrak{K}^{k}(\lambda,N)$ is given by the following
absolutely convergent expansion

$\displaystyle\mathfrak{K}^{k}(\lambda,N)$ $\displaystyle=$
$\displaystyle{\cal P}^{k}_{n}(\lambda,N,J)+{\cal
Q}^{k}_{n}(\lambda,N,J)+{\cal R}^{k}_{n}(\lambda,N,J),$ (58)
$\displaystyle{\cal P}^{k}_{n}(\lambda,N,J)$ $\displaystyle=$
$\displaystyle\sum_{{G\text{ labeled ribbon graph }\atop\text{ with $k$
cilia,}}\atop e(G)\leq
n}\hskip-8.5359pt\frac{(-\lambda)^{e(G)}N^{\chi(G)}}{v(G)!}\hskip-2.84544pt\prod_{f\in
b(G)}\hskip-2.84544pt\operatorname{Tr}[(JJ^{\dagger})^{c(f)}],$ (59)
$\displaystyle\hskip-8.5359pt{\cal Q}^{k}_{n}(\lambda,N,J)$ $\displaystyle=$
$\displaystyle\sum_{{(G,T)\text{ LVR graph }\atop\text{ with $k$ cilia}}\atop
e(T)=n+1}{\cal A}^{k}_{(G,T)}(\lambda,N,J),$ (60) $\displaystyle{\cal
R}^{k}_{n}(\lambda,N,J)$ $\displaystyle=$ $\displaystyle\sum_{{T\text{ LVR
tree }\atop\text{ with $k$ cilia}}\atop e(T)\geq n+2}{\cal
A}^{k}_{T}(\lambda,N,J).$ (61)

This expansion is analytic for any $\lambda\in{\cal C}$ and the remainder at
order $n$ obeys, for $\sigma$ constant large enough, the analog of (96)

$|{\cal
R}^{k}_{n}(\lambda,N,J)|=\big{|}\mathfrak{K}^{k}(\lambda,N,J)-\sum_{m=0}^{n}a_{m}(N,J)\lambda^{m}\big{|}\leq\sigma^{n}\,[(p-1)n]!\,|\lambda|^{n+1},$
(62)

uniformly in $N\in{\mathbb{N}}^{*}$, J such that
$\|J^{\dagger}J\|<\epsilon_{\lambda}$. Therefore it obeys the theorem stated
in the Appendix of this article (Borel-LeRoy-Nevanlinna-Sokal) with $q\to
p-1$, $z\to\lambda$, $\omega\to\big{\\{}N,J\big{\\}}$, whenever
$N\in{\mathbb{N}}^{*},\|J^{\dagger}J\|<\epsilon_{\lambda}$.

As a corollary we state an analyticity result and a Borel summability for the
constructive expansion of scalar cumulants.

######  Corollary 1.

Let $k\leq k_{\max}$. The expansion

${\mathfrak{K}}^{k}_{\pi}(\lambda,N)=\sum_{T\text{ LVR tree with $2k$ cilia
}}{\cal A}^{\pi}_{T}(\lambda,N)\;$ (63)

defines an analytic function of $\lambda\in{\cal C}$. Moreover, each term in
this sum is bounded as:

$\big{|}{\cal
A}^{\pi}_{T}(\lambda,N)\big{|}\leq\frac{N^{2-|\pi|}|\lambda|^{e(T)}\,(k!)^{2}\,2^{2k}}{(\cos\frac{\arg\lambda}{(p-1)})^{2e(T)+k}\,v(T)!}\;,$
(64)

where $|\pi|$ is the number of integers in the partition $\pi$. This expansion
reads:

${\mathfrak{K}}^{k}_{\pi}(\lambda,N)=\sum_{G\text{ labeled ribbon graph with
$k$ cilia,}\atop\text{ broken faces corresponding to $\pi$, and $e(G)\leq
n$}}\frac{(-\lambda)^{e(G)}N^{\chi(G)}}{v(G)!}+{\cal
R}^{k}_{\pi,n}(\lambda,N)\;,$ (65)

where ${\cal R}^{k}_{\pi,n}(\lambda,N)$ is a sum over LVR graphs with $k$
cilia, at least $n+1$ edges and at most $n+1$ loop edges. This remainder is
uniformly analytic for $\lambda\in{\cal C}$, and it obeys the bound, for
$\sigma$ constant large enough,

$\displaystyle\Big{|}{\cal R}^{k}_{\pi,n}(\lambda,N)\Big{|}$
$\displaystyle\leq$
$\displaystyle\;\sigma^{n}\,N^{2-|\pi|}\>[(p-1)n]!\;|\lambda|^{n+1}.$ (66)

So it obeys the theorem stated in the Appendix of this article (Borel-LeRoy-
Nevanlinna-Sokal) with $q\to p-1$, $z\to\lambda$, $\omega\to N$ whenever
$N\in{\mathbb{N}}^{*}$.

##  5 Proof of Theorem 1.

###  5.1 Strategy

Theorem 1 contains three pieces, respectively indexed by ${\cal
P}^{k}_{n}(\lambda,N,J)$, ${\cal Q}^{k}_{n}(\lambda,N,J)$ and ${\cal
R}^{k}_{n}(\lambda,N,J)$, because $\mathfrak{K}^{k}(\lambda,N)$ is a sum of
three pieces. So the proof can be decomposed into three parts:

  * •

the one who concerns ${\cal P}^{k}_{n}(\lambda,N,J)$,

  * •

the one who concerns ${\cal Q}^{k}_{n}(\lambda,N,J)$,

  * •

and the part that concerns the remainder at order $n$, ${\cal
R}^{k}_{n}(\lambda,N,J)$.

For the first part, the one who concerns

$\sum_{{G\text{ labeled ribbon graph }\atop\text{ with $2k$ cilia
}}\atop|e(G)|\leq
n}\hskip-8.5359pt\frac{(-\lambda)^{|e(G)|}N^{\chi(G)}}{|v(G)|!}\hskip-2.84544pt\prod_{f\in
b(G)}\hskip-2.84544pt\operatorname{Tr}(JJ^{\dagger})^{c(f)}$ (67)

the proof is rather trivial: it is absolutely convergent expansion and this
expansion is analytic for any $\lambda\in{\cal C}$ since it is is not only
analytic but polynomial. It remain to prove the two other parts. Here we go.

###  5.2 Proof of the part concerning ${\cal Q}^{k}_{n}(\lambda,N,J)$

This part concerns a sum over labeled ribbon graphs with $k$ cilia of LVR
amplitudes ${\cal A}_{(G,T)}(\lambda,N,J)$. But in the definition of ${\cal
A}_{(G,T)}(\lambda,N,J)$ given by (41), the only part depending on $J$, is
$(J^{\dagger}J)^{\eta_{c}}$. Since for $1\leq k\leq k_{max}$,
$(J^{\dagger}J)^{\eta_{c}}$ is evidently bounded by $\|J^{\dagger}J\|^{k}$ we
do not worry about this part. It holds.

We turn now to the part no longer dependent on $J$, hence dependent of
holomorphic matrix calculus and to contour integrals like [20]; therefore no
wonder we are going to make a heavy use of the notations and the results of
[20]. It is crucial to make the distinction between $\operatorname{Tr}$ and
$\operatorname{Tr}_{\otimes}$. To simplify the notations of this subsection we
often forget the dependency on $\lambda$ when no confusion is possible. For
example we often write simply $\Sigma(M)$ or even $\Sigma$ for
$\Sigma(\lambda,M)$, where $\Sigma(\lambda,M)$ is defined by (51).

Figure 4: A _finite_ keyhole contour $\Gamma^{f}_{r,R,\psi}$ encircling a
segment on the real positive $Ox$ axis, which includes the spectrum of $X$.
The spectrum of $X$ lies on a real axis positive segment, like the one shown
in boldface.

Given a holomorphic function $f$ on a domain containing the spectrum of a
square matrix $X$, Cauchy’s integral formula yields a convenient expression
for $f(X)$:

$f(X)=\oint_{\Gamma}du\frac{f(u)}{u-X},$ (68)

provided the contour $\Gamma$ is a _finite_ keyhole contour enclosing all the
spectrum of $X$ (see Figure 4).

In [20] it is established that

$A(\lambda,X)=\oint_{\Gamma}du\;a(\lambda,u)\,\frac{1}{u-X},$ (69)

where $a(\lambda,u)=uT_{p}(-\lambda u^{p-1})$ (see (2),(13)). On the other
hand by a useful lemma also proven in [20], we know that

$\frac{\partial A}{\partial X}=\big{[}{\bf
1}_{\otimes}+\lambda\Sigma(\lambda,X)\big{]}^{-1}=\big{[}{\bf
1}_{\otimes}+\lambda\Sigma\big{]}^{-1}.$ (70)

Then we can write the matrix derivative acting on a resolvent. We obtain by
wrtiting again the superscript $i$

$\frac{\partial A}{\partial X^{i}}=\big{[}{\bf
1}_{\otimes}+\lambda\Sigma^{i}\big{]}^{-1}=\oint_{\Gamma}du\;a(\lambda,u)\frac{1}{u-X^{i}}\otimes\frac{1}{u-X^{i}}.$
(71)

Now we reapply the holomorphic calculus, but in different ways666Our choices
below are made in order to allow for the bounds of Section 5.3. depending on
the term chosen in the sum over $k$.

  * •

For $k=0$, we apply the holomorphic calculus to the right
$\frac{A^{p-1}(\lambda,X)}{u-X}$ factor, with a contour $\Gamma_{2}$
surrounding $\Gamma_{0}$ for a new variable called $v_{2}$, and we rename $u$
and $\Gamma_{0}$ as $v_{1}$ and $\Gamma_{1}$ (see Figure 5),

  * •

for $k=p-1$, we apply the holomorphic calculus to the left
$\frac{A^{p-1}(\lambda,X)}{u-X}$ factor, with a contour $\Gamma_{2}$
surrounding $\Gamma_{0}$ for a new variable called $v_{2}$, and we rename $u$
and $\Gamma_{0}$ as $v_{1}$ and $\Gamma_{1}$; we obtain a contribution
identical to the previous case,

  * •

in all other cases, hence for $1\leq k\leq p-2$, we apply the holomorphic
calculus both to left and right factors in the tensor product, with two
variables $v_{1}$ and $v_{2}$ and two equal contours $\Gamma_{1}$ and
$\Gamma_{2}$ enclosing enclose the contour $\Gamma_{0}$.

Figure 5: A keyhole contour $\Gamma_{1}$ encircling a keyhole contour
$\Gamma_{0}$.

The first $\frac{\partial}{\partial X}$ derivative is a bit special as it
destroys forever the logarithm in ${\cal S}(\lambda,X)$ and gives

$\bigg{[}\frac{\partial}{\partial
X}\bigg{]}\operatorname{Tr}_{\otimes}\log\big{[}{\bf
1}_{\otimes}+\Sigma\big{]}=\big{[}{\bf
1}_{\otimes}+\Sigma\big{]}^{-1}\frac{\partial\Sigma}{\partial X}\;.$ (72)

Recall $X_{l}=MM^{\dagger}$, $X_{r}=M^{\dagger}M$, (16) and recall, in (41),
the part no longer dependent on $J$:

$\frac{(-\lambda)^{e(G)}N^{v(G)-e(G)}}{v(G)!}\int dw_{T}\partial_{T}\int
d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)\prod_{f\in
f(G)}\operatorname{Tr}\mathop{\prod}\limits_{c\in\partial
f}^{\longrightarrow}\big{[}{\bf 1}_{\otimes}+\Sigma(\lambda,M)\big{]}^{-1}.$
(73)

Let us for the moment concentrate about the part depending on
$\operatorname{Tr}\mathop{\prod}\limits_{c\in\partial
f}^{\longrightarrow}\big{[}{\bf 1}_{\otimes}+\Sigma(\lambda,M)\big{]}^{-1}$
Defining the loop resolvent

${\cal
R}(v_{1},v_{2},M,M^{\dagger}):=\Big{[}\operatorname{Tr}\frac{1}{v_{1}-MM^{\dagger}}\Big{]}\Big{[}\operatorname{Tr}\frac{1}{v_{2}-M^{\dagger}M}\Big{]},$
(74)

we obtain

$\displaystyle\frac{\partial{\cal S}}{\partial\lambda}$ $\displaystyle=$
$\displaystyle-\oint_{\Gamma_{1}}dv_{1}\oint_{\Gamma_{2}}dv_{2}\Big{\\{}\oint_{\Gamma_{0}}du\;a(\lambda,u)\sum_{{\mathfrak{k}}=1}^{p-2}\frac{\partial_{\lambda}[\lambda
a^{\mathfrak{k}}(\lambda,v_{1})a^{p-{\mathfrak{k}}-1}(\lambda,v_{2})]}{(v_{1}-u)(v_{2}-u)}\,$
(75)
$\displaystyle\qquad+\;2a(\lambda,v_{1})\frac{\partial_{\lambda}\big{[}\lambda
a^{p-1}(\lambda,v_{2})\big{]}}{v_{1}-v_{2}}\Big{\\}}{\cal
R}(v_{1},v_{2},M,M^{\dagger}).$

In the manner of [20] we can now commute the functional integral and the
contour integration. This results in

$\displaystyle\int dw_{T}\partial_{T}\int
d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)\prod_{f\in
f(G)}\operatorname{Tr}\mathop{\prod}\limits_{c\in\partial
f}^{\longrightarrow}\big{[}{\bf 1}_{\otimes}+\Sigma(\lambda,M)\big{]}^{-1}$
(76) $\displaystyle=\sum_{T}\int\\{dw_{T}dtdudv\\}\Phi_{n}\int
d\mu_{C\\{x^{T}\\}}(M)\partial_{T}^{M}{\cal
R}^{k}_{n}\Big{|}_{x_{ij}=x_{ij}^{T}(w)},$

where

$\displaystyle\partial_{T}^{M}$ $\displaystyle:=$
$\displaystyle\prod_{(i,j)\in
T}\operatorname{Tr}_{\otimes}\Big{[}\frac{x_{ij}^{T}+x_{ji}^{T}}{2}\frac{\partial}{\partial
M_{i}}\frac{\partial}{\partial M^{\dagger}_{j}}\Big{]},$ (77)
$\displaystyle{\cal R}^{k}_{n}$ $\displaystyle:=$
$\displaystyle\prod_{i=k+1}^{n}{\cal
R}(v^{i}_{1},v^{i}_{2},M_{i},M_{i}^{\dagger}),$ (78)

and the symbol $\int\\{dw_{T}dtdudv\\}\Phi_{n}$ stands for

$\displaystyle\int\\{dw_{T}dtdudv\\}\Phi_{n}$ $\displaystyle=$
$\displaystyle\prod_{i,j\in
T}\int_{0}^{1}dw_{ij}\prod_{i=1}^{n}\Bigg{[}\int_{0}^{\lambda}dt^{i}\oint_{\Gamma^{i}_{1}}dv^{i}_{1}\oint_{\Gamma^{i}_{2}}dv^{i}_{2}$
(79)
$\displaystyle\Big{\\{}\oint_{\Gamma^{i}_{0}}du^{i}\phi(t^{i},u^{i},v^{i}_{1},v^{i}_{2})+\psi(t^{i},v^{i}_{1},v^{i}_{2})\Big{\\}}\Bigg{]}.$

The trace $\operatorname{Tr}_{\otimes}$ in (77) can also be thought as two
independent traces $\operatorname{Tr}$ associated to ordinary loops (hence the
name “loop vertex representation”). In (5.2) they are only coupled through the
scalar factors of (79). The nice property of this LVR representation is that
it does not break the symmetry between the two factors of the tensor product
in (71).

The condition on the contours $\Gamma^{f}_{r_{j},\psi_{j},R_{j}}$ for
$j=0,1,2$, can be written

$\displaystyle
0<\psi_{0}<\min(\psi_{1},\psi_{2})\leq\max(\psi_{1},\psi_{2})<\delta,$ (80)
$\displaystyle 0<r_{0}<\min(r_{1},r_{2});\quad\|MM^{\dagger}\|+1\leq
R_{0}<\min(R_{1},R_{2}).$ (81)

${\cal S}$ is _not uniformly bounded in $MM^{\dagger}$_ but grows
logarithmically at large $\|MM^{\dagger}\|$. However it _fully disappear in
the LVR formulas below_ , because these formulas do not use ${\cal S}$ but
derivatives of ${\cal S}$ with respect to the field $M$ or $M^{\dagger}$.
Hence we may use _infinite_ contours $\Gamma^{\infty}_{r,\psi}$ which are
completely independent of $\|MM^{\dagger}\|$ [20].

The outcome of applying $\partial^{M}_{T}$ to ${\cal R}^{k}_{n}$ is a bit
difficult to write, but the combinatorics has been treated in [20]. For a
single loop the Faà di Bruno formula allows to write this outcome as a sum
over a set $\Pi^{q,\bar{q}}_{r}$ of Faà di Bruno terms, each one of these with
a factor 1:

$\displaystyle\frac{\partial^{r}}{\partial M_{1}\cdots\partial M_{q}\partial
M^{\dagger}_{1}\cdots\partial M^{\dagger}_{\bar{q}}}\frac{1}{v-X}\\!$
$\displaystyle=$
$\displaystyle\\!\hskip-2.84544pt\sum_{\pi\in\Pi^{q,\bar{q}}_{r}}\;\operatorname{Tr}\Big{[}O^{\pi}_{0}\sqcup
O^{\pi}_{1}\sqcup\cdots\sqcup O^{\pi}_{r}\Big{]}.$ (82)

In the sum (82) there are exactly $r$ symbols $\sqcup$, separating $r+1$
corner operators $O^{\pi}_{c}$.

The result of this computation is obtained by identifying the two ends of each
pair of $\sqcup$ symbols along each edge of $T$. This pairing of the $2n-2$
$\sqcup$ symbols then exactly glue the $2n$ traces of the tensor products
present in the $n$ vertices into $n+1$ traces. These corner operators can be
of four different types, either resolvents $\frac{1}{v-X}$, $M$-resolvents
$\frac{1}{v-X}M$, $M^{\dagger}$-resolvents $M^{\dagger}\frac{1}{v-X}$, or the
identity operator ${\bf 1}$. We call $r_{\pi}$, $r^{M}_{\pi}$,
$r^{M^{\dagger}}_{\pi}$ and $i_{\pi}$ the number of corresponding operators in
$\pi$. By a lemma proven in [20] we know

$|\Pi^{q,\bar{q}}_{r}|\leq 2^{r}r!,\quad r_{\pi}=1+i_{\pi},\quad
r^{M}_{\pi}+r^{M^{\dagger}}_{\pi}=r-2i_{\pi}.$ (83)

Applying (82) at each of the two loops of each loop vertex, we get for any
tree $T$

$\partial_{T}^{M}\operatorname{Tr}\frac{1}{v-X}=\prod_{i=1}^{n}\Big{\\{}\prod_{j=1}^{2}\Big{[}\sum_{\pi_{j}^{i}\in\Pi_{r^{i}_{j}}^{q^{i}_{j},\bar{q}^{i}_{j}}}\operatorname{Tr}\big{(}O^{\pi_{j}^{i}}_{0}\sqcup
O^{\pi_{j}^{i}}_{1}\sqcup\cdots\sqcup
O^{\pi_{j}^{i}}_{r^{i}_{j}}\big{)}\Big{]}\Big{\\}}$ (84)

where the indices of the previous (82) are simply all decomposed into indices
for each loop $j=1,2$ of each loop vertex $i=1,\cdots,n$.

Exactly as in [20], we simply glue the $\sqcup$ symbols of (84) into $n+1$
traces $\operatorname{Tr}$. This is the fundamental common feature of the LVE-
LVR. Each trace acts on the product of all corners operators $O^{c}$
cyclically ordered in the way obtained by turning around the connected
components $\bar{T}$. Hence we obtain, with hopefully transparent notations,

$\displaystyle\partial_{T}^{M}{\cal R}^{k}_{n}\Big{|}_{x_{ij}=x_{ij}^{T}(w)}$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\prod_{j=1}^{2}\sum_{\pi_{j}^{i}\in\Pi_{r^{i}_{j}}^{q^{i}_{j},\bar{q}^{i}_{j}}}\big{[}\operatorname{Tr}\prod_{c\;\circlearrowleft\;\bar{T}}O^{c}(M^{\dagger}M)^{\eta_{c}}\big{]}.$
(85)

###  5.3 Proof of the part concerning ${\cal R}^{k}_{n}(\lambda,N,J)$

For the remainder part ${\cal R}^{k}_{n}(\lambda,N,J)$, it is a sum over the
_LRV tree amplitudes_ with $2k$ cilia and $e(T)\geq n+2$, and therefore we can
use (85). Hence we write

$\displaystyle{\cal R}^{k}_{n}(\lambda,N,J)\\!$ $\displaystyle=$
$\displaystyle\\!\hskip-14.22636pt\sum_{{T\text{ LVR tree }\atop{{\text{ with
}}2k\text{ cilia}},}\atop e(T)\geq
n+2}\int\\{dw_{T}dtdudv\\}\Phi_{n}F^{k}_{T}(\lambda,N,J,v)\hskip-5.69046pt\prod_{f\in
b(G)}\hskip-2.84544pt\operatorname{Tr}[(JJ^{\dagger})^{c(f)}],$ $\displaystyle
F^{k}_{T}(\lambda,N,v)$ $\displaystyle=$ $\displaystyle\hskip-5.69046pt\int
d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)\prod_{i=1}^{n}\prod_{j=1}^{2}\sum_{\pi_{j}^{i}\in\Pi_{r^{i}_{j}}^{q^{i}_{j},\bar{q}^{i}_{j}}}\big{[}\operatorname{Tr}\prod_{c\;\circlearrowleft\;\bar{T}}O^{c}(M^{\dagger}M)^{\eta_{c}}\big{]}.$
(86)

We now bound the functional integral. Since there are exactly $n+1$ traces,
the factors $N$ _exactly cancel_ , all operator norms now commute and taking
into account (83) we are left with

$|F^{k}_{T}(\lambda,N,v)|\leq K^{n}\int
d\mu_{C\\{x_{ij}^{\cal{T}}\\}}(M)\prod_{i=1}^{n}r_{i}!\Bigl{[}\prod_{c\in\bar{T}}\|O^{c}(M^{\dagger}M)^{\eta_{c}}\|\Big{]}_{x_{ij}=\frac{x_{ij}^{T}(w)+x_{ji}^{T}(w)}{2}}.$
(87)

where, like in [20], $K$ is a constant.

Exactly as in [20], using that
$\sup\\{\|M\|,\|M^{\dagger}\|\\}\leq\|M^{\dagger}M\|^{1/2}$, it is easy to now
bound, for $v$’s on these keyhole contours, the norm of resolvent factors such
as $\|\frac{1}{v^{i}_{j}-X^{i}}\|$ by a constant times $(1+|v^{i}_{j}|)^{-1}$
and the norm of resolvent factors such as $\|\frac{1}{v^{i}_{j}-X^{i}}M^{i}\|$
or $\|{M^{i}}^{\dagger}\frac{1}{v^{i}_{j}-X^{i}}\|$ by a constant times
$(1+|v^{i}_{j}|)^{-1/2}$. Plugging into (87) we can use again (83) to prove
that we get exactly a decay factor $(1+|v^{i}_{j}|)^{-(1+r^{i}_{j}/2)}$ for
each of the $2n$ loops. The corresponding bound being _uniform_ in all
$\pi,\\{w\\},\\{M\\}$, and since the integral $\int d\mu_{C\\{x^{T}\\}}$ is
_normalized_ , we get

$|F^{k}_{T}(\lambda,N,v)|\leq
K^{n}\;\prod_{i=1}^{n}\Big{\\{}r_{i}!\prod_{j=1}^{2}(1+|v^{i}_{j}|)^{-(1+r^{i}_{j}/2)}\Big{\\}}.$
(88)

Recall that with our notations, $r_{i}=r^{i}_{1}+r^{i}_{2}$. Since all
integrals with respect to $w$ are _normalized_ , i.e.

$\int dw_{T}\phi(w_{T})=\prod_{(i,j)\in
T}\int_{0}^{1}dw_{ij}\phi(\\{w_{ij}\\})\leq\|\phi(\\{w_{ij}\\})\|,$ (89)

we have simply to bound

$\displaystyle\int\\{dw_{T}dtdudv\\}\Phi_{n}\leq\|\int\\{dtdudv\\}\Phi_{n}\|.$
(90)

Now we bound the contour integral bound $\int\\{dtdudv\\}\Phi_{n}$ exactly
like in [20]. Finally since each vertex has at least one contour operator, the
number of $|\lambda|^{\frac{1}{4p^{2}}}$ factors in the bound is at least $n$.
Taking into account that the number of (labeled) trees $T$ is bounded by
$K^{n}n!$ for some constant $K$, we arrive at

$\displaystyle{\cal R}^{k}_{n}(\lambda,N,J)$ $\displaystyle=$ $\displaystyle
N^{k}\|J^{\dagger}J\|^{k}\sum_{n=k}^{\infty}K^{n}|\lambda|^{n+2+\frac{n}{4p^{2}}}\quad$
(91) $\displaystyle\leq$ $\displaystyle K^{k}N^{k}\|J^{\dagger}J\|^{k},$ (92)

uniformly in $N\in{\mathbb{N}}^{*}$, J such that
$\|J^{\dagger}J\|<\epsilon_{\lambda}$.

###  5.4 Conclusion

So where is, should we say, the crux of Theorem 1? It’s in the “Borel-LeRoy
part” of the perturbative expansion. For the perturbative expansion, defined
in (59),

$\displaystyle\hskip-34.14322pt\sum_{{G\text{ labeled ribbon
graph}\atop\text{with $k$ cilia},}\atop e(G)\leq
n}\hskip-8.5359pt\frac{(-\lambda)^{e(G)}N^{\chi(G)}}{v(G)!}\hskip-2.84544pt\prod_{f\in
b(G)}\hskip-2.84544pt\operatorname{Tr}(J^{\eta_{c}})+\hskip-5.69046pt\sum_{{(G,T)\text{
LVR graph }\atop\text{ with $k$ cilia,}}\atop e(G)=n+1}\hskip-8.5359pt{\cal
A}^{k}_{(G,T)}(\lambda,N,J),$ (93)

the analytic part of the proof of Theorem 1 is obvious, since this part is a
_polynomial_ with respect to $\lambda$; but this polynomial is of order

$e(G)\leq(p-1)e(T)\leq(p-1)(n+1),\quad$ (94)

so, when we arrived at proposition 2, more precisely at (96), we inevitably
transform a factor $(qn)!$ into a factor $[(p-1)n]!$.

Then Corollary 1 would follow easily from the proof of Theorem 1 and [19].

##  6 Appendix: Borel-LeRoy-Nevanlinna-Sokal theorem

We recall the following theorem [27, 28, 29].

$\begin{array}[]{cc}\includegraphics[width=113.81102pt]{Boreldisc.pdf}&\includegraphics[width=190.63338pt]{Borelstrip.pdf}\\\
{\cal D}_{R}&\Sigma_{\sigma}\end{array}$
Figure 6: Domain of analyticity of $F$ and of its Borel transform for $q=1$.

######  Theorem 2.

Let $q\in{\mathbb{N}^{*}}$. Let $F_{\omega}(z)$ be a family of analytic
functions on the domain

$D_{R}=\\{z:\Re
z^{-\frac{1}{q}}>(2R)^{-1}\\}=\\{z:|z|<(2R)^{q}\cos^{q}(\frac{\arg z}{q})\\}$
(95)

depending on some parameter $\omega\in\Omega$, and such that, for some
$\sigma\in{\mathbb{R}}_{+}$,

$|R_{n}(z)|=\big{|}F_{\omega}(z)-\sum_{m=0}^{n}a_{m}(\omega)z^{m}\big{|}\leq\sigma^{n}(qn)!\,|z|^{n+1}$
(96)

uniformly in $D_{R}$ and $\omega\in\Omega$. Then the formal expansion

$\sum_{n=0}^{\infty}s^{qn}\frac{a_{n}(\omega)}{(qn)!}$ (97)

is convergent for small $s$ and determines a function $B_{\omega}(s^{q})$
analytic in

$\Sigma_{\sigma}=\\{s:{\text{d}ist}(s,{\mathbb{R}}_{+})<\sigma^{-1}\\}$ (98)

and such that

$|B_{\omega}(s^{q})|\leq B\exp\big{(}\frac{|s|}{R}\big{)}$ (99)

uniformly for $\Sigma_{\sigma}$ (in (99) $B$ is a constant, that is, it is
independent of $\omega$). Moreover, setting $t=s^{q}$,

$F_{\omega}(z)=\frac{1}{qz}\int_{0}^{\infty}B_{\omega}(t)\;\Big{(}\frac{t}{z}\Big{)}^{\frac{1}{q}-1}\exp\bigg{(}-\Big{(}\frac{t}{z}\Big{)}^{\frac{1}{q}}\bigg{)}\;dt$
(100)

for all $z\in D_{R}$. Conversely, if $F_{\omega}(z)$ is given by (100), with
the above properties for $B_{\omega}(s^{q})$, then it satisfies remainder
estimates of the type (96) uniformly, in any $D_{r}$ such that $0<r<R$, and in
$\omega\in\Omega$.

For Theorem 1 in the core of this article, change $q\to p-1$, $z\to\lambda$,
$\omega\to\big{\\{}N,J\big{\\}}$, whenever
$N\in{\mathbb{N}}^{*},\|J^{\dagger}J\|<\epsilon_{\lambda}$. For Corollary 1,
change $q\to p-1$, $z\to\lambda$, $\omega\to N$ whenever
$N\in{\mathbb{N}}^{*}$.

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