# Towards a Simple Framework of Skill Transfer Learning for
Robotic Ultrasound-guidance Procedures

Tsz Yan Leung1 and Miguel Xochicale2 1 King’s College London, UK
<EMAIL_ADDRESS>Currently University College London, UK. Previously
King’s College London, UK<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a simple framework of skill transfer learning for
robotic ultrasound-guidance procedures. We briefly review challenges in skill
transfer learning for robotic ultrasound-guidance procedures. We then identify
the need of appropriate sampling techniques, computationally efficient neural
networks models that lead to the proposal of a simple framework of skill
transfer learning for real-time applications in robotic ultrasound-guidance
procedures. We present pilot experiments from two participants (one
experienced clinician and one non-clinician) looking for an optimal scanning
plane of the four-chamber cardiac view from a fetal phantom. We analysed
ultrasound image frames, time series of texture image features and quaternions
and found that the experienced clinician performed the procedure in a quicker
and smoother way compared to lengthy and non-constant movements from non-
clinicians. For future work, we pointed out the need of pruned and quantised
neural network models for real-time applications in robotic ultrasound-
guidance procedure. The resources to reproduce this work are available at
https://github.com/mxochicale/rami-icra2023.

##  I INTRODUCTION

Ultrasound (US) imaging is a popular imaging modality because of its
affordability, non-ionising imaging, and real-time capabilities. Recently, the
field of ultrasound-guidance procedures has been advanced with the development
of robotic ultrasound systems that range from tele-operated, semi-autonomous
and fully autonomous [1, 2, 3]. However, there are still scientific and
technical challenges in robotic ultrasound-guidance procedures: (a)
traditional imaging is user-dependent, skill-dependant and device-dependent
[4], (b) traditional hardware for human motion tracking is usually expensive
or cumbersome [5], and (c) frameworks are designed for specific types of
sensors, clinical US devices, robots and operating systems [6]. Learning
ultrasound skills from sonographers that look for the optimal scanning plane
(OSP) is still an open challenge in robotic ultrasound-guidance procedures
[1]. It is hypothesised that robotic ultrasound-guided procedures would
require to be simple, less expensive and less cumbersome for skill transfer
learning. Hence, in this paper, we are proposing a simple framework of skill
transfer learning for robotic ultrasound-guidance procedures. This paper is
divided into robotic ultrasound-guidance procedures, framework for sonographer
skill transfer learning, results and conclusions with future work.

##  II ROBOTIC ULTRASOUND-GUIDANCE PROCEDURES

Robotic ultrasound systems are actively investigated for teleoperatation,
semi-autonomous and fully autonomous modalities [2]. For instance, Deng et al.
proposed a multi-modal task learning architecture for ultrasound scanning
skills with input data from ultrasound images, force and probe pose, stating
the challenge of real-time guidance due to computational signal processing
[1]. Robotic US-guidance radiation therapy for lesion in abdomen has been
successful using CNN-based search. However, treatment plans without US-robot
guidance are still superior to robotic US-guidance because of the quality of
acquired ultrasound images [3].

Figure 1:  (a) Ultrasound-guidance procedures: sonographer operating an
ultrasound machine with fetal phantom and sensor fusion signals from inertial
sensors and ultrasound imaging; (b) Simple framework for skill transfer
learning: collecting experience with sensors ($Pt_{n}$ pose and $St$ Signal),
sampling method for fusion sensor ($\Delta_{t}$), and identified the need of
computational efficient neural network model ($\Omega_{\theta}$), and output
for high-dimensional model [1], and (c) Robotic ultrasound-guidance
procedures: transformations, graphical user interface and simulation using
robotic US-guidance light-weight 7 degrees-of-freedom robot (KUKA LBR Med 7)
[3, 7].

##  III FRAMEWORK FOR SONOGRAPHER SKILL TRANSFER LEARNING

Modelling the optimal scanning plane (OSP) requires ultrasound images, spacial
locations and anatomical understanding from experienced sonographers (Fig 1a)
[1, 2]. Current systems to model OPS are expensive and cumbersome leading to
the need of smaller and low-cost systems [5]. One potential avenue to reduce
cost and ergonomic form factor is (a) the use of Inertial Measurement Units
(IMU) to track probe position during US scanning procedures [8] and (b) the
application of the appropriate fusion techniques of ultrasound video signal
and motion signal from IMU for probe guidance [9], and (c) computationally
efficient neural networks models [1]. Hence, we introduce a framework for
sonographer skills transfer learning, considering: sampling technique and
fusion sensor techniques (IMU and US) and discuss the need of real-time
guidance with pruned and quantised neural network models, feature extraction
and transfer learning for robotic-ultrasound-guidance procedures (Fig 1b, c).

##  IV EXPERIMENTS: DESIGN AND RESULTS

###  IV-A Experiment setup

Considering fetal biometry parameters from the NHS 20-week screening scan
protocol and National Health Service (NHS) foetal anomaly screening programme
(FASP) [10], we conducted a pilot experiment with two participants (one non-
clinical and one clinical with 10 years of experience in echocardiography).
During the experiment, we asked participants to find the optimal scanning
plane for the four chamber view of foetal ultrasound examination phantom
(”SPACE FAN-ST”, Kyoto Kagaku Co., Ltd, Kyoto, Japan). We attached inertial
sensors (LPMS-B2, LP-Research Inc., Tokyo, Japan) to a convex US probe to
track the pose of the ultrasound image from a clinical ultrasound device (EPIQ
7G, Koninklijke Philips N.V., Amsterdam, Netherlands) with the use of a CPU
laptop computer that streamed images via a USB framegrabber (Mirabox).

###  IV-B Experimental results

Fig 2 shows the time series from one experienced clinician and one non-
clinician while looking for an optimal scanning plane of the four-chamber
view. It can be noted that the expert took less time (1600 and 2500 samples)
to find an optimal scanning plane whereas non-clinician took a greater time
(5000 and 10000 samples). Similarly, the time-series for the experienced
clinician appear to be more smooth and consistent compared to the jerkiness of
non-clinical participant.

Figure 2:  Ultrasound image frames, image distributions, image texture
features (energy, Angular Second Momentum ASM, and homogeneity) and pose time-
series (quaternions) from one experienced clinician (clinician01) and one non-
clinician (non-clinician01) in two trials (trial01 and trial02), looking for
four-chamber optimal scanning plane.

##  V CONCLUSIONS AND FUTURE WORK

We have presented a simple framework of skill transfer learning for robotic
ultrasound-guidance procedures. We presented sensor fusion methods and
sampling rate techniques for optimal scanning plane of the four-chamber view
from two participants (one experienced clinician and one non-clinicians),
where experienced clinician showed an smother and quicker procedure compare to
a lengthy and non-constant movement of non-clinician. For future work, we
pointed out the need of pruned and quantised neural network models for real-
time applications in robotic ultrasound-guidance procedure.

## ACKNOWLEDGMENT

Thanks to Tsz Yan Leung for her excellent research work during her M.Sc. in
Medical Physics and Engineering in 2022 at King’s College London. Thanks to
Nhat Phung for volunteering as experienced sonographer in the experiments to
identify optimal scanning plane of fetal four-chamber views at St Thomas’
Hospital.

## References

  * [1] X. Deng, Z. Lei, Y. Wang, and M. Li, “Learning ultrasound scanning skills from human demonstrations,” 2021. [Online]. Available: https://arxiv.org/abs/2111.09739
  * [2] F. von Haxthausen, S. Böttger, D. Wulff, J. Hagenah, V. García-Vázquez, and S. Ipsen, “Medical robotics for ultrasound imaging: Current systems and future trends,” _Current Robotics Reports_ , vol. 2, no. 1, pp. 55–71, Mar 2021. [Online]. Available: https://doi.org/10.1007/s43154-020-00037-y
  * [3] S. Gerlach, T. Hofmann, C. Fürweger, and A. Schlaefer, “Ai-based optimization for us-guided radiation therapy of the prostate,” _International Journal of Computer Assisted Radiology and Surgery_ , vol. 17, no. 11, pp. 2023–2032, Nov 2022. [Online]. Available: https://doi.org/10.1007/s11548-022-02664-6
  * [4] J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: Theoretical considerations and initial test,” _International Journal of Imaging Systems and Technology_ , vol. 8, no. 1, pp. 38–44, 1997.
  * [5] F. F. Dressler, C. Gratzke, A. Miernik, and D. S. Schoeb, “Track and teach: Identifying key movement patterns in endoscopic transurethral enucleation of the prostate,” _Urologia Internationalis_ , vol. 105, no. 9-10, pp. 835–845, 2021. [Online]. Available: https://www.karger.com/DOI/10.1159/000514596
  * [6] B. Niu, D. Yang, P. Wang, H. Yang, L. Zhang, Y. Gu, and L. Jiang, “Virtual-force-guided intraoperative ultrasound scanning with online lesion location prediction: A phantom study,” _The International Journal of Medical Robotics and Computer Assisted Surgery_ , vol. n/a, no. n/a, p. e2491. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/rcs.2491
  * [7] S. Ipsen, D. Wulff, I. Kuhlemann, A. Schweikard, and F. Ernst, “Towards automated ultrasound imaging—robotic image acquisition in liver and prostate for long-term motion monitoring,” _Physics in Medicine and Biology_ , vol. 66, no. 9, p. 094002, apr 2021. [Online]. Available: https://dx.doi.org/10.1088/1361-6560/abf277
  * [8] R. Prevost, M. Salehi, S. Jagoda, N. Kumar, J. Sprung, A. Ladikos, R. Bauer, O. Zettinig, and W. Wein, “3d freehand ultrasound without external tracking using deep learning,” _Medical Image Analysis_ , vol. 48, pp. 187–202, 2018\. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1361841518303712
  * [9] R. Droste, L. Drukker, A. T. Papageorghiou, and J. A. Noble, “Automatic probe movement guidance for freehand obstetric ultrasound,” in _Medical Image Computing and Computer Assisted Intervention – MICCAI 2020_. Cham: Springer International Publishing, 2020, pp. 583–592.
  * [10] “Guidance. 20-week screening scan. gov.uk.” https://www.gov.uk/government/publications/fetal-anomaly-screening-programme-handbook/20-week-screening-scan, accessed: 2023-03-15.


# Coalescent processes emerging from large deviations

Ethan Levien  Dartmouth College, Department of Mathematics
<EMAIL_ADDRESS>
###### Abstract

The classical model for the genealogies of a neutrally evolving population in
a fixed environment is due to Kingman. Kingman’s coalescent process, which
produces a binary tree, universally emerges from many microscopic models in
which the variance in the number of offspring is finite. It is understood that
power-law offspring distributions with infinite variance can result in a very
different type of coalescent structure with merging of more than two lineages.
Here we investigate the regime where the variance of the offspring
distribution is finite but comparable to the population size. This is achieved
by studying a model in which the log offspring sizes have a stretched
exponential form. Such offspring distributions are motivated by biology, where
they emerge from a toy model of growth in a heterogenous environment, but also
mathematics and statistical physics, where limit theorems and phase
transitions for sums over random exponentials have received considerable
attention due to their appearance in the partition function of Derrida’s
Random Energy Model (REM). We find that the limit coalescent is a
$\beta$-coalescent – a previously studied model emerging from evolutionary
dynamics models with heavy-tailed offspring distributions. We also discuss the
connection to previous results on the REM.

## 1 Introduction

Evolution is, in large part, shaped by a tension between two opposing
“forces”: neutral genetic drift and selection. Neutral genetic drift refers to
random changes in the genetic composition of a population due to chance events
in the deaths and reproduction of individuals. Selection, in contrast, results
from a deterministic bias towards fitter individuals. A major objective of
evolutionary biology is to determine the relative impact of these forces. In
order to achieve this, we need to understand how different microscopic
mechanisms manifest in macroscopic observables, such that the frequency of a
genotype or the shape of the genealogical tree.

Much of our understanding of the interplay between genetic drift and selection
comes from the Wright-Fisher model [NW18, DD08], or its counterpart with
overlapping generations, the Moran model. In both models, the source of noise
is the random sampling of individuals from generation to generation. In the
large population size limit this sampling noise has a variance which is
inversely proportional to $N$, the population size. When genetic drift is the
sole source of changes in the composition of the population – that is, in
neutrally evolving populations – $N$ sets the characteristic time-scale of the
evolutionary dynamics.

The true population size is rarely related in a simple way to the variance in
genotype frequency, as predicted by the Wright-Fisher model. Therefore, one
usually thinks of an $N$ as an _effective population size_ measuring the
overall strength of genetic drift, rather than the literal number of cells.
This has motivated the question: What determines the effective population
size? Of particular interest is the question of how non-genetic variation
between individuals shapes the effective population size [JLA23, LMKA21]. J.H.
Gillespie was one of the first to address this question by considering a model
in which the number of offspring from each individual is a random variable
having finite variance. He showed that in this context the effective
population size is obtained by scaling the true population size by the
variance in offspring [Gil73, SK12, Sch15].

We now have a more general and mathematically rigorous understanding of this
problem which is based on the Cannings model [Can74]. Beginning with $N$
labeled cells, trajectories of the Cannings model are constructed by
generating an exchangeable random vector $(\nu_{1,k},\dots,\nu_{N,k})$
satisfying $\sum_{i=1}^{N}\nu_{i,k}=N$ for each $k\in\mathbb{N}$. $\nu_{i,k}$
represents the number of offspring in generation $k$ which descend from
individual $i$ in the $(k-1)$th time-step. We will henceforth omit the
subscript $k$ and it should be understood that all quantities associated with
a generation are implicitly dependent on $k$.111The term _generation_ is used
to refer to a time-step in the Cannings model. Inspired by [Sch03, Hal18,
OH21], we will focus on the particular case where $\nu_{i}$ is obtained by
first generating iid random variables $\\{U_{i}\\}_{i=1}^{N}$ representing the
number of offspring of each individual _before_ resampling, and then sampling
the resulting offspring pool, without replacement, to obtain the individuals
in the next generation. Conditional on the offspring numbers
$U_{1},\dots,U_{N}$, $\nu_{i}$ follows a hypergeometric distribution:

$\mathbb{P}\left(\nu_{i}=n|\\{U_{i}\\}_{i=1}^{N}\right)=\frac{1}{{S_{N}\choose
N}}{U_{i}\choose n}{S_{N}-U_{i}\choose N-n}.$ (1)

This formulation bears a close resemblance to the model studied by Gillespie
[Gil73] and is an example of a Generalized Wright-Fisher model defined in
[DEP11].

One limit theorem for the Cannings model concerns the behavior of genotype
frequencies over long time-scales in large populations. By long time-scales,
we mean on the order of the _coalescent time_ , defined as the average number
of generations we must travel backwards in time to find a common ancestor for
two randomly selected individuals in the same generation. The coalescent time
is equivalent to the effective population size in some cases, but is more
general in the sense that the definition does not require a mapping to the
Wright-Fisher model. Under the assumption that the variance in the number of
offspring produced by each individual is finite, the time-rescaled genotype
frequencies converge as $N\to\infty$ (in the Skorokhod sense – see [Ker22,
EK09]) to the well-known Wright-Fisher diffusion (WFD). [MS01, Gil74]. In the
WFD, the change in the frequency, $X(t)$ of a genotype over a time interval
$dt\ll 1$ is Normally distributed with mean zero and variance
$X(t)(1-X(t))dt$.

There is similar limit Theorem, due to Kingman [Kin82], for the genealogical
trees in the Cannings model. These trees can be generated by the stochastic
process known as a _coalescent processes_ which, roughly speaking, is
specifies which lineages have merged $k$ generations back in time from our
original sample. Under the condition that the genotype frequencies converge to
the WFD, the time-rescaled coalescent process converges to a continuous time
process known now as the Kingman Coalescent. The genealogical tree produced by
this process is almost surely a binary tree where pairs of lineages merge at a
rate one, and importantly, the probably of more than two lineages merging in a
single instant is zero. Similar results exist for other microscopic models
(e.g. the Moran process) and a large body of work focuses on understanding how
the microscopic details of the process shape the coalescent time, or effective
population size [Gil74, JLA23, Cha09, Gil04, Gil01].

The WFD/Kingman models are not always sufficient to capture the dynamics of
real evolution. This is due to, for example, _multiple merger_ coalescents
appearing in experimental data [TL14, SHJ19]. In multiple merger coalescents
there is a non-negligible probability to observe more than two individuals
sharing a common ancestor in a single unit of time (e.g. a generation in the
model) on time-scales of order of the coalescent time. In [Sch03],
Schweinsberg explored the question of whether such genealogies can emerge from
neutral evolution by studying the Cannings model with power-law distributions.
A number of papers have since investigated the role of power-law tail
offspring distribution in generating multiple merger coalescents and non-
diffusive genotype frequency fluctuations [Hal18, OH21, CGCSWB22]. The
genotype frequency dynamics which emerge from power law offspring – known as
$\Lambda$-Fleming Viot processes – are non-diffusive processes which, in
general, have discontinuous sample paths. With the notable exception of
[CGCSWB22], most previous work has focused on how multiple merger coalescents
emerge when the variance in offspring is infinite. The assumption of infinite
variance is a mathematical convenience, which may be justified in some cases
e.g. for certain models of dormancy [CGCSWB22, WV19] and rare mutations
[LD43]. What remains unclear is which coalescent processes emerge when the
offspring distribution has a variance which is finite, yet large enough
relative to the population size to give rise to multiple merger coalescents.

In this paper, in order to understand the role of large but finite offspring
variability, we investigate the limit processes which emerge when the
population size and offspring variability are simultaneously taken to be
large. Similar limits were investigated in [CGCSWB22], but we focus on a more
specific scaling between population size and offspring variability, which
allows us to obtain precise descriptions of the limit coalescents. The
offspring distribution we consider and our scaling assumption are both
inspired by prior work on the Random Energy Model (REM) of disordered systems.

Our main result (Theorem 2) says that the limit processes emerging form the
genealogies of the Cannings model under this scaling limit, called
$\beta$-coalescents, are the same as the coalescent processes emerging from
power law offspring. Our model is parameterized by a scaling rate which is
analogous to the temperature of the REM. Just as in the REM, we find there are
two critical points. Below the lower critical point there is no continuous
time limit process, while between the two critical points one finds multiple
merger coalescents. However, while the lower critical point corresponds
exactly to the lower critical point of the REM marking the transition to the
“frozen” state, the upper critical point does have the obvious interpretation
in the context of the REM (which would be to separate the regimes of strong
and weak-self averaging of the partition function). Our results complement and
expand upon the existing connection between coalescent theory and the REM,
which was made by Bolthausen and Sznitman in [BS98].

### 1.1 Organization of this paper

This paper is organized as follows. In Section 2.1 we describe the model under
consideration, which is a particular instance of the Cannings model. In
Sections 2.2 and 2.3 we review some background of coalescent theory and review
what is known about this model. In Sections 3 we present our main result which
concerns the limiting coalescent when both the population size and offspring
variation are taken to be large. We also discuss the relationship between our
results and [CGCSWB22]. Section 4 is devoted to the Random Energy Model and
the connection between our results and the thermodynamic limit of the REM.

## 2 Background

Throughout this paper, we use the following standard notation. $g\sim f$ means
$f/g\to 1$, $a\wedge b=\min\\{a,b\\}$, $a\vee b=\max\\{a,b\\}$,
$(n)_{k}=n(n-1)\cdots(n-k+1)$ and $[n]=\\{1,2,\dots,n\\}$.

### 2.1 Exponential offspring model

In order to study the situation where variation in offspring numbers is
finite, but large relative to the population size, we set

$U_{i}=e^{\zeta\Phi_{i}},\quad i\in[N]$ (2)

where $\\{\Phi_{i}\\}_{i=1}^{N}$ are iid and $\zeta$ is a deterministic
parameter. Since we are interested in the large noise limit, we will
eventually take $\zeta\to\infty$. In the remainder of this paper, in order to
avoid ambiguity we will often replace $i$ with $1$ when referencing elements
of an exchangeable random vector. Note that, technically, $U_{1}$ should take
values in $\mathbb{Z}$, however in the large noise limit the distinction is
not relevant. Moreover, taking $\zeta\to\infty$ implies
$\mathbb{E}[U_{1}]\to\infty$, so we can assume that

$\mathbb{E}[U_{1}]>1.$ (3)

Offspring distribution of the form 2 were also considered in [SJHW23] within
the context of a model including heritability and [CGCSWB22] in a model of
dormancy.

It is biologically sensible to work with variation on an exponential scale
whenever the organisms in question proliferate exponentially between
bottlenecks. In this context, $\zeta\Phi_{i}$ is the product of the duration
of growth between bottlenecks and the time-averaged growth rate for the
offspring of the $i$th cell. For example, imagine an asexual population that
is subject to successive cycles of growth and dilution in a heterogeneous
environment. An example is a microbial pathogen, such as _Mycobacterium
tuberculosis_ [Gag18, FVJH+21]. Suppose that after passing through a
bottleneck, each of the $N$ cells occupies a spatially distinct location.
Then, due to heterogeneity in the environment, the offspring of each cell will
proliferate at different rates, $\Phi_{i}$. If $\zeta$ is the duration of the
growth phase, the total number of offspring from the $i$th cell (before
resampling) will be of the form 2. Alternatively, one could imagine that it is
the duration of the growth phase which is random – for example, due to a
period of dormancy before growth – while the growth rates are fixed. In this
case we would use $\zeta$ to denote the growth rates and $\Phi_{i}$ the
duration of the growth phase. Such a dormancy model was studied in [CGCSWB22].

We focus on the case where $\Phi_{1}$ has a stretched exponential
distribution, which essentially means the large deviations of $\Phi_{1}$ are
scale-invariant. To be precise, let

$W(\phi)\equiv-\ln P(\Phi_{1}>\phi).$

We assume $W(\phi)$ is smooth and bounded as $\phi\to 0$ and, inspired by
[Eis83, BABM05], assume

$W(\phi)\sim W^{*}(\phi)\equiv\frac{1}{q}\phi^{q},\quad q\geq 1.$ (4)

as $\phi\to\infty$. The pre-factor of $1/q$ is arbitrary, since this could be
absorbed into $\zeta$, but this particular form will lead to some more elegant
analytical formula. Note that we can always set $\mathbb{E}[\Phi_{1}]=0$
without loss of generality, since the contribution from
$e^{\zeta\mathbb{E}[\Phi_{1}]}$ cancels in the ratio $U_{1}/S_{N}$. Note that
offspring distribution satisfying Equations 3 and 4 include the special cases
where $U_{1}$ is exponential $(q=1)$ and lognormal $(q=2)$.

### 2.2 Characterization of limit coalescent processes

For a realization of the Cannings model with population size $N$, the
corresponding discrete coalescent process on a sample of size $n$, denoted by
$(\Psi_{n,k}^{N})_{k\geq 0}$, describes the genealogical tree obtained by
following each sampled individual’s ancestors back through time and grouping
branches when individuals share a common ancestor. This process is shown in
Figure 1. We can intuitively understand $\Psi_{n,k}^{N}$ as the state of this
tree $k$ generations back from the time our original sample was taken. To
define $\Psi_{n,k}^{N}$ more mathematically, let $\mathcal{P}_{n}$ denote the
space of partitions on $[n]=\\{1,\dots,n\\}$; that is, each element of
$\mathcal{P}_{n}$ is a set of disjoint subsets of $[n]$ whose union is $[n]$.
Then $\Psi_{n,k}^{N}$ is a Markov chain taking values in $\mathcal{P}_{n}$ and
$\Psi_{n,0}^{N}$ is the partition into singletons; that is,
$\Psi_{n,0}^{N}=\\{\\{1\\},\\{2\\},\dots,\\{n\\}\\}$. For $k>0$, indices
$i,j\in[n]$ are in the same block of the partition if and only if the $i$th
and $j$th individuals in the original sample share an ancestor $k$ generation
in the past.

The continuous time coalescent processes which emerge as limits of
$\Psi_{n,k}^{N}$ from any exchangeable Cannings model have been characterized
in [MS01]. The authors consider the large-$N$ limit of the time-scaled
coalescent process

$\Psi_{n}^{N}(t)=\Psi_{n,\lfloor t/c_{N}\rfloor}^{N}$

where $c_{N}$ is the probability two random selected individuals in one
generation share an ancestor in the previous generation. Observe that
$c_{N}^{-1}$ is simply the coalescent time, since the number of generations to
the most recent common ancestor of two random selected individuals from the
current generation follows a geometric distribution with parameter $c_{N}$.
Conditional on $\\{\nu_{i}\\}$, the chance for two individuals to both be
descendants of the first individual in the previous generation is
$(\nu_{1}/N)(\nu_{1}-1)/(N-1)$. Multiplying by $N$ and averaging over all
possible $\\{\nu_{i}\\}_{i=1}^{N}$ gives

$c_{N}=\frac{\mathbb{E}\left[\nu_{1}(\nu_{1}-1)\right]}{N-1}.$ (5)

We emphasize that, given the distribution of $U_{i}$, it is not
straightforward to compute $c_{N}$ because it has a nonlinear dependence on
the sum $S_{N}$.

The precise notion of convergence considered is in the Skorohod sense 222Let
${\mathbb{D}}_{\mathcal{P}_{n}}[0,\infty)$ denote the space of cádlág
functions (right continuous functions whose left limit exits) from
$[0,\infty)$ to $\mathcal{P}_{n}$. The results of [MS01] concern instances
where $(\Psi_{n}^{N}(t))_{t\geq 0}$ has a weak limit
${\mathbb{D}}_{\mathcal{P}_{n}}[0,\infty)$ under the Skorohod toplogy
$J_{1}$.– see [EK09] for details. If $(\Psi_{n}(t))_{t\geq 0}$ converges to a
continuous time process $(\Psi_{n}^{N}(t))_{t\geq 0}$ in the Skorohod sense,
the possible transitions that can occur in the process $\Psi_{n}(t)$ involve
$\sum_{i=1}^{r}k_{i}$ of the $n=\sum_{i=1}^{r}k_{i}+s$ blocks in
$\mathcal{P}_{n}$ merging into $r$ lineages by collapsing into groups of sizes
$k_{1},\dots,k_{r}$ while $s$ lineages remain unchanged. Such events are
called $(n,k_{1},\dots,k_{r};s)$-collisions. The rates of these collisions,
which uniquely determine the law of $\Psi_{n}(t)$, will be denoted by
$\lambda_{n,k_{1},\dots,k_{r};s}$ for $n>2$.

Figure 1: (left) A simulation of the Cannings model. Squares indicate
individuals at the beginning of each generation and the protruding lines are
their offspring. The thick red lines indicate an example of a discrete
genealogy, or coalescent $\Psi_{5,k}^{5}$, obtained from a sample of the final
population of labeled cells. For example,
$\Psi_{5,1}^{5}=\\{\\{1\\},\\{2,3\\},\\{4,5\\}\\}$. (right) A larger
simulation of a continuous time coalescent process which is obtained in the
limit of the model on the left.

In order to relate the merge rates $\lambda_{n,k_{1},\dots,k_{r};s}$ to the
distribution of $\nu_{i}$, one notes that after conditioning on
$\\{\nu_{i}\\}_{i=1}^{N_{\zeta}}$ the chance that $r$ groups of sizes
$k_{1},\dots,k_{r}$ each descended from individuals $1,\dots,r$ in the
previous generation is

$\frac{1}{(N)_{\sum_{i=1}^{r}k_{i}}}\prod_{i=1}^{r}(\nu_{i})_{k_{i}}.$

It follows that

$\lambda_{n,k_{1},\dots,k_{r};0}=\lim_{\zeta\to\infty}c_{N}^{-1}N^{r-n}\mathbb{E}\left[\prod_{i=1}^{r}(\nu_{i})_{k_{i}}\right].$
(6)

In particular, if this limit exists and $c_{N}^{-1}\to\infty$, the time-
rescaled coalescent $\Psi_{n}^{N}(t)$ converges to a process $\Psi_{n}(t)$
with merger rates given by Equation 6. The rates for $s>0$ can be obtained via
a certain recursive relation discussed in [Sch01].

Of particular relevance for our results are those coalescent processes for
which there are multiple mergers, but not simultaneous multiple mergers – in
this case $\lambda_{n,k_{1},\dots,k_{r},s}=0$ unless $r=1$. Such coalescents
are known as $\Lambda$-coalescents and are defined by the merger rates

$\lambda_{n,k}=\lim_{N\to\infty}c_{N}^{-1}N^{1-n}\mathbb{E}\left[(\nu_{1})_{k}\right].$
(7)

Any rates defined in this way will satisfy the consistency condition

$\lambda_{n,k}-\lambda_{n+1,k}=\lambda_{n+1,k+1},$ (8)

which is proved in [Pit99]. The intuition behind Equation 8 is: The difference
between the rate to see $k$ mergers in a sample of size $n$ and $n+1$ is
entirely caused by mergers of $k+1$ lineages in the sample of $n+1$, since
these look like $k$ mergers when restricted to the sample of size $n$. Pitman
also proved that any triangular array
$\\{\lambda_{n,k}\\}_{n=1,\dots,\infty,k=1,\dots,n}$ satisfying Equation 8 has
a representation in terms of a positive measure
$\Lambda:[0,1]\to\mathbb{R}_{\geq 0}$ via the relation

$\lambda_{n,k}=\int_{0}^{1}z^{k-2}(1-z)^{n-k}\Lambda(dz).$ (9)

Thus, there is a correspondence between coalescent process and positive
measures on the unit interval.

A simple criteria for the convergence to a $\Lambda$-coalescent is found by
noting that if $\lambda_{2,2,2;s}=0$ then there are no simultaneous multiple
mergers, since these would contribute to this rate. Therefore, if $c_{N}\to 0$
and

$\lim_{N\to\infty}c_{N}^{-1}\frac{\mathbb{E}[(\nu_{1})_{2}(\nu_{2})_{2}]}{N^{2}}=0$
(10)

the limit process of the genealogy $(\Psi_{n,k}^{N})_{k\geq 0}$ is a
$\Lambda$-coalescent [MS01]. The following proposition, which follows from
Propositions 1 and 3 of [Sch03], summarizes these observations.

######  Proposition 1.

If $c_{N}\to 0$ and Equation 10 is satisfied, then as $N\to\infty$,
$(\Psi_{n}^{N}(t))_{t\geq 0}=(\Psi_{n,\lfloor t/c_{N}\rfloor}^{N})_{t\geq 0}$
converges (in the Skorohod sense) to a $\Lambda$-coalescent
$(\Psi_{n}(t))_{t\geq 0}$ with merger rates $\lambda_{n,k}$ given by Equation
7 for $k=n$ and Equation 8 otherwise.

### 2.3 Known result for $q=1$

We now return to the Cannings model with offspring distributions given by
Equation 4. When $q=1$, $\Phi_{i}$ has exponential tails and the offspring
sizes, $U_{i}$, have power law tails

$P(U_{i}>u)=P\left(\Phi_{i}>\frac{1}{\zeta}\ln u\right)\sim Cu^{-\alpha_{1}}$

for $\alpha_{1}=1/\zeta$ and some constant $C>0$. The large $N$ limit of the
Cannings model with power law offspring (and $\zeta$ fixed) is covered by the
main result of [Sch03], which we have stated below in an abbreviated form

######  Theorem 1.

Assuming Equation 2.3 holds, then as $N\to\infty$ we have

  * •

When $2\leq\alpha_{1}$, $\Psi_{n,\lfloor t/c_{N}\rfloor}^{N}$ converges to the
$\Lambda$-coalescent with $\Lambda=\delta$, which is called the Kingman
coalescent.

  * •

For $1\leq\alpha_{1}<2$, $\Psi_{n,\lfloor t/c_{N}\rfloor}^{N}$ converges a
$\Lambda$-coalescent where $\Lambda$ given by a $\beta$ distribution ${\rm
Beta}(2-\alpha_{1},\alpha_{1})$. It follows from Equation 9 that the merger
rates are

$\lambda_{n,k}=\frac{B(k-\alpha_{1},n-k+\alpha_{1})}{B(2-\alpha_{1},\alpha_{1})}$
(11)

where $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ is the beta function. This
process is a called a $\beta$-coalescent with parameter $\alpha_{1}$.

  * •

For $\alpha_{1}<1$, lineages will coalesce in $O(N^{0})$ and hence there is no
continuous time limit process for any rescaling of time. We refer to [Sch03]
for a detailed description of this process.

This Theorem is closely related to the Generalized CLT (GCLT) for the sum
$S_{N}$ – see [Hal18, OH21]. Recall that the GCLT tells us there are two
critical points for the limit law of $S_{N}$, one where the CLT breaks down
($\alpha_{1}=2$) and another where the LLN breaks down($\alpha_{1}=1$) - see
e.g. [Ami20b, Ami20a, Nol20]. In Theorem 1, these critical points correspond
to the appearance of multiple mergers and a disappearance of any continuous
time limit process respectively.

## 3 Limit coalescent for large but finite offspring variability

### 3.1 Scaling assumption

Our main result concerns the case where $q>1$. We reiterate that in this case
the variance is finite for all $\zeta$, and therefore in the large $N$ limit
(with $\zeta$ fixed) the convergence is to the WFD for the Allele frequencies
and Kingman Coalescent for the genealogies. For any fixed $N$, if $\zeta$ is
large enough the WFD/Kingman models are of course going to provide a very poor
approximation. In order to better understand exactly what happens when $\zeta$
is large, but finite, we set $N=N_{\zeta}$ where $N_{\zeta}$ grows with
$\zeta$ in such a way that there is a well-defined limit process as
$\zeta\to\infty$. The appropriate scaling is related to the thermodynamic
limit of the REM [Eis83, BABM05]; see Section 4.

To state our scaling assumption, we define the _cumulant generating function_
,

$r_{\zeta}=\ln\mathbb{E}[e^{\zeta\Phi_{i}}].$

To simplify some formulas later on, we define
$A_{\zeta}\equiv\mathbb{E}[S_{\zeta}]=N_{\zeta}e^{r_{\zeta}}$ with

$S_{\zeta}\equiv S_{N_{\zeta}}=\sum_{i=1}^{N_{\zeta}}e^{\zeta\Phi_{i}}.$

The relationship between $r_{\zeta}$ and $W^{*}(z)$ is crucial for our
analysis. Using the Laplace method [DB81], which amounts to evaluating
$\mathbb{E}[e^{\zeta\Phi_{i}}]$ where the integral attains its maximum, it can
be shown that $r_{\zeta}$ is asymptotically the convex conjugates of $W^{*}$.
As a result, these functions are related according to the Legendre transform
[Tou05]:

$r_{\zeta}\sim r_{\zeta}^{*}=\inf_{z\geq 0}\\{z\zeta-W^{*}(z)\\}.$

It then follows from Equations 4 that

$r_{\zeta}^{*}=\frac{1}{q^{\prime}}\zeta^{q^{\prime}}$ (12)

where $q^{\prime}$ is the so-called dual exponent $q^{\prime}=q/(q-1)$. The
equivalence between Equations 12 and 4 is a special case of Kasahara–de
Bruijn’s exponential Tauberian Theorem [Mik99].

It follows from Equation 12 that all the moments of $U_{i}$ grow as
$\zeta^{q^{\prime}}$, since

$\mathbb{E}[U_{i}^{m}]=\mathbb{E}[e^{m\zeta\Phi}]=e^{r_{m\zeta}}\sim
e^{m^{q^{\prime}}r_{\zeta}^{*}}$

Intuitively, if $\ln N_{\zeta}$ grows slower than $\zeta^{q^{\prime}}$ then
$N_{\zeta}$ cannot keep up with the variation as $\zeta$ increases and no
continuous time limit process will exists. This serves as motivation for the
scaling assumption,

$N_{\zeta}=e^{\tau r_{\zeta}}.$ (13)

Here, $\tau$ is a control parameter closely related to temperature in the
Random Energy Model. Limit theorems for the sum $S_{\zeta}$ under this scaling
assumption are proved in [BABM05]. The authors show that, much like in the
GCLT, $S_{\zeta}$ has two critical points, one where the CLT breaks down and
one where the LLN breaks down. Moreover, the limit law of $S_{\zeta}$ is
scale-invariant (see Theorem 4 in Section 4), which strongly suggests a
connection between the limit coalescent processes obtained from offspring
distribution of the form 2 under the scaling assumption 13, and those
described by Theorem 1.

### 3.2 Main result

The following theorem generalizes Theorem 1 to the case $q>1$. Interestingly,
we find that the $\beta$-coalescent emerges universally from exponentially
large offspring variation.

######  Theorem 2.

Assume Equations 4 and 13 holds with $q>1$. Let

$\alpha_{q}=(W^{*})^{\prime}\left((1+\tau)r_{1}^{*}\right)=\left(\frac{1+\tau}{q^{\prime}}\right)^{q/q^{\prime}}.$
(14)

and $\Psi_{n,k}^{\zeta}=\Psi_{n,k}^{N_{\zeta}}$. Then

  * •

For $1<\alpha_{q}<2$, as $\zeta\to\infty$ the discrete coalescent process
$(\Psi_{n,\lfloor t/c_{\zeta}\rfloor}^{\zeta})_{t\geq 0}$ converges to a
$\beta$-coalescent with parameter $\alpha_{q}$.

  * •

For $\alpha_{q}>2$, $(\Psi_{n,\lfloor t/c_{\zeta}\rfloor}^{\zeta})_{t\geq 0}$
converges to the Kingman coalescent.

In Figure 2, the region $1<\alpha_{q}<2$ is plotted for various values of $q$.
By moving upwards through these regions that we obtain $\beta$-coalescents. As
expected, for larger $q$, the region becomes more tilted towards the right,
since a larger value of $\zeta$ is needed to obtain a continuous-time limit
process for the same $N$. As $q\to 1$, the region becomes a vertical strip
between $1/\zeta=1$ and $1/\zeta=2$, hence Theorem 1 is retrieved in this
limit. The regime $\alpha_{q}\leq 1$ requires a different treatment not
covered by this result, but is closely related to calculations of the Gibbs
measure for the REM at low temperature, as we discuss in Section 4.

Figure 2: (left) The region $1<\alpha<2$ in $\ln N-\zeta$ space for different
values of $q$. (right) A diagram of the idea behind the definition of
$\alpha$.

We will Theorem 2 in Section 5. Here, we provide an outline of the derivation
which we will be useful when making the comparison to results for the REM.
Following [Sch03], note that we can replace $\nu_{1}$ with $NU_{1}/S_{N}$ when
computing averages (this is justified by Lemma 3). Therefore

$\mathbb{E}\left[(\nu_{1})_{k}\right]\sim\mathbb{E}\left[\nu_{1}^{k}\right]\sim
N_{\zeta}^{k}\mathbb{E}\left[\left(\frac{U_{1}}{S_{\zeta}}\right)^{k}\right].$
(15)

A LLN for $S_{\zeta}$ (Lemma 2) allows us to replace $S_{\zeta}$ with
$U_{1}+N_{\zeta}e^{r_{\zeta}}$ (this is justified by Lemma 6). Hence, letting
$f_{U}(u)$ denote the density of $U_{i}$ and replacing sums with integrals
(this is justified rigorously by [Sch03, Lemma 12]), we have

$\displaystyle
N_{\zeta}^{k}\mathbb{E}\left[\left(\frac{U_{1}}{S_{\zeta}}\right)^{k}\right]$
$\displaystyle\sim
N_{\zeta}^{k}\int_{1}^{\infty}\frac{u^{k}}{(u+Ne^{r_{\zeta}})^{k}}f_{U}(u)du$
(16)
$\displaystyle=e^{r_{\zeta}}N_{\zeta}^{k+1}\int_{(A_{\zeta})^{-1}}^{\infty}\frac{z^{k}}{(z+1)^{k}}f_{U}(zA_{\zeta})dz.$
(17)

To obtain the last equation we have changed variables $z=u/A_{\zeta}$. If
$f_{U}$ is a power law, then integral can be evaluated and is given in terms
of $\Gamma$-functions – this is one way to prove Theorem 1, although a
different approach is taken in [Sch03]. The essence of our argument is that
when we evaluate this integral we can replace $f_{U}(zA_{\zeta})$ with
$e^{W^{*}(\zeta^{-1}(\ln A_{\zeta}z))}$ and then neglect the higher order
terms in

$\displaystyle W^{*}\left(\frac{\ln A_{\zeta}z}{\zeta}\right)$
$\displaystyle=W^{*}\left(\frac{\ln
A_{\zeta}}{\zeta}\right)+\frac{1}{\zeta}(W^{*})^{\prime}\left(\frac{\ln
A_{\zeta}}{\zeta}\right)\ln z+\cdots$
$\displaystyle=\frac{(1+\tau)^{q}}{q(q^{\prime})^{q}}\zeta^{(q^{\prime}-1)q}+\left(\frac{1+\tau}{q^{\prime}}\right)^{q/q^{\prime}}\zeta^{(q^{\prime}-1)(q-1)-1}\ln
z+\cdots.$
$\displaystyle=\frac{(1+\tau)^{q}}{q(q^{\prime})^{q}}\zeta^{q^{\prime}}+\alpha_{q}\ln
z+\cdots.$

Here, we have used the relations

$(q-1)(q^{\prime}-1)=1,\quad(q^{\prime}-1)q=q^{\prime}$

to simplify the exponents of $\zeta$. Since the coefficient of $\ln z$ is
independent of $\zeta$, $f_{U}$ is approximately a decaying power law with
exponent $\alpha_{q}$, defined by Equation 14. Making this replacement and
evaluating the integral leads to the following Lemma, which says that
$\mathbb{E}\left[(\nu_{1})_{k}\right]$ is dominated by the event $U_{1}>N$
when $k>\alpha_{q}$.

######  Lemma 1.

For $1<\alpha_{q}<k$, $k\in{\mathbb{N}}$ we have

$\mathbb{E}[(\nu_{1})_{k}]\sim\alpha_{q}B(\alpha_{q},k-\alpha_{q})N_{\zeta}^{k}\mathbb{P}(e^{\zeta\Phi_{1}}>A_{\zeta}),\quad\zeta\to\infty.$
(18)

If $\alpha_{q}<2$, then from Lemma 1 we have

$c_{\zeta}\sim\alpha_{q}B(\alpha_{q},2-\alpha_{q})N_{\zeta}\mathbb{P}(e^{\zeta\Phi_{1}}>A_{\zeta})$

and then from Equation 7 and another application of Lemma 1,

$\lambda_{n,n}=\frac{B(n-\alpha_{q},\alpha_{q})}{B(2-\alpha_{q},\alpha_{q})},\quad
n\geq 2.$

These are precisely the merger rates of the $\beta$-coalescent and due to the
consistency condition 8, uniquely determine the rates $\lambda_{n,k}$ for
$k<n$.

On the other hand, when $\alpha_{q}>k$, $\mathbb{E}[\nu^{k}]$ is no longer
dominated by the tail and we can replace $S_{\zeta}$ with $A_{\zeta}$ in
Equation 15, leading to

$\mathbb{E}[\nu_{1}^{k}]\sim\frac{N_{\zeta}^{k}\mathbb{E}[e^{k\zeta\Phi_{1}}]}{A_{\zeta}^{k}}.$

Thus if $\alpha_{q}>2$,

$c_{\zeta}\sim\frac{N_{\zeta}}{A_{\zeta}^{2}}\mathbb{E}[e^{2\zeta\Phi_{1}}]\sim
e^{(2^{q^{\prime}}-\tau-2)r^{*}_{\zeta}}.$ (19)

In this regime, $c_{\zeta}$ decays slower than
$N^{1-n}\mathbb{E}[\nu_{1}^{n}]$ for all $n>2$ and hence all the
$\lambda_{n,n}$ except $\lambda_{2,2}$ vanish.

### 3.3 Relationship to the result of [CGCSWB22]

We now remark on the relationship between Theorem 2 and the results of
[CGCSWB22]. In their model, it is assumed that at the beginning of each
generation (referred to as the spring in [CGCSWB22]), individuals experience a
period of dormancy during which no reproduction occurs. At random times,
individuals awaken from dormancy and reproduce according to a Yule process –
that is, new individuals are spawned from existing ones at a (deterministic)
rate until the end of the generation. The authors also allow for a period
(called the summer) during which all individuals are awake and reproducing,
although we will neglect this for the present discussion.

To make the connection to our model, let $\zeta$ rate at which cells divide
after the period of dormancy has ended and let $\Phi_{i}$ denote the duration
of the $i$th cell’s growth phase (i.e. the difference between the total time
between bottlenecks and the dormancy period). In the dormancy model, it
follows from properties of Yule processes that $U_{1}|\Phi_{1}$ is a geometric
random variable with parameter $e^{-\zeta\Phi_{1}}$, hence

$P\left(U_{1}>e^{\zeta\phi}|\Phi_{1}\right)=\left(1-e^{-\zeta\Phi_{1}}\right)^{\lfloor
e^{\zeta\phi}\rfloor}$

and $E[U_{1}|\Phi_{1}]=e^{\zeta\Phi_{1}}$. In contrast, in our model $U_{1}$
is deterministic after conditioning on $\Phi_{1}$. This distinction should
have no effect on the limit coalescent under our scaling assumption, since
$\left(1-e^{-\zeta\Phi_{1}}\right)^{\lfloor e^{\zeta\phi}\rfloor}$ can be
replaced with $1_{\Phi_{1}>\phi}$ in the large $\zeta$ limit, and therefore

$\displaystyle P\left(\frac{1}{\zeta}\ln U_{1}>\phi\right)$
$\displaystyle=\mathbb{E}\left[P\left(\frac{1}{\zeta}\ln
U_{1}>\phi|\Phi_{1}\right)\right]$
$\displaystyle=\mathbb{E}\left[P\left(U_{1}>e^{\zeta\phi}|\Phi_{1}\right)\right]\sim
P\left(\Phi_{1}>\phi\right).$

At least heuristically, this justifies the replacement of $U_{1}$ with
$e^{\zeta\Phi_{1}}$ when calculating asymptotic behavior of the coalescent
process. However, we have neglected variation $U_{1}|\Phi_{1}$ in order to
simplify the derivations in the present paper.

The main results of [CGCSWB22] concern the limit genealogies and are closely
related to ours. Theorem 1.3 concerns the situation where the period of growth
after dormancy is exponentially distributed and is therefore very similar to
Theorem 1 in the present paper. Their more general result, stated below,
characterizes the possible forms of $\Lambda$ which can emerge as limits of
the discrete coalescents in the dormancy model.

######  Theorem 3 (Proposition 1.8 and Theorem 1.7 from [CGCSWB22]).

Suppose $U_{i}$ are of the form 2. For any $\Lambda$-coalescent that emerges
as the limit of $(\Psi_{n,\lfloor t/c_{\zeta}\rfloor}^{\zeta})_{t\geq 0}$ in
the dormancy model described above, $\Lambda$ has the form

$\Lambda(dx)=b_{0}\delta_{0}(dx)+b_{1}\delta_{1}(dx)+h(x)dx$

where $\delta_{x}$ is the point mass at $x$ and $h$ is a probability density
on $(0,1)$ with the representation

$h(y)=\frac{y^{2}}{(1-y)^{2}}g\left(\frac{y}{1-y}\right)$ (20)

for a monotone function $g$ satisfying

$\int_{0}^{\infty}g(v)(1\wedge v^{2})dv<\infty.$

Our main result, Theorem 2, says that under the scaling assumption 13 the
limit coalescent is of the form described by Theorem 3 with
$g(v)=v^{-1-\alpha_{q}}$ and $b_{0}=b_{1}=0$.

## 4 Interpretation in the REM

### 4.1 Background on the REM

The REM was introduced by Derrida in [Der81] as a toy model of spin glasses.
As with other models of magnetic systems, the state space is the
$n$-hypercube, $\mathcal{C}_{n}\equiv\\{-1,1\\}^{n}$ and each configuration
$\sigma\in\mathcal{C}_{n}$ is assigned an energy, $E_{\sigma}$. When in
equilibrium with a reservoir of temperature $T$, the steady-state distribution
of configurations is given by the Boltzmann distribution $P_{\sigma}\propto
e^{-\beta E_{\sigma}}$ where $\beta^{-1}=T$ is the inverse temperature
(assuming $k_{B}=1$ for notational simplicity). The quantity of central
interest in (equilibrium) statistical mechanics is the free energy,

$F_{n}=\ln\sum_{\sigma\in\mathcal{C}_{n}}^{2^{n}}e^{-\beta E_{\sigma}}.$ (21)

It is said that the thermodynamic limit exists if the limit of $F_{n}/n$
exists, and by differentiating the _free energy density_ ,
$\psi\equiv-\lim_{n\to\infty}F_{n}/(\beta n)$, with respect to temperature (or
other model parameters) various _thermodynamic_ relations are obtained [Gol18,
Bov06]. In spin glass models, $E_{\sigma}$ is itself taken to be a random
variable, and the free energy density is computed from the mean free energy,
$\mathbb{E}[F_{n}]$. The hope is always that the thermodynamics emerging from
a random energy field are typical of systems with very unstructured energy
landscapes. For mathematical convenience
$\\{E_{\sigma}\\}_{\sigma\in\mathcal{C}_{n}}$ is usually taken to be a
Gaussian random field, and the simplest such model is of course found by
taking $E_{\sigma}$ to be iid. In order for the thermodynamic limit to exist
in this case, the variance of $E_{\sigma}$ must grow proportional to $n$ –
this is precisely Derrida’s REM, which can be understood as the limit of a
very rugged energy landscape.

Since there are no correlations between energies, we can abandon the hypercube
structure and identify the state space with $[2^{n}]$, writing $E_{i}$ for the
$i$th energy level. Following previous convections, we suppose that $E_{i}/n$
has variance $1/2$. Then, when $\Phi_{i}$ are Gaussian with unit variance, by
setting

$\displaystyle n$ $\displaystyle=\log_{2}N_{\zeta}$ $\displaystyle\beta$
$\displaystyle=\sqrt{\frac{2\ln 2}{\tau}}$ $\displaystyle E_{i}$
$\displaystyle=-\sqrt{\frac{n}{2}}\Phi_{i}$

we see that the partition function in the REM with these parameters is equal
to the total number of offspring in the Cannings model

$S_{\zeta}=\sum_{i=1}^{N_{\zeta}}e^{\zeta\Phi_{i}}=\sum_{i=1}^{2^{n}}e^{-\beta
E_{i}}.$

Note that the negative sign in front of $E_{i}$ is inconsequential because we
have assumed $\mathbb{E}[\Phi_{i}]=0$ and can simply change the sign in
Equation 4.

Interestingly, even in this simple model one finds a phase transition, which
occurs at a critical value $\beta_{c}=\sqrt{2\ln 2}$, or in our notation,
$\tau=1$ (with $q=2$). The nature of this transition can be understood by
examining the average number of configurations for which
$E_{i}\in[n\varepsilon,n(\varepsilon+d\varepsilon)]$, which we denote by
$\mathcal{N}(d\varepsilon)$ [MM09]. Loosely speaking,
$\mathcal{N}(d\varepsilon)$ is on the order of [Kis15]

$2^{n}e^{-n\varepsilon^{2}}={\rm
exp}\left\\{n\left[\frac{\beta_{c}^{2}}{2}-\varepsilon^{2}\right]\right\\}.$

This approximation makes sense only when $\varepsilon<\beta_{c}/\sqrt{2}$,
since for large $n$ there are virtually no configurations with
$\varepsilon>\beta_{c}/\sqrt{2}$. With these heuristics, one can identify two
regimes for the _entropy density_ ,
$s(\varepsilon)\equiv\lim_{n\to\infty}\frac{1}{n}\ln\mathcal{N}(d\varepsilon)$:
$s(\varepsilon)=\beta_{c}^{2}/2-\varepsilon^{2}$ when
$|\varepsilon|\leq\sqrt{2}\beta_{c}$ and $s(\varepsilon)=-\infty$ when
$|\varepsilon|>\sqrt{2}\beta_{c}$. Meanwhile, the free energy density can be
obtained as

$\psi(\beta)=\lim_{n\to\infty}\frac{1}{\beta
n}\int_{-\infty}^{\infty}e^{-\beta
n\varepsilon}\mathcal{N}(d\varepsilon)=\lim_{n\to\infty}\frac{1}{\beta
n}\ln\int_{-\infty}^{\infty}e^{n(s(\varepsilon)-\beta\varepsilon)}d\varepsilon.$

and hence $\psi(\beta)$ and $s(\varepsilon)$ are convex conjugates of each
other.333In the terminology of large deviation theory, $\phi(\beta)$ is the
scaled cumulant generating function associated with the normalized energy
$\varepsilon$, whose large deviation rate function is $s(\varepsilon)$
[Tou09]. A short calculation using the Laplace method yields Derrida’s result:

$\psi(\beta)=\left\\{\begin{array}[]{cc}-\frac{\beta}{4}-\frac{\beta_{c}^{2}}{2\beta}&\beta\leq\beta_{c}\\\
-\ln 2&\beta>\beta_{c}\end{array}\right..$ (22)

Intuitively, when $\beta\leq\beta_{c}$ the variation in energies is small
enough that the LLN can be applied to the partition function. Indeed, one way
to arrive at $\psi(\beta)$ in the high temperature phase is to replace
$\mathbb{E}[\ln S]$ with $\ln\mathbb{E}[S]$, since

$\mathbb{E}\left[\sum_{i=1}^{2^{n}}e^{-\beta
E_{i}}\right]=2^{n}e^{\beta^{2}n/4}=\exp\left\\{\frac{\beta_{c}^{2}}{2}+\frac{\beta^{2}}{4}\right\\}.$

In this regime, we say that the system is self-averaging. On the other hand,
when $\beta>\beta_{c}$, the system enters a so-called “frozen” phase where the
partition function is dominated by the extremal statistical weights.

The arguments above can be generalized to the sum $S_{\zeta}$ for any $q>1$ –
for example, see [Eis83]. Of central importance for the application to
coalescent theory is following LLN numbers, which tells us the partition
function of the REM (with non-gaussian energy distribution) is self-averaging
in the high temperature regime.

######  Lemma 2 (Theorem 2.1 from [BABM05]).

Let $q>1$ and assume

$\tau>\frac{1}{q-1}.$

Then for all $\varepsilon>0$,

$\lim_{\zeta\to\infty}\mathbb{P}\left(\left|\frac{S_{\zeta}}{A_{\zeta}}-1\right|>\varepsilon\right)=0.$

In the context of the Cannings model, the self-averaging of $S_{\zeta}$ allows
us to make the approximation used in Equation 16 and ensures there is a
continuous time limit process.

### 4.2 Limit theorem for the partition function

A richer mathematical structure to the REM is revealed by a careful study of
the fluctuations in the partition function. This is closely related to the
GCLT for iid sums over scale-invariant random variables, where one can
distinguish between regimes of weak and strong self-averaging. The former
refers to the case where there is a LLN, but no CLT for the sum. The precise
limit Theorem for Gaussian energy distributions is stated in [Bov06] and the
extension to distributions of the form 4 can be found in [BABM05]. We now
state (a slightly watered down) version of their result.

######  Theorem 4 (Theorem 2.3 from [BABM05]).

Let

$\tilde{\alpha}_{q}=\left(\frac{q}{q^{\prime}}\tau\right)^{1/q^{\prime}}.$
(23)

and

$Z_{\zeta}=\frac{S_{\zeta}-A_{\zeta}}{B_{\zeta}}.$ (24)

where $B_{\zeta}$ is defined by

$\tau r^{*}_{\zeta}=W^{*}\left(\frac{\ln B_{\zeta}}{\zeta}\right).$ (25)

  * •

For $1<\tilde{\alpha}_{q}<2$, $Z_{\zeta}$ converges in distribution to an
$\alpha$-stable random variable $Z_{\tilde{\alpha}}$ for which the
characteristic function is

$\varphi_{\tilde{\alpha}}(u)\equiv
E\left[e^{iZ_{\tilde{\alpha}_{q}}u}\right]=\exp\left\\{-\Gamma(1-\tilde{\alpha}_{q})|u|^{\tilde{\alpha}_{q}}e^{-i\pi\tilde{\alpha}_{q}{\rm
sgn}(u)/2}\right\\}$ (26)

where the parameter, $\tilde{\alpha}_{q}$, is given by Equation 23.

  * •

For $\tilde{\alpha}_{q}>2$, $U_{\zeta}$ obeys a central limit theorem, meaning
that it converges to a Gaussian when rescaled by the standard deviation.

Briefly, the $\alpha$-stable distributions mentioned in this result are
defined by the property that they are equal in distribution to linear
combinations of realizations of themselves: $Z$ is $\alpha$-stable if and only
if there are constant $c,d,e$ such that $Z=cZ_{1}+dZ_{2}+e$ for random two
variables $Z_{1}$ and $Z_{2}$ equal in distribution to $Z$. The GCLT states
that $\alpha$-stable distributions arise as limits of sums of iid random
variables whose variances are not necessarily finite [Ami20b, Zol86]. Hence,
our result is playing the role of Theorem 4 for the Cannings model by
expanding Theorem 1 to the “large but finite” regime.

Much like Theorem 2, the idea behind Theorem 4 is that in the transition
regime ($1<\tilde{\alpha}<2$), the sum will be dominated by the maximum. From
Equation 13 and Equation 4, we have

$\displaystyle\mathbb{P}\left({\rm max}_{1\leq i\leq N}U_{i}>u\right)$
$\displaystyle=1-\left(1-\mathbb{P}\left(e^{\zeta\Phi}>u\right)\right)^{N}$
$\displaystyle\sim N\mathbb{P}\left(e^{\zeta\Phi}>u\right)\sim e^{\tau
r_{\zeta}-W^{*}(\ln u/\zeta)}.$

The left-hand side will approach one as $\zeta\to\infty$, so in order to
obtain a well-defined limit we need to look at deviations on a scale
$B_{\zeta}$, which is increasing with $\zeta$. To this end, we replace $u$
with $uB_{\zeta}$ in the exponent of Equation 4.2 and make the approximation

$\displaystyle\tau r_{\zeta}-W^{*}\left(\frac{\ln u+\ln
B_{\zeta}}{\zeta}\right)\approx\tau r_{\zeta}-W^{*}\left(\frac{\ln
B_{\zeta}}{\zeta}\right)-\frac{\ln u}{\zeta}(W^{*})^{\prime}\left(\frac{\ln
B_{\zeta}}{\zeta}\right).$ (27)

In order for this to have a large $\zeta$ limit, the first two terms must
cancel, indicating that $B_{\zeta}$ should satisfy Equation 25. Since
$B_{\zeta}=O(\zeta^{q^{\prime}})$, it follows from Equation 25 that the
coefficient of $\ln u$ in Equation 27 is independent of $\zeta$ and given by

$\tilde{\alpha}_{q}=\frac{1}{\zeta}(W^{*})^{\prime}\left(\frac{\ln
B_{\zeta}}{\zeta}\right)$ (28)

which simplifies to Equation 23. This plays the role of $\alpha$ in our
analysis of the coalescent process. The two parameters agree only at
$\alpha=\tilde{\alpha}=1$, which is the transition to the frozen state; see
Figure 3. Interestingly, this implies the regime of weak-self averaging in
Theorem 4 ($1<\tilde{\alpha}_{q}<2$) does not exactly correspond to the regime
of multiple merger coalescents in Theorem 2 ($1<\alpha_{q}<2$), although the
two coincide in the limit $q\to 1$.

Figure 3: $\alpha$ compared to $\tilde{\alpha}$ as a function of the tail
exponent $q$ for different values of $\tau$. The lower limit of the plot is
the critical point $\alpha=1$ where both expressions agree. Above this point
$\alpha>\tilde{\alpha}$ for all $q$.

### 4.3 Gibbs measure

In addition to the partition function, there is an interest in understanding
fluctuations in the statistical weights

$P_{\sigma}=\frac{e^{\zeta\Phi_{\sigma}}}{S_{\zeta}},$ (29)

in the REM and other disordered systems models. In the large-$\zeta$ limit
these weights approach a measure on the infinite dimensional hypercube
$\mathcal{C}_{\infty}=\\{-1,1\\}^{\mathbb{Z}}$ called the _Gibbs measure_ –
see [RAS15] for an introduction to this formalism. The question of how the
Gibbs measure fluctuates between replicates of the energies is closely related
to the coalescent process, since $\\{U_{i}/S\\}_{i=1,\dots,N_{\zeta}}$
(asymptotically) have the same distribution as $P_{\sigma}$ when the
configurations are projected to the one dimensional lattice
$[2^{n}]=\\{1,\dots,2^{n}\\}$. The projected Gibbs measure approaches the
Lebesgue measure on the unit interval in the high temperature regime [Bov06].
However, the main focus in this context of the REM appears to have been the
low temperature regime, $\tilde{\alpha}_{q}<1$.

There is already an established connection between coalescent processes and
the Gibbs measure via Derrida’s generalized REM (GREM). In this model, the
energies are no longer independent, but drawn from a Gaussian random field on
$\mathcal{C}_{n}$ whose correlation function depends on a certain ultrametric
distance between configurations – see for a precise description [Ber09]. This
ultrametric distance induces a hierarchical structure to the configuration
space. Ruelle gave a mathematical formulation of Derrida’s models [Rue87] and
the connection to coalescent processes was made in [BS98] in the context of
their abstract cavity method. The authors show that by sampling configurations
from the Gibbs measure and constructing a genealogical tree based on the
hierarchies induced by the distance, one obtains (up to a time-change) a
continuous time coalescent process now referred to as the Bolthausen-Sznitman
coalescent. This processes is nothing but the $\beta$ coalescent with
$\tilde{\alpha}_{q}=1$.

To understand what Theorem 2 tells us about $P_{\sigma}$, suppose we sample
two configurations $i$ and $j$ from the equilibrium distribution and set

$\varrho_{i,j}=1_{\\{i=j\\}},$ (30)

which Derrida refers to this as the replica overlap. It is not too difficult
to see that $\mathbb{E}[\rho_{1,2}]$ is asymptotic to the inverse coalescent
time, $c_{\zeta}$: First, notice that conditional on
$\\{\Phi_{i}\\}_{i\in[N]}$ the distribution of $\varrho_{1,2}$ is (see [DM21])

$\mathbb{E}[\varrho_{1,2}|\\{\Phi_{k}\\}_{k\in\mathcal{C}_{n}}]=\sum_{i=1}^{2^{n}}\left(\frac{e^{\zeta\Phi_{i}}}{S_{\zeta}}\right)^{2}$
(31)

The joint distribution of the terms in the sum is asymptotic to that of
$\\{(\nu_{i}/N_{\zeta})^{2}\\}_{i\in[N]}$. Therefore, recalling Equation 5, we
can see that $\mathbb{E}[\rho_{1,2}]\sim c_{\zeta}$. It is well known that in
the high temperature regime ($\tilde{\alpha}>1$)

$\mathbb{E}[\varrho_{1,2}]\to 0,$ (32)

while in the low temperature phase Derrida derives an expression in terms of
the Beta functions.

Now suppose we sample configurations of the Gibbs measure from $n$ replicates
of the REM and consider the event that these configurations are not unique and
instead $k_{i}>1$ come from configuration $i$ for $i=1,\dots,r$ with
$\sum_{i=1}^{r}k_{i}=n$. Such events are simply the analogues of
$(k_{1},\dots,k_{r};0)$-collisions defined in Section 2.2, and they occur with
probability

$N_{\zeta}^{r-n}\mathbb{E}\left[\prod_{i=1}^{r}\left(P_{\sigma}\right)_{k_{i}}\right].$
(33)

This is asymptotic to the expression appearing in the definition of
$\lambda_{n,k_{1},\dots,k_{r};s}$ (Equation 6) and in the REM is related to
the higher order replica overlaps. We can then ask what chance of these events
is when we take $1/c_{\zeta}$ replicates, which yields
$\lambda_{n,k_{1},\dots,k_{r};s}$. Therefore, Theorem 2 tells us there is a
phase transition in the typical composition of our samples at the critical
point $\alpha=2$. As we have explained above, this transition happens at a
lower temperature (higher $\beta$) than the breakdown of the CLT for the
partition function at $\tilde{\alpha}_{q}=2$.

## 5 Proof of Theorem 2

In order to simplify notation in the proofs, we set $\alpha=\alpha_{q}$.

###### Proof of Lemma 1.

The idea of the proof is that most of the contribution to the moments of
$\nu_{1}$ come from the event $\nu_{1}>A_{\zeta}$. Let $f_{\Phi_{1}}$ denote
the density of $\Phi_{1}$; hence

$\int_{\phi}^{\infty}f_{\Phi_{1}}(x)dx=e^{-W(\phi)}.$

Define

$\eta_{\zeta}(z)=\frac{1}{\zeta}\ln(A_{\zeta}z).$

By Lemma 6 and Lemma 4 in the A,

$\displaystyle E[\nu_{1}^{k}]$ $\displaystyle\sim
N^{k}\int_{\zeta^{-1}\ln(1)}^{\infty}\frac{e^{k\zeta\phi}A_{\zeta}^{-k}}{(e^{\zeta\phi}A_{\zeta}^{-1}+1)^{k}}f_{\Phi_{1}}(\phi)d\phi$
$\displaystyle=e^{k\tau
r_{\zeta}}\int_{A_{\zeta}^{-1}}^{\infty}\frac{z^{k-1}}{(z+1)^{k}}\frac{1}{\zeta}W^{\prime}\left(\eta(\zeta,z)\right)e^{-W\left(\eta(\zeta,z)\right)}dz$

where we have changed variables $z=u/A_{\zeta}$. This is the step which breaks
down for $\alpha\leq 1$, since Lemma 6 uses Lemma 2.

Now define $R_{\zeta}(z)$ and $K_{\zeta}(z)$ by

$\displaystyle W\left(\eta_{\zeta}(z)\right)$
$\displaystyle=W\left(\eta_{\zeta}(1)\right)+\hat{\alpha}_{\zeta}\ln
z+R_{\zeta}\left(z\right)$ (34) $\displaystyle
W^{\prime}\left(\eta_{\zeta}(z)\right)$
$\displaystyle=\zeta\hat{\alpha}_{\zeta}+\zeta K_{\zeta}(z)$ (35)

where

$\hat{\alpha}_{\zeta}=\zeta^{-1}W^{\prime}\left(\eta_{\zeta}(1)\right).$ (36)

Observe that $\hat{\alpha}_{\zeta}\sim\alpha$ and by Equation 4 there is a
constant $C^{\prime}$ such that

$K_{\zeta}(z)\leq\frac{C^{\prime}\ln z}{\zeta}$

for large enough $\zeta$.

Because $R_{\zeta}(z)>0$ for large enough $\zeta$ for all $z$,

$\displaystyle\int_{A_{\zeta}^{-1}}^{\infty}\frac{z^{k-1}}{(z+1)^{k}}\frac{1}{\zeta}W^{\prime}\left(\eta_{\zeta}(z)\right)e^{-W\left(\eta_{\zeta}(z)\right)}dz$
(37) $\displaystyle\quad\leq
e^{-W(\eta_{\zeta}(1))}\int_{A_{\zeta}^{-1}}^{\infty}\frac{z^{k-1-\hat{\alpha}_{\zeta}}}{(z+1)^{k}}\frac{1}{\zeta}W^{\prime}\left(\eta_{\zeta}(z)\right)z.$
(38)

Assume $\zeta$ is large enough that
$|\hat{\alpha}_{\zeta}-\alpha|<\varepsilon$. Then

$\displaystyle\int_{A_{\zeta}^{-1}}^{\infty}\frac{z^{k-1-\hat{\alpha}_{\zeta}}}{(z+1)^{k}}\frac{W^{\prime}\left(\eta_{\zeta}(z)\right)}{\zeta}dz$
$\displaystyle\leq\int_{0}^{\infty}\frac{z^{k-1-\alpha(\tau)+\varepsilon}}{(z+1)^{k}}\frac{W^{\prime}\left(\eta_{\zeta}(z)\right)}{\zeta}dz.$
(39)

As $\epsilon\to 0$, using that the logarithmic term vanishes,

$\displaystyle\int_{0}^{\infty}\frac{z^{k-1-\alpha+\varepsilon}}{(z+1)^{k}}\frac{W^{\prime}\left(\eta_{\zeta}(z)\right)}{\zeta}dz$
$\displaystyle\to\alpha\int_{0}^{\infty}\frac{z^{k-1-\alpha}}{(z+1)^{k}}dz$
(40)
$\displaystyle=\alpha\frac{\Gamma(\alpha)\Gamma(k-\alpha)}{\Gamma(k)}=\alpha
B(\alpha,k-\alpha).$ (41)

Similarly, for large enough $\zeta$ and any $L$

$\displaystyle\int_{A_{\zeta}^{-1}}^{\infty}\frac{z^{k-1}}{(z+1)^{k}}\frac{W^{\prime}\left(\eta_{\zeta}(z)\right)}{\zeta}e^{-W\left(\eta_{\zeta}(z)\right)}dz$
(42) $\displaystyle\quad\geq
e^{-R_{\zeta}(L)-W(\eta_{\zeta}(1))}\int_{A_{\zeta}^{-1}}^{L}\frac{z^{k-1-\alpha-\varepsilon}}{(z+1)^{k}}\frac{W^{\prime}\left(\eta_{\zeta}(z)\right)}{\zeta}dz$
(43) $\displaystyle\quad\sim
e^{-R_{\zeta}(L)-W(\eta_{\zeta}(1))}\alpha\int_{A_{\zeta}^{-1}}^{L}\frac{z^{k-1-\alpha}}{(z+1)^{k}}dz$
(44)

We can expand $R_{\zeta}$ as

$R_{\zeta}(L_{\zeta})=\frac{1}{2}\left(\frac{\ln
L_{\zeta}}{\zeta}\right)^{2}W^{\prime\prime}\left(\frac{\ln
A_{\zeta}}{\zeta}\right)+\frac{1}{6}\left(\frac{\ln
L_{\zeta}}{\zeta}\right)^{3}W^{\prime\prime\prime}\left(\frac{\ln
A_{\zeta}}{\zeta}\right)+\cdots.$ (45)

Therefore, by the definition of $W^{*}$ (Equation 4),

$\frac{d^{n}}{dz^{n}}W^{*}(z)=(q-1)\cdots(q-n+1)z^{q-n}$ (46)

which means the coefficient of $\ln L_{\zeta}/\zeta$ grows as a power law in
$\zeta$ with exponent

$(q-n)(q^{\prime}-1)=\frac{q-n}{q-1}.$

Now set $L_{\zeta}=B\zeta^{q^{\prime}/4}$ for some constant $B$. The $\zeta$
dependence of the $n$th term is

$n(q^{\prime}/2-1)+(q^{\prime}-1)(q-n)=\frac{4-3n}{4}q^{\prime},$

so in the large $\zeta$ limit $R_{\zeta}(L_{\zeta})$ will vanish. In summary,

$\int_{A_{\zeta}^{-1}}^{\infty}\frac{z^{k-1}}{(z+1)^{k}}\frac{W^{\prime}\left(\eta_{\zeta}(z)\right)}{\zeta}e^{-W\left(\eta_{\zeta}(z)\right)}dz\sim
e^{-W^{*}(\eta_{\zeta}(1))}\alpha B(\alpha,k-\alpha)$ (47)

Finally, note that

$e^{-W(\eta_{\zeta}(1))}=\mathbb{P}\left(e^{\zeta\Phi_{1}}>A_{\zeta}\right),$
(48)

so

$\mathbb{E}[(\nu_{1})_{k}]\sim\alpha
B(\alpha,k-\alpha)N_{\zeta}^{k}\mathbb{P}(e^{\zeta\Phi_{1}}>A_{\zeta}).$ (49)

∎

If $\alpha>k$, the integral derived above diverges and the bounds are no
longer useful. The divergence comes from the very small values of $z$ (meaning
small $\Phi$ relative to $A_{\zeta}$) when we make the linear approximation.
In this case, we have

$\displaystyle\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{(e^{\zeta\Phi_{1}}+A_{\zeta})^{k}}\right]$
$\displaystyle\sim\frac{1}{A_{\zeta}^{k}}\mathbb{E}[e^{k\zeta\Phi_{1}}]$ (50)
$\displaystyle\sim e^{-\left(k(1+\tau)-k^{q^{\prime}}\right)r_{\zeta}^{*}}.$
(51)

Note that $\left(k(1+\tau)-k^{q^{\prime}}\right)r_{1}^{*}\geq
W^{*}((1+\tau)r_{1}^{*})$. In fact, these are equal exactly at $\alpha_{q}=2$.

###### Proof of Theorem 2.

Lemma 1 implies

$c_{\zeta}\sim\alpha
B(\alpha,2-\alpha)N_{\zeta}\mathbb{P}(e^{\zeta\Phi_{1}}>A_{\zeta})\sim\alpha
B(\alpha,2-\alpha)e^{-W^{*}\left(\eta_{\zeta}(1)\right)+\tau r_{\zeta}^{*}}.$
(52)

Simplifying the exponent, we have

$\displaystyle W^{*}\left(\eta_{\zeta}(1)\right)+\tau r_{\zeta}^{*}$
$\displaystyle=W^{*}\left(\frac{1}{\zeta}(1+\tau)r_{\zeta}^{*}\right)+\tau
r^{*}(\zeta)$ (53)
$\displaystyle=\left(\alpha^{q^{\prime}/q}-\frac{1}{q^{\prime}}-\frac{\alpha^{q^{\prime}}}{q}\right)\zeta^{q^{\prime}}.$
(54)

The prefactor of $\zeta^{q^{\prime}}$ evaluates to $0$ at $\alpha=1$. Since
$q>1$,

$\frac{d}{d\alpha}\left(\alpha^{q^{\prime}/q}-\frac{1}{q^{\prime}}-\frac{\alpha^{q^{\prime}}}{q}\right)=\frac{\alpha^{q^{\prime}/q}-\alpha^{q^{\prime}}}{\alpha(q-1)}<0,$
(55)

so that $c_{\zeta}^{-1}\to\infty$ for $\alpha>1$.

The merger rates in Equation 11 follow from Lemma 1, so it remains to check
10. Applying Lemma 5 and 1,

$\displaystyle\mathbb{E}[\nu_{1}^{2}\nu_{2}^{2}]$ $\displaystyle\sim
N_{\zeta}^{4}\mathbb{E}\left[\frac{U_{1}^{2}U_{2}^{2}}{S_{\zeta}^{4}}\right]\leq
N_{\zeta}^{4}\mathbb{E}\left[\frac{e^{\zeta(\Phi_{1}+\Phi_{2})}}{(e^{2\zeta\Phi_{1}}\vee
A_{\zeta}^{2})(e^{2\zeta\Phi_{1}}\vee A_{\zeta}^{2})}\right]$ (56)
$\displaystyle=N_{\zeta}^{4}\mathbb{E}\left[\frac{e^{2\zeta\Phi_{1}}}{e^{2\zeta\Phi_{1}}\vee
A_{\zeta}^{2}}\right]^{2}\leq(A^{\prime})^{2}N_{\zeta}^{2}c_{\zeta}^{2}$ (57)

Notice that we are again using the idea that the expectation is dominated by
the event $e^{\zeta\Phi_{1}}>A_{\zeta}$, since the final expression above is
asymptotic to
$(A^{\prime})^{2}N_{\zeta}^{4}\mathbb{P}(e^{\zeta\Phi_{1}}>A_{\zeta})^{2}$. It
follows that

$c_{\zeta}\frac{\mathbb{E}[\nu_{1}^{2}\nu_{2}^{2}]}{N_{\zeta}^{2}}=c_{\zeta}\to
0.$ (58)

When $\alpha>2$, Lemma 1 and Equation 19 implies

$c_{\zeta}^{-1}\frac{\mathbb{E}\left[\nu_{1}^{3}\right]}{N_{\zeta}^{2}}\sim\frac{\mathbb{E}\left[\nu_{1}^{3}\right]}{N_{\zeta}\mathbb{E}[\nu_{1}^{2}]}=\frac{N_{\zeta}^{2}\mathbb{P}(e^{\zeta\Phi_{1}}>A_{\zeta})}{\mathbb{E}[\nu_{1}^{2}]}\to
0.$ (59)

It follows that $\lambda_{3,3}\to 0$, and by Equation 8 $\lambda_{n,n}=0$ for
$n>3$.

∎

## 6 Discussion

In this article we have studied the asymptotics of genealogies in the Cannings
model when both the population sizes and offspring variation are
simultaneously taken to be large. Such limits are not covered by previous
results for scale invariant offspring distribution, since in that setting the
offspring variation is infinite for finite $N$. Our analysis rests on a
certain scaling scaling assumption under which the total number of offspring
produced in a generation is equivalent to the partition function of the REM
and the offspring numbers $\nu_{i}$ are related to the Gibbs measure. As with
the REM, competition between fluctuations in growth rates (energies in the
REM) with averaging over an increasing system-size (our log population size)
leads to a form of weak self-averaging and anomalous scaling of the coalescent
time.

Our main finding is that the $\beta$-coalescent – a previous studied model of
coalescence in populations where variation in offspring is infinite – also
emerge from models where the tail of the offspring distribution is thin, but
large fluctuations are not too unlikely. This is related to the existing limit
theorems for the Cannings model proved in [Sch03] in the same way that the
fluctuation theorem in [BABM05] is related to the GCLT for iid sums. Our
result does not describe the critical point and low temperature regime,
although at least for $q=2$ these can be likely be deduced from previous
results on the REM – see [Bov06]. The limit coalescent processes are the
discrete-time $\Xi$-coalescents with simultaneous multiple mergings of
lineages described in [MS01].

Biologically, Theorem 2 suggests the $\beta$-coalescent serves as a more
universal description of neutral evolution in the presence of highly skewed
offspring distributions than might be expected. This also indicates that
little information about the demographic structure of a population is
contained in the coalescent process itself.

### Acknowledgments

I am thankful to Daniel Weissman, Takashi Okada, Linh Huynh, Oskar Hallatschek
and Ariel Amir for helpful discussions and feedback on early versions of this
work. Three anonymous referees provided detailed feedback which greatly
improved the quality of this manuscript. Finally, I would like to thank the
organizers of the conference “Evolution Beyond the Classical Limits” held at
the Banff International Research Station (BIRS), where preliminary work on
this manuscript was carried out.

##  Appendix A Additional Lemmas

We will assume $U_{i}>1$ and therefore $S_{N}>N$. It is straightforward to
obtain the results for the more general case where only Equation 3 holds.

######  Lemma 3.

For $0<\varepsilon<x$

$\limsup_{\zeta\to\infty}N_{\zeta}T_{\zeta}\mathbb{P}\left(\frac{\nu_{1}}{N_{\zeta}}>x\right)\leq\limsup_{\zeta\to\infty}NT\mathbb{P}\left(\frac{U_{1}}{S_{\zeta}}\geq
x-\varepsilon\right)$ (60)

and

$\limsup_{\zeta\to\infty}N_{\zeta}T_{\zeta}\mathbb{P}\left(\frac{\nu_{1}}{N_{\zeta}}>x\right)\geq\liminf_{\zeta\to\infty}N_{\zeta}T_{\zeta}\mathbb{P}\left(\frac{U_{1}}{S_{\zeta}}\geq
x+\varepsilon\right)$ (61)

###### Proof.

Following the proof of [Sch03, Theorem 4] we condition on $\\{X_{i}\\}$ to
obtain

$\displaystyle\lim_{\zeta\to\infty}NT\mathbb{P}(\nu_{1}/N>x)$
$\displaystyle=\lim_{\zeta\to\infty}NT\mathbb{E}[\mathbb{P}(\nu_{1}/N>x|\\{X_{i}\\})]$
$\displaystyle=\lim_{\zeta\to\infty}NT\mathbb{E}[\mathbb{P}(\nu_{1}/N>x|\\{X_{i}\\})1_{\\{X_{1}/S\leq
x-\varepsilon\\}}]$
$\displaystyle\quad\quad\quad+\lim_{\zeta\to\infty}NT\mathbb{E}[\mathbb{P}(\nu_{1}/N>x|\\{X_{i}\\})1_{\\{X_{1}/S\geq
x-\varepsilon\\}}]$

Using that $\nu_{1}|\\{X_{i}\\}$ has a hypergeometric distribution with
parameters $(S,X_{1},N)$,

$\displaystyle\mathbb{E}[\mathbb{P}(\nu_{1}/N>x|\\{X_{i}\\})1_{\\{X_{1}/S\leq
x-\varepsilon\\}}]$
$\displaystyle<\mathbb{P}(\nu_{1}/N>x|X_{1}=S(x-\varepsilon))\mathbb{P}(X_{1}/S\leq
x-\varepsilon)$
$\displaystyle=\mathbb{P}(\nu_{1}>Nx|X_{1}/S+\varepsilon=x)\mathbb{P}(X_{1}/S\leq
x-\varepsilon)$ $\displaystyle\leq e^{-2\varepsilon^{2}N}$

Since $T$ grows sub-exponentially with $N$,

$\displaystyle\limsup_{\zeta\to\infty}NT\mathbb{P}(\nu_{1}/N>x)$
$\displaystyle=\limsup_{\zeta\to\infty}NT\mathbb{E}[\mathbb{P}(\nu_{1}/N>x|\\{X_{i}\\})1_{\\{X_{1}/S\geq
x-\varepsilon\\}}]$
$\displaystyle\leq\limsup_{\zeta\to\infty}NT\mathbb{P}(X_{1}/S\geq
x-\varepsilon)$

which is the upper bound.

The lower bound is obtained similarly using the variable $1_{\\{X_{1}/S\geq
x+\varepsilon\\}}$. ∎

######  Lemma 4 (Similar to Lemma 6 of [Sch03]).

For $r\geq 1$ and $k_{1},\dots,k_{r}\geq 2$,

$\mathbb{E}\left[\prod_{i=1}^{r}(\nu_{i})_{k_{i}}\right]\sim
N^{\sum_{i=1}^{r}k_{i}}\mathbb{E}\left[\frac{e^{\sum_{i=1}^{r}k_{i}\zeta\Phi_{i}}}{S_{\zeta}^{\sum_{i=1}^{r}k_{i}}}\right]$

###### Proof.

Let $M_{k_{1},k_{2}}^{i}$ be the event that individuals $k_{1}$ through
$k_{2}$ descent from individual $i$ in the previous generation and set

$J_{k_{1},\dots,k_{r}}=\bigcup_{i=1}^{r}M_{\sum_{j\leq
i}k_{j},\sum_{j<i+1}k_{j}}^{i}.$

As $N\to\infty$ (or $\zeta\to\infty$),

$\displaystyle\mathbb{P}(J_{k_{1},\dots,k_{r}})$
$\displaystyle=\frac{1}{(N)_{\sum_{i=1}^{r}k_{i}}}\mathbb{E}\left[(\nu_{1})_{k_{1}}\cdots(\nu_{r})_{k_{r}}\right]$
$\displaystyle\sim\frac{1}{N^{\sum_{i=1}^{r}k_{i}}}\mathbb{E}\left[\prod_{i=1}^{r}\nu_{i}^{k_{i}}\right]$

Since $S_{N}\geq N$,

$\displaystyle P(J_{k_{1},\dots,k_{r}})$
$\displaystyle=\mathbb{E}\left[\mathbb{P}\left(J_{k_{1},\dots,k_{r}}\Big{|}\\{\Phi_{i}\\}_{i=1}^{N}\right)\right]=\mathbb{E}\left[\frac{e^{\sum_{i=1}^{r}k_{i}\zeta\Phi_{i}}}{S_{\zeta}^{\sum_{i=1}^{r}k_{i}}}\right],$
(62)

which implies the result. ∎

######  Lemma 5 (Similar to Lemma 8 of [Sch03]).

For each $k\geq 1$ there is a constant $B_{k}$ such that

$\mathbb{E}[\nu^{k}]\geq
B_{k}N_{\zeta}^{k}\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{e^{k\zeta\Phi_{1}}\vee
A_{\zeta}^{k}}\right].$

###### Proof.

By Lemma 4,

$\displaystyle\mathbb{E}[\nu^{k}]$ $\displaystyle\sim
N^{k}\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{S_{\zeta}^{k}}\right]$
$\displaystyle\geq
N^{k}\mathbb{E}\left[\frac{e^{2\zeta\Phi_{1}}}{(e^{\zeta\Phi_{1}}+2(A_{\zeta}-\mu_{\zeta}))^{k}}1_{\\{\sum_{i=2}^{N_{\zeta}}e^{\zeta\Phi_{i}}\leq
2(A_{\zeta}-\mu_{\zeta})\\}}\right]$
$\displaystyle=N^{k}\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{(e^{\zeta\Phi_{1}}+2(A_{\zeta}-\mu_{\zeta}))^{k}}\right]\mathbb{P}\left(\sum_{i=2}^{N_{\zeta}}e^{\zeta\Phi_{i}}\leq
2(A_{\zeta}-\mu_{\zeta})\right)$

By the law of large numbers

$\mathbb{P}\left(\sum_{i=2}^{N_{\zeta}}e^{\zeta\Phi_{i}}\leq
2(A_{\zeta}-\mu_{\zeta})\right)\geq\frac{1}{2}$

for large enough $\zeta$, therefore

$\displaystyle
N^{k}\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{S_{\zeta}^{k}}\right]$
$\displaystyle\geq\frac{N^{k}}{2}\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{(e^{\zeta\Phi_{1}}+2(A_{\zeta}-\mu_{\zeta}))^{k}}\right]$
(63)
$\displaystyle\geq\frac{N^{k}}{2^{k}}\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{(e^{k\zeta\Phi_{1}}+A_{\zeta})^{k}}\right].$
(64)

The result follows from

$(e^{\zeta\Phi_{1}}+A_{\zeta})^{k}\leq 2^{k}(e^{k\zeta\Phi_{1}}\vee
A_{\zeta}^{k})$ (65)

∎

######  Lemma 6.

Assume Equation 4 and Equation 13. Then for $1<\alpha$

$\mathbb{E}\left[\frac{e^{k\zeta\Phi}}{\left(\sum_{i=1}^{N_{\zeta}}e^{\zeta\Phi_{1}}\right)^{k}}\right]\sim\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{\left(e^{\zeta\Phi_{1}}+A_{\zeta}\right)^{k}}\right].$

###### Proof.

The idea of the proof is based on [Sch03] combined with Lemma 2. By the Lemma
2, for any $\epsilon>0$ and $\delta$ such that
$(1-\delta)\mathbb{E}[X_{i,N}]>1$, we can pick $\zeta$ large enough that

$\mathbb{P}\left(1-\delta\leq\frac{S_{\zeta}}{A_{\zeta}}\leq
1+\delta\right)>1-\varepsilon.$ (66)

If $\delta$ and $N$ are such that Equation 66 holds,

$\displaystyle\mathbb{E}\left[\frac{e^{k\zeta\Phi}}{\left(\sum_{i=1}^{N_{\zeta}}e^{\zeta\Phi_{1}}\right)^{k}}\right]$
$\displaystyle=\mathbb{E}\left[\frac{e^{k\zeta\Phi}}{\left(\sum_{i=1}^{N_{\zeta}}e^{\zeta\Phi_{1}}\right)^{k}}1_{\left\\{\sum_{i=2}^{N_{\zeta}}e^{\zeta\Phi_{i}}<(1-\delta)NA_{\zeta}\\}\right\\}}\right]$
$\displaystyle\quad\quad\quad+\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{\left(\sum_{i=1}^{N_{\zeta}}e^{\zeta\Phi_{i}}\right)^{k}}1_{\left\\{\sum_{i=2}^{N_{\zeta}}e^{\zeta\Phi_{i}}\geq(1+\delta)NA_{\zeta}\right\\}}\right]$
$\displaystyle\leq
4\varepsilon\mathbb{E}\left[\frac{e^{k\zeta\Phi}}{\left(e^{\zeta\Phi_{1}}+N_{\zeta}\right)^{k}}\right]+\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{\left(e^{\zeta\Phi_{1}}+(1-\delta)A_{\zeta}\right)^{k}}\right]$

and

$\displaystyle\mathbb{E}\left[\frac{e^{k\zeta\Phi}}{\left(\sum_{i=1}^{N_{\zeta}}e^{\zeta\Phi_{1}}\right)^{k}}\right]\geq(1-\varepsilon)\mathbb{E}\left[\frac{e^{k\zeta\Phi_{1}}}{\left(e^{\zeta\Phi_{1}}+(1+\delta)A_{\zeta}\right)^{k}}\right].$
(67)

The result follows after taking $\varepsilon,\delta\to 0$. ∎

## References

## References

  * [Ami20a] Ariel Amir.  An elementary renormalization-group approach to the generalized central limit theorem and extreme value distributions.  Journal of Statistical Mechanics: Theory and Experiment, 2020(1):013214, 2020.
  * [Ami20b] Ariel Amir.  Thinking Probabilistically: Stochastic Processes, Disordered Systems, and Their Applications.  Cambridge University Press, 2020.
  * [BABM05] Gérard Ben Arous, Leonid V Bogachev, and Stanislav A Molchanov.  Limit theorems for sums of random exponentials.  Probability theory and related fields, 132:579–612, 2005.
  * [Ber09] Nathanaël Berestycki.  Recent progress in coalescent theory.  arXiv preprint arXiv:0909.3985, 2009.
  * [Bov06] Anton Bovier.  Statistical mechanics of disordered systems: a mathematical perspective, volume 18.  Cambridge University Press, 2006.
  * [BS98] Erwin Bolthausen and Alain-Sol Sznitman.  On ruelle’s probability cascades and an abstract cavity method.  Communications in mathematical physics, 197:247–276, 1998.
  * [Can74] Chris Cannings.  The latent roots of certain markov chains arising in genetics: a new approach, i. haploid models.  Advances in Applied Probability, 6(2):260–290, 1974.
  * [CGCSWB22] Fernando Cordero, Adrián González Casanova, Jason Schweinsberg, and Maite Wilke-Berenguer.  $\Lambda$-coalescents arising in a population with dormancy.  Electronic Journal of Probability, 27:1–34, 2022.
  * [Cha09] Brian Charlesworth.  Effective population size and patterns of molecular evolution and variation.  Nature Reviews Genetics, 10(3):195–205, 2009.
  * [DB81] Nicolaas Govert De Bruijn.  Asymptotic methods in analysis, volume 4.  Courier Corporation, 1981.
  * [DD08] Richard Durrett and Richard Durrett.  Probability models for DNA sequence evolution, volume 2.  Springer, 2008.
  * [DEP11] Ricky Der, Charles L Epstein, and Joshua B Plotkin.  Generalized population models and the nature of genetic drift.  Theoretical population biology, 80(2):80–99, 2011.
  * [Der81] Bernard Derrida.  Random-energy model: An exactly solvable model of disordered systems.  Physical Review B, 24(5):2613, 1981.
  * [DM21] Bernard Derrida and Peter Mottishaw.  One step replica symmetry breaking and overlaps between two temperatures.  Journal of Physics A: Mathematical and Theoretical, 54(4):045002, 2021.
  * [Eis83] Theodor Eisele.  On a third-order phase transition.  1983\.
  * [EK09] Stewart N Ethier and Thomas G Kurtz.  Markov processes: characterization and convergence.  John Wiley & Sons, 2009.
  * [FVJH+21] Luca Freschi, Roger Vargas Jr, Ashaque Husain, SM Mostofa Kamal, Alena Skrahina, Sabira Tahseen, Nazir Ismail, Anna Barbova, Stefan Niemann, Daniela Maria Cirillo, et al.  Population structure, biogeography and transmissibility of mycobacterium tuberculosis.  Nature communications, 12(1):6099, 2021.
  * [Gag18] Sebastien Gagneux.  Ecology and evolution of mycobacterium tuberculosis.  Nature Reviews Microbiology, 16(4):202–213, 2018.
  * [Gil73] John H Gillespie.  Natural selection with varying selection coefficients–a haploid model.  Genetics Research, 21(2):115–120, 1973.
  * [Gil74] John H Gillespie.  Natural selection for within-generation variance in offspring number.  Genetics, 76(3):601–606, 1974.
  * [Gil01] John H Gillespie.  Is the population size of a species relevant to its evolution?  Evolution, 55(11):2161–2169, 2001.
  * [Gil04] John H Gillespie.  Population genetics: a concise guide.  JHU press, 2004.
  * [Gol18] Nigel Goldenfeld.  Lectures on phase transitions and the renormalization group.  CRC Press, 2018.
  * [Hal18] Oskar Hallatschek.  Selection-like biases emerge in population models with recurrent jackpot events.  Genetics, 210(3):1053–1073, 2018.
  * [JLA23] Farshid Jafarpour, Ethan Levien, and Ariel Amir.  Evolutionary dynamics in non-markovian models of microbial populations.  Physical Review E, 108(3):034402, 2023.
  * [Ker22] Julian Kern.  The skorokhod topologies: What they are and why we should care.  arXiv preprint arXiv:2210.16026, 2022.
  * [Kin82] John FC Kingman.  On the genealogy of large populations.  Journal of applied probability, 19(A):27–43, 1982.
  * [Kis15] Nicola Kistler.  Derrida’s random energy models: From spin glasses to the extremes of correlated random fields.  Correlated Random Systems: Five Different Methods: CIRM Jean-Morlet Chair, Spring 2013, pages 71–120, 2015.
  * [LD43] Salvador E Luria and Max Delbrück.  Mutations of bacteria from virus sensitivity to virus resistance.  Genetics, 28(6):491, 1943.
  * [LMKA21] Ethan Levien, Jiseon Min, Jane Kondev, and Ariel Amir.  Non-genetic variability in microbial populations: survival strategy or nuisance?  Reports on progress in physics, 84(11):116601, 2021.
  * [Mik99] Thomas Mikosch.  Regular variation, subexponentiality and their applications in probability theory, volume 99.  Eindhoven University of Technology Eindhoven, The Netherlands, 1999.
  * [MM09] Marc Mezard and Andrea Montanari.  Information, physics, and computation.  Oxford University Press, 2009.
  * [MS01] Martin Mohle and Serik Sagitov.  A classification of coalescent processes for haploid exchangeable population models.  Annals of Probability, pages 1547–1562, 2001.
  * [Nol20] John P Nolan.  Univariate stable distributions.  Springer Series in Operations Research and Financial Engineering, DOI, 10:978–3, 2020.
  * [NW18] Richard A Neher and Aleksandra M Walczak.  Progress and open problems in evolutionary dynamics.  arXiv preprint arXiv:1804.07720, 2018.
  * [OH21] Takashi Okada and Oskar Hallatschek.  Dynamic sampling bias and overdispersion induced by skewed offspring distributions.  Genetics, 219(4):iyab135, 2021.
  * [Pit99] Jim Pitman.  Coalescents with multiple collisions.  Annals of Probability, pages 1870–1902, 1999.
  * [RAS15] Firas Rassoul-Agha and Timo Seppäläinen.  A course on large deviations with an introduction to Gibbs measures, volume 162.  American Mathematical Soc., 2015.
  * [Rue87] David Ruelle.  A mathematical reformulation of derrida’s REM and GREM.  Communications in Mathematical Physics, 108:225–239, 1987.
  * [Sch01] Jason Schweinsberg.  Coalescents with simultaneous multiple collisions.  University of California, Berkeley, 2001.
  * [Sch03] Jason Schweinsberg.  Coalescent processes obtained from supercritical galton–watson processes.  Stochastic processes and their Applications, 106(1):107–139, 2003\.
  * [Sch15] Sebastian J Schreiber.  Unifying within-and between-generation bet-hedging theories: an ode to jh gillespie.  The American Naturalist, 186(6):792–796, 2015.
  * [SHJ19] Andrew M Sackman, Rebecca B Harris, and Jeffrey D Jensen.  Inferring demography and selection in organisms characterized by skewed offspring distributions.  Genetics, 211(3):1019–1028, 2019.
  * [SJHW23] Arno Siri-Jégousse and Alejandro H. Wences.  Exchangeable coalescents beyond the cannings class.  arXiv preprint arXiv:2212.02154, 2023.
  * [SK12] Jostein Starrfelt and Hanna Kokko.  Bet-hedging—a triple trade-off between means, variances and correlations.  Biological Reviews, 87(3):742–755, 2012.
  * [TL14] Aurélien Tellier and Christophe Lemaire.  Coalescence 2.0: a multiple branching of recent theoretical developments and their applications.  Molecular Ecology, 23(11):2637–2652, 2014.
  * [Tou05] Hugo Touchette.  Legendre-fenchel transforms in a nutshell.  URL http://www. maths. qmul. ac. uk/~ ht/archive/lfth2. pdf, 2005\.
  * [Tou09] Hugo Touchette.  The large deviation approach to statistical mechanics.  Physics Reports, 478(1-3):1–69, 2009.
  * [WV19] Erik S Wright and Kalin H Vetsigian.  Stochastic exits from dormancy give rise to heavy-tailed distributions of descendants in bacterial populations.  Molecular Ecology, 28(17):3915–3928, 2019.
  * [Zol86] Vladimir M Zolotarev.  One-dimensional stable distributions, volume 65.  American Mathematical Soc., 1986.