# Counting Functions for Random Objects in a Category

Brandon Alberts

###### Abstract

In arithmetic statistics and analytic number theory, the asymptotic growth
rate of counting functions giving the number of objects with order below $X$
is studied as $X\to\infty$. We define general counting functions which count
epimorphisms out of an object on a category under some ordering. Given a
probability measure $\mu$ on the isomorphism classes of the category with
sufficient respect for a product structure, we prove a version of the Law of
Large Numbers to give the asymptotic growth rate as $X$ tends towards $\infty$
of such functions with probability $1$ in terms of the finite moments of $\mu$
and the ordering. Such counting functions are motivated by work in arithmetic
statistics, including number field counting as in Malle’s conjecture and point
counting as in the Batyrev-Manin conjecture. Recent work of Sawin–Wood gives
sufficient conditions to construct such a measure $\mu$ from a well-behaved
sequence of finite moments in very broad contexts, and we prove our results in
this broad context with the added assumption that a product structure in the
category is respected. These results allow us to formalize vast heuristic
predictions about counting functions in general settings.

##  1 Introduction

Distributions of arithmetic objects are commonly studied via counting
functions. A classic example is the prime number theorem, which determines the
asymptotic growth rate of

$\pi(X)=\\#\\{p\text{ prime }\mid p\leq X\\}\sim\frac{X}{\log X}$

as $X$ tends to infinity. Counting functions on subsets of prime numbers is
one way that we work to understand the distribution of the prime numbers.
Cramér made rigorous a heuristic argument of Gauss for predicting the
distribution of prime numbers in [Cra94], whereby a sequence of independent
variables $X_{n}$ indexed by natural numbers and valued in $\\{0,1\\}$ are
considered, for which $X_{n}=1$ with probability $\frac{1}{\log(n)}$ as
suggested by the prime number theorem. By asking the same distribution
questions of the sequence $X_{n}$ instead of the sequence of prime numbers,
Cramér is able to prove that this sequence satisfies a number of properties
conjectured to be true for the primes with probability $1$. See [Gra95] for a
summary of Cramér’s random model and some improvements.

In arithmetic statistics, other objects are frequently studied in this way.
Given an ordering of the objects satisfying a Northcott property, one can ask
for the asymptotic growth rate of the counting function

$\displaystyle\\#\\{\text{object}\mid\text{order(object)}\leq X\\}$ (1)

as $X$ tends to $\infty$. The standard examples are points on schemes (or
stacks) bounded by height, whose asymptotic growth rates are predicted by the
Batyrev-Manin conjecture [FMT89, BM90], and counting $G$-extensions of a
global field $K$ ordered by discriminant, whose asymptotic growth rates are
predicted by Malle’s conjecture [Mal02, Mal04].

Malle’s conjecture in particular is closely related to the study of class
group statistics, where one asks for the distribution of the $p$-parts of the
class group of an extension $K/\mathbb{Q}$ as $K$ varies over some family of
extensions ordered by discriminant. This distribution is very often predicted
to agree with a true probability distribution on random groups. For a non-
exhaustive list of examples for statistics of very general unramified objects
over a family of global fields see [CL84, FW89, BBH17, LWZB19]. Despite this
close relation, there has been no attempt to make predictions for Malle’s
counting function using similar random structures. Like with Cramér’s model,
it is reasonable to consider that there exists a random group with some
discriminant structure which models the absolute Galois group of the global
field $K$ for which Malle’s conjecture holds with probability $1$. Potentially
the same is true for other counting functions in arithmetic statistics.

Another eye-catching feature of Malle’s conjecture is the Galois
correspondence. $G$-extensions $L/K$ are in a $1$-to-$|\textnormal{Aut}(G)|$
correspondence with surjective homomorphisms
$\textnormal{Gal}(\overline{K}/K)\to G$. The recent work of Sawin–Wood [SW22]
solved the moment problem for random objects in a category, where they proved
in great generality that given a sequence $M_{G}$ on (isomorphism classes of)
the category of finite objects $C$ that “do not grow too fast”, there exists a
measure $\mu$ on (isomorphism classes of) the category of pro-objects
$\mathcal{P}$ whose finite moments are given by $M_{G}$. In the setting of
objects in a category, this is taken to mean

$\int_{\mathcal{P}}\\#{\rm Epi}(\mathscr{G},G)\ d\mu(\mathscr{G})=M_{G}$

for each finite object $G$. Here we see an immediate similarity with Malle’s
conjecture - Malle’s conjecture can be interpreted as an asymptotic count for
the number of epimorphisms from the absolute Galois group of $K$ to a fixed
finite group. This suggests that the setting Sawin–Wood work in, which they
state was built with statistics of unramified objects in mind, is also an
excellent setting to model Malle’s conjecture. By extension, random models for
other counting functions may also exist in the setting considered by
Sawin–Wood as long as they can be interrpretted as counting epimorphisms.

The goal of this paper is to consider counting functions in the broad setting
described by Sawin–Wood. The remainder of this introduction is separated into
two subsections - the first defining a counting function with respect to an
ordering for counting epimorphisms in a category (see Definition 1.1), and the
second for stating Law of Large Numbers results to determine the asymptotic
growth rate of these counting functions with probability $1$ (see Theorem
1.3). We will prove the Law of Large Numbers for random objects in a category
in as great of generality as possible, with the intention of allowing these
results to easily translate to a variety of other counting functions in other
settings. In a forthcoming paper [Alb23], the author will apply these results
to the case of Malle’s conjecture to both recreate and improve on existing
predictions for the asymptotic growth rate. This will be done by constructing
a category of “groups with local data”, applying the results of [SW22] to
construct a measure on this category, and applying the results of this paper
to prove that 100% of random groups with local data satisfy Malle’s
conjecture.

We adopt the notation and terminology of [SW22] throughout. Let $C$ be a
diamond category in the sense of [SW22, Definition 1.3] with countably many
isomorphism classes, and suppose $\mu$ is a probability measure (so the whole
space has measure $1$) on the isomorphism classes of corresponding pro-
objects, $\mathcal{P}$, under the level topology with finite moments $M_{G}$
for each $G\in C/\cong$. Sawin–Wood give sufficient conditions for a sequence
$M_{G}$ to correspond to such a measure in [SW22, Theorem 1.7 and 1.8]. Our
results do not require $\mu$ to be constructed in this way, but the author
expects that most interesting examples will come from the existence results
proven by Sawin–Wood.

###  1.1 Counting Functions

To translate the setting given by [SW22] to that of counting functions more
closely resembling (1), we very generally define an ordering to be a sequence
of functions $f_{n}:C/\cong\to\mathbb{C}$ indexed by positive integers $n$. We
are motivated by classical orderings such as the discriminant and height
functions, which correspond to $f_{n}$ being the characteristic function of an
increasing chain of finite sets $A_{n}\subseteq C/\cong$, i.e. a chain of
subsets for which $A_{n}\subseteq A_{n+1}$. In the general format described in
(1), we would choose the ordering

$f_{n}({\rm object})=\begin{cases}1&{\rm order}({\rm object})\leq n\\\
0&\text{else}.\end{cases}$

The property that such functions have finite support is called the Northcott
Property, and is known to hold for discriminant and height orderings among
many others.

######  Definition 1.1.

The counting function on a pro-object $\mathscr{G}\in\mathcal{P}$ ordered by
$f_{n}:C/\cong\to\mathbb{C}$ is defined by

$\displaystyle N(\mathscr{G},f_{n})=\sum_{G\in C/\cong}f_{n}(G)\\#{\rm
Epi}(\mathscr{G},G)$

as a function of $n$, when the series is convergent.

In the classical setting, a Northcott Property is used to guarantee that the
counting function is well-defined by forcing the series to be finite. We
remark that in the classical setting ${\rm order}({\rm object})$ is taken to
be bounded above by a real number $X$, while Definition 1.1 restricts to only
using integer bounds, $n$. This will not make a difference for classical
orderings, as the (norm of the) discriminant and height functions are integer-
valued. However, it is an artifact of the methods we use that we ask the
sequence $(f_{n})_{n=1}^{\infty}$ to be indexed by a _countable_ set rather
than the uncountable set of positive real numbers.

In our general setting, we may relax the Northcott requirement and still
guarantee an (almost everywhere) well-defined counting function. Fix a
probability measure $\mu$ on the category of pro-objects $\mathcal{P}$ with
finite moments $M_{G}$. For convenience, we package the sequence $M_{G}$ as a
discrete measure $M:C/\cong\to\mathbb{R}_{\geq 0}$ with $M(\\{G\\})=M_{G}$
(note that while we require $\mu$ to be a probability measure, $M$ need not
be).

######  Definition 1.2.

Fix a probability measure $\mu$ on the category of pro-objects $\mathcal{P}$
with finite moments $M_{G}$. We call $f_{n}$ an $L^{1}$-ordering with respect
to $\mu$ if

$\int_{C}|f_{n}|\ dM=\sum_{G\in C/\cong}|f_{n}(G)|M_{G}<\infty.$

In other words, the $f_{n}$ are $L^{1}$-functions for the discrete measure $M$
induced by $\mu$. We will often omit “with respect to $\mu$” if the
probability measure is clear from context.

We will prove in Lemma 3.1 that $N(\mathscr{G},f_{n})$ is well-defined as a
function of $n$ for almost all $\mathscr{G}$ whenever $f_{n}$ is an
$L^{1}$-ordering (i.e., there exists a measure $1$ set of $\mathscr{G}$ on
which the counting function is well-defined as a function on the positive
integers), and that the expected value is given by

$\displaystyle\int_{\mathcal{P}}N(\mathscr{G},f_{n})\
d\mu(\mathscr{G})=\int_{C}f_{n}\ dM.$ (2)

In the classical case, if $f_{n}$ has finite support in an increasing chain as
$n$ tends towards $\infty$ then the counting function $N(\mathscr{G},f_{n})$
is a sum of increasing length depending on $n$, where the summands
$f_{n}(G)\\#{\rm Epi}(\mathscr{G},G)$ are random variables as $\mathscr{G}$
varies according to $\mu$. This is precisely the setting of the Law of Large
Numbers, which suggests a much stronger result of the form

$\frac{N(\mathscr{G},f_{n})}{\displaystyle\int_{C}f_{n}\ dM}\longrightarrow 1$

as $n\to\infty$ with probability $1$ in some appropriate sense. Typically,
there is some requirement of pairwise independence between the summands in
order to prove such a statement (see [SS93] for the classical statement, and
[Sen13] for a brief history of the Law of Large Numbers, including cases that
relax the requirement of pairwise independence). The work of Korchevsky–Petrov
[KP10] on nonegative random variables, building on work in [Ete83a, Ete83b,
Pet09a, Pet09b], is particularly relevant to these counting functions, as
$\\#{\rm Epi}(\mathscr{G},G)$ is always nonegative. We will prove several
different versions of the Law of Large Numbers for $N(\mathscr{G},f_{n})$ of
varying strengths under sufficiently nice orderings.

###  1.2 Main Results

Take $C$ to be a diamond category with countably many isomorphism classes,
$\mathcal{P}$ the category of pro-objects, and $\mu$ a probability measure on
$\mathcal{P}$ with finite moments $M_{G}$. Theorem 1.3 is our primary
probabilistic result, while Theorems 1.4 and 1.5 are our primary category
theoretic results.

######  Theorem 1.3 (Law of Large Numbers in a Category).

Let $f_{n}:C/\cong\to\mathbb{R}$ be a real-valued $L^{1}$-ordering for which
$\liminf_{n\to\infty}\left\lvert\int_{C}f_{n}dM\right\rvert>0$. Suppose there
exists an integer $k$ and a non-decreasing function
$\gamma:\mathbb{N}\to\mathbb{R}^{+}$ for which
$\lim_{t\to\infty}\gamma(t)=\infty$, and

$\displaystyle\int_{\mathscr{G}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\
dM\right\rvert^{k}d\mu(\mathscr{G})=O\left(\frac{\left\lvert\int_{C}f_{n}\
dM\right\rvert^{k}}{\gamma(n)}\right).$ (3)

Then each of the following hold:

  1. (i)

(Weak Law of Large Numbers)

$\frac{N(\mathscr{G},f_{n})}{\displaystyle\int_{C}f_{n}\
dM}\overset{p.}{\longrightarrow}1$

as $n\to\infty$, where the “p.” stands for converges in probability with
respect to $\mu$.

  2. (ii)

(Strong Law of Large Numbers) If we additionally assume that
$\sum_{n=1}^{\infty}\frac{1}{\gamma(n)}<\infty$ then

$\frac{N(\mathscr{G},f_{n})}{\displaystyle\int_{C}f_{n}\
dM}\overset{a.s.}{\longrightarrow}1$

as $n\to\infty$, where the “a.s.” stands for converges almost surely with
respect to $\mu$.

  3. (iii)

(Strong Law of Large Numbers) If we additionally assume that

     * •

$f_{n}$ is nonegative,

     * •

the counting function $n\mapsto N(\mathscr{G},f_{n})$ is almost everywhere
nondecreasing, and

     * •

$\gamma(n)=\psi(\int_{C}f_{n}dM)$ for a nondecreasing function
$\psi:\mathbb{R}\to\mathbb{R}^{+}$ for which
$\sum_{n=1}^{\infty}\frac{1}{n\psi(n)}<\infty$,

then

$\frac{N(\mathscr{G},f_{n})}{\displaystyle\int_{C}f_{n}\
dM}\overset{a.s.}{\longrightarrow}1$

as $n\to\infty$, where the “a.s.” stands for converges almost surely with
respect to $\mu$.

The primary new contribution of Theorem 1.3 is the connection between counting
functions and the Law of Large Numbers. The probabilistic content is largely
standard, and in fact the proof of Theorem 1.3(iii) is essentially the same as
that of [KP10]. We follow a standard technique for bounding probabilities
using Chebyshev’s law and the first Borel-Cantelli lemma. Most standard
references will have some version of this technique, sometimes in the context
of the “method of moments”, such as [Fis11, Chapter 4]. This technique is
ubiquitous to the point that examples can be found in proofs on Wikipedia
[wik22] and a number of Math StackExchange posts such as [Mic17, Ele18].

If the $f_{n}$ are specifically characteristic functions with finite support
in an increasing chain then Theorem 1.3(iii) is a special case of [KP10,
Theorem 3]. We state and prove these results separately from work in [KP10] to
allow for a more general class of orderings, although the proof requires no
new ideas. For example, we can now directly consider orderings of the form

$f_{n}({\rm object})=\begin{cases}1&n<{\rm order}({\rm object})\leq 2n\\\
0&\text{else}.\end{cases}$

This type of ordering gives counting functions $N(\mathscr{G},f_{n})$ that
count objects landing in the moving interval $(n,2n]$. These sorts of counting
functions are used to avoid errant behavior for small objects, and can be
useful for proving averaging results.

In cases for which
$\liminf_{n\to\infty}\left\lvert\int_{C}f_{n}dM\right\rvert=0$, there is
something we can say using the more general results Theorem 4.1 and Theorem
4.2 we prove in the main body of the paper. This requires a finer study of the
rate of convergence, as we are no longer allowed to have $\int_{C}f_{n}dM$ on
the denominator for such cases.

The bound on $k^{\rm th}$ moments in (3) is a concrete way to state that the
random variables are “close enough to independent” for the Law of Large
Numbers. Our primary categorical result is on bounding this quantity in terms
of purely categorical structures of $C$ and $\mathcal{P}$ together with the
discrete moments of the ordering $f_{n}$.

Given an ordering $f_{n}:C/\cong\to\mathbb{C}$, we extend the ordering
multiplicatively to the product category $f_{n}:C^{k}/\cong\to\mathbb{C}$ by

$f_{n}(G_{1},...,G_{k})=f_{n}(G_{1})\cdots f_{n}(G_{k}).$

We also define the discrete measure $M^{(j)}:C^{j}/\cong\to\mathbb{R}_{\geq
0}\cup\\{\infty\\}$ given by the mixed moments of the measure $\mu$ as

$M_{(G_{1},...,G_{j})}^{(j)}=\int_{\mathcal{P}}\prod_{i=1}^{j}\\#{\rm
Epi}(\mathscr{G},G_{i})\ d\mu(\mathscr{G}).$

This is sufficient information to prove a bound for (3) for $k=2$.

######  Theorem 1.4.

Let $f_{n}:C/\cong\to\mathbb{R}$ be a real-valued $L^{1}$-ordering. Then

$\int_{\mathscr{G}}\left\lvert N(\mathscr{G},f_{n})-\int_{C}f_{n}\
dM\right\rvert^{2}d\mu(\mathscr{G})\ll_{k}\max_{j\in\\{1,2\\}}\int_{C^{2}\setminus
E(2,M)}|f_{n}|\ dM^{(j)}(dM)^{2-j},$

where $E(2,M)\subseteq C^{2}$ is the full subcategory of pairs
$(G_{1},G_{2})\in C^{2}$ such that

  * (a)

the epi-product $G_{1}\times_{\rm epi}G_{2}$ exists (defined in Definition
5.1), and

  * (b)

$M_{G_{1}\times_{\rm epi}G_{2}}=M_{G_{1}}M_{G_{2}}$.

Taking away the full subcategory $E(2,M)$ of $C^{2}$ is the new content, and
allows for bounds on the higher moments that can be computed solely from data
given by $M$, $C$, and $f_{n}$ without reference to the measure $\mu$. The
epi-product in condition (a) is a special type of product structure in a
category, satisfying a similar universal property to the direct product with
every morphism in the diagram being an epimorphism (Definition 5.1). We will
show that the existence of an epi-product together with (b) implies the random
variables $\\#{\rm Epi}(\mathscr{G},G_{1})$ and $\\#{\rm
Epi}(\mathscr{G},G_{2})$ are uncorrelated. This will be key for bounding the
moments. It will often be the case that determining the existence or
nonexistence of epi-products is easier than approaching the bound (3) in
Theorem 1.3 directly. We include some examples using this approach in Section
6.

We will prove a similar bound for (3) when $k>2$ is even, but this will
require more assumptions on the category. These extra assumptions can prove
useful for applying Theorem 1.4, as they give us extra tools for calculating
the integral with respect to $M^{(2)}$. Lemma 5.3 will be very useful in this
regard, and we give an example demonstrating its use in Section 6.

######  Theorem 1.5.

Suppose $C$ is a category with for which every morphism factors uniquely (up
to isomorphism) as a composition of an epimorphism with a monomorphism. Let
$f_{n}:C/\cong\to\mathbb{R}$ be a real-valued $L^{1}$-ordering and $k$ a
positive integer. Then

$\int_{\mathscr{G}}\left\lvert N(\mathscr{G},f_{n})-\int_{C}f_{n}\
dM\right\rvert^{2k}d\mu(\mathscr{G})\ll_{k}\max_{j\in\\{1,...,2k\\}}\int_{C^{2k}\setminus
E(2k,M)}|f_{n}|\ dM^{(j)}(dM)^{2k-j},$

where $E(2k,M)\subseteq C^{2k}$ is the full subcategory of tuples
$(G_{1},G_{2},...,G_{2k})$ for which there exists an index $i$ such that for
each subset $A\subseteq\\{1,...,k\\}$ with $i\not\in A$

  * (a)

the product object $G_{A}:=\prod_{m\in A}G_{m}$ exists in $C^{2k}$,

  * (b)

$G_{A}$ has finitely many subobjects up to isomorphism,

  * (c)

the epi-product $G_{i}\times_{\rm epi}H$ exists for each subobject
$H\hookrightarrow G_{A}$ (defined in Definition 5.1), and

  * (d)

$M_{G_{i}\times_{\rm epi}H}=M_{G_{i}}M_{H}$ for each subobject
$H\hookrightarrow G_{A}$.

Unique factorization of morphisms is known for many categories of interest,
including finite sets, finite groups, and finite modules over a ring. In fact,
each of these categories automatically satisfies parts (a) and (b) as well.
Thus, in numerous categories of interest, Theorem 1.5 requires no extra input
over Theorem 1.4.

We also prove an upper bound utilizing simpler categorical structures that
holds without the restrictions of Theorem 1.3.

######  Corollary 1.6.

Let $C$ be a diamond category which has countably many isomorphism classes,
$\mu$ be a probability measure on the isomorphism classes of the corresponding
category of pro-objects $\mathcal{P}$ with finite moments $M_{G}$, and
$f_{n}:C/\cong\to\mathbb{C}$ an $L^{1}$-ordering.

Then for any $\epsilon>0$ and any positive integer $k$

$\frac{N(\mathscr{G},f_{n})}{\displaystyle
n^{\frac{1+\epsilon}{k}}\max_{j\in\\{1,...,k\\}}\left\\{\int_{C^{k}}|f_{n}|\
dM^{(j)}(dM)^{k-j}\right\\}^{1/k}}\overset{a.s.}{\longrightarrow}0$

as $n\to\infty$, where the “a.s.” stands for converges almost surely with
respect to $\mu$.

Corollary 1.6 is motivated by Malle’s predicted weak upper bound, although the
correspondence is less than obvious. In the classical case $f_{n}$ is a
nonegative function so that $\int f_{n}\ dM=\int|f_{n}|\ dM$, and when $j=1$
the denominator is given by

$n^{\frac{1+\epsilon}{k}}\int_{C}f_{n}\ dM.$

Ignoring the other values of $j$ and taking $k$ sufficiently large, Corollary
1.6 appears to give an upper bound of the form

$N(\mathscr{G},f_{n})\ll n^{\epsilon}\int_{C}f_{n}\ dM$

with probability $1$, where we recall that $n$ is playing the role of $X$ in
Malle’s counting function. Of course, we cannot actually ignore the other
values of $j$, and depending on $C$ and $M$ it is possible for these bounds to
be worse.

Theorem 1.3 and Corollary 1.6 can be used to provide evidence for conjectures
about counting functions in great generality. We state this as an explicit
heuristic:

######  Heuristic 1.7 (Vast Counting Heuristic).

Let $C$ be a diamond category which has countably many isomorphism classes and
let $\mu$ be a probability measure on the isomorphism classes of the
corresponding category of pro-objects $\mathcal{P}$ with finite moments
$M_{G}$. Choose an $L^{1}$-ordering $f_{n}$.

Let $\mathbb{G}$ be an element of $\mathcal{P}/\cong$ for which
$N(\mathbb{G},f_{n})$ is well-defined. If $\mathbb{G}$ is an object we expect
to behave typically∗ with respect to the probability measure $\mu$, such as an
object coming from arithmetic, then

  * (i)

(Strong form) If $f_{n}$ satisfies the bound in (3) for some positive integer
$k$, then

$N(\mathbb{G},f_{n})\sim\int_{C}f_{n}\ dM$

as $n\to\infty$.

  * (ii)

(Weak form) For each $\epsilon>0$ and each positive integer $k$

$N(\mathbb{G},f_{n})\ll
n^{\frac{1+\epsilon}{k}}\max_{j\in\\{0,...,k\\}}\left(\int_{C^{j}}|f_{n}|\
dM^{(j)}(dM)^{k-j}\right)^{1/k}.$

The asterisk on “typically” is meant to highlight that this word is where the
intricacies of counting conjectures lie, and in particular is in recognition
of counter-examples to Malle’s conjecture and the like. For number field
counting, Malle’s original conjecture is incorrect due to behavior coming from
roots of unity that Malle did not initially consider. For the purpose of a
general statement, this can be interpreted as saying “if there are no
obstructions, this is what we expect”. Heuristic 1.7 provides a baseline for
counting predictions when we believe there are no “non-random” structures of
$\mathbb{G}$ missing from the measure $\mu$ on the category $\mathcal{P}$.
Issues, like those occurring in Malle’s conjecture from roots of unity, might
be fixed by either (a) incorporating the missing structure into a modified
version of the category (this is analogous to the approach commonly take in
the study of Malle’s conjecture), or (b) excluding epimorphisms from the count
which are affected by the missing structure (this is the approach taken for
the Batyrev-Manin conjecture).

The literature on proving new cases of the Law of Large Numbers is vast, and
it is possible that Theorem 1.3 may be extended with finer probabilistic
arguments. From a categorical perspective, we’ve stated Theorem 1.4, Theorem
1.5, Corollary 1.6 and Heuristic 1.7 in the greatest generality possible so
that they might be applied to make predictions on a variety of different
counting functions in arithmetic statistics independent of the author’s
immediate knowledge. We hope that these results will be useful as a starting
point for many future predictions for the asymptotic growth rates of counting
functions.

###  1.3 Layout of the Paper

We begin in Section 2 with a discussion of what we mean by an asymptotic
version of the Law of Large Numbers. This is conceptually important for
understanding the methods of this paper, as the references to the Law of Large
Numbers are not just in analogy. Korchevskey–Petrov’s result [KP10, Theorem 3]
is a particular example of the type of Law of Large Numbers result we
consider.

We prove well-definedness of the counting function in Section 3. We give the
probabilistic proofs of Theorem 1.3 and Corollary 1.6 in Section 4. These
proofs closely follow the methods of [KP10]. We define epi-products in Section
5, prove some useful features of these objects, and prove Theorems 1.4 and
1.5.

Lastly, we include Section 6 to work out a couple of important special cases.
The arithmetic statistics questions that primarily motivated the author will
require independent work to set up the necessary category, measure, and
orderings which the author will present in a forthcoming paper [Alb23]. We
instead include simpler examples where these steps are either shorter or done
by previous work such as in [SW22], and we pay particular attention to when
epi-products exist in such cases. Despite the easier nature of these examples,
we recover existing important random models from number theory as special
cases of Theorem 1.3 and Heuristic 1.7.

## Acknowledgments

The author would like to thank Melanie Matchett Wood for feedback and helpful
discussions on the mechanics on [SW22].

##  2 An asymptotic version of the Law of Large Numbers

Given a sequence of independent, identically distributed random variables
$\\{X_{i}\\}$ the Law of Large Numbers says that we expect
$\frac{1}{n}(X_{1}+\cdots+X_{n})$ to converge to the expected value of $X_{i}$
in some sense. A version of this statement can sometimes also hold for random
variables which are neither independent nor identically distributed. If the
variables are not identically distributed, then $\mathbb{E}[X_{i}]$ may vary
depending on $i$, which necessarily alters the statement we want to make. We
will instead be comparing

$\displaystyle\sum_{i=1}^{n}X_{i}$ with
$\displaystyle\sum_{i=1}^{n}\mathbb{E}[X_{i}]$

as $n\to\infty$. When the random variables are pairwise independent,
identically distributed, and have bounded variance in some sense, the
difference of these is known to grow slower than $n$ almost surely by
Kolmogorov’s strong law [Ete81, SS93]. Rather than a convergence statement, we
will state the Law of Large Numbers as an asymptotic statement. The conclusion
of Kolmogorov’s strong law would then be written as

${\rm
Prob}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}-\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[X_{i}]=o(1)\text{
as }n\to\infty\right)=1.$

We can understand this term from the perspective of asymptotic vounting
functions as picking out a “main term + O(error term)”. Kolmogorov’s strong
law can then be understood as

$\sum_{i=1}^{n}X_{i}=\sum_{i=1}^{n}\mathbb{E}[X_{i}]+o(n)$

as $n\to\infty$ with probability $1$. It is often the case with the Law of
Large Numbers that such results are written as

$\frac{\sum_{i=1}^{n}X_{i}-\sum_{i=1}^{n}\mathbb{E}[X_{i}]}{n}\overset{a.s.}{\longrightarrow}0.$

The statements of Theorem 1.3 can be interpretted as something similar to
this. $N(\mathscr{G},f_{n})$ is defined to be a sum of random variables
$f_{n}(G)\\#{\rm Epi}(\mathscr{G},G)$, so if we label $X_{G}=f_{n}(G)\\#{\rm
Epi}(\mathscr{G},G)$ we can understand the counting function as

$N(\mathscr{G},f_{n})=\sum_{G,f_{n}(G)\neq 0}X_{G}.$

The ordering is analogous to a statement of the form “${\rm order}(G)<n$”,
although our results apply to more general orderings. We can then see that
$N(\mathscr{G},f_{n})$ behaves like a sum of random variables $X_{G}$, whose
length is growing with $n$. The philosophy of the Law of Large Numbers
suggests that we should expect this to be close to $\sum_{G,f_{n}(G)\neq
0}\mathbb{E}[X_{G}]=\int_{C}f_{n}\ dM$. However, there is no reason to expect
an error term like $o(n)$ to hold. This isn’t precisely what we want, we are
interested in a statement of the form

$N(\mathscr{G},f_{n})=\int_{C}f_{n}\ dM+o\left(\int_{C}f_{n}\ dM\right)$

with probability $1$. This error does not necessarily agree with $o(n)$.

We consider two examples to demonstrate the intricacies of the error term.

  * •

Consider the following example: Let $X$ be a random variable equal to $1$ with
probability $1/2$ and $0$ with probability $1/2$, and consider the dependent
sequence $X_{n}=\frac{1}{n}X$. Classical SLLN certainly holds for this
sequence, as

$\displaystyle\left\lvert\sum_{i=1}^{n}X_{i}-\sum_{i=1}^{n}\mathbb{E}[X_{i}]\right\rvert$
$\displaystyle\leq\sum_{i=1}^{n}X_{i}+\sum_{i=1}^{n}\frac{1}{2i}$
$\displaystyle\leq\sum_{i=1}^{n}\frac{1}{i}+\sum_{i=1}^{n}\frac{1}{2i}$
$\displaystyle=\sum_{i=1}^{n}\frac{1}{3i}$ $\displaystyle=O(\log n)=o(n).$

However, the $X_{n}$ are identically 0 with probability $1/2$, so

${\rm Prob}\left(\sum_{i=1}^{n}X_{i}\sim\sum_{i=1}^{n}\mathbb{E}[X_{i}]\text{
as }n\to\infty\right)\leq\frac{1}{2}.$

This example satisfies the $o(n)$ error term, but it is certainly not what we
want. What is happening here is that

$\sum_{i=1}^{n}\mathbb{E}[X_{i}]=o(n)$

right from the start, so the classical error for the Law of Large Numbers does
not actually tell us much about how close $\sum X_{i}$ is to
$\sum\mathbb{E}[X_{i}]$.

  * •

If we instead work backwards, we might ask for error terms which are smaller
than

$o\left(\sum_{i=1}^{n}\mathbb{E}[X_{i}]\right),$

so that the sum of expected values can rightly be recognized as the main term.
In the classical setting with identically distributed random variables with
nonzero expected value $E$, these notions coincide as

$o\left(\sum_{i=1}^{n}\mathbb{E}[X_{i}]\right)=o(En)=o(n).$

In fact, this is known more generally. Korchevsky–Petrov [KP10, Theorem 3]
give sufficient conditions for when

$\frac{S_{n}-\mathbb{E}[S_{n}]}{\mathbb{E}[S_{n}]}\overset{a.s.}{\longrightarrow}1$

as $n\to\infty$, when $S_{n}=\sum_{i=1}^{n}X_{n}$ is a sum of _nonegative_
random variables. Theorem 1.3(iii) is based on their result.

However, in the case that $\mathbb{E}[X_{i}]=0$ for all $i$ this is asking for
a trivial error term and a trivial main term. This is certainly an issue, as
in the classical case with a sequence of independent identically distributed
random variables with mean $0$ the central limit theorem forces the error term
to be about $\sqrt{n}$ (i.e., not $o(0)$). The case that all expected values
are zero is important in the classical study and applications of the Law of
Large Numbers, so we do not want to exclude this case.

The condition $\liminf_{n\to\infty}\left\lvert\int_{C}f_{n}dM\right\rvert>0$
in Theorem 1.3 is a consequence of the second bullet above. This corresponds
to the condition that $\mathbb{E}[S_{n}]\neq 0$ for all but finitely many $n$,
so that we can make sense of having $\mathbb{E}[S_{n}]$ on the denominator.
This issue can alternatively be fixed avoided by relaxing what we want for an
error term. Some options might include
$o\left(\sum_{i=1}^{n}\mathbb{E}[|X_{i}|]\right)$ to guarantee the sum is
nonzero unless $X_{i}$ are identically zero, or something of the form
$o\left(n^{1/2+\epsilon}\right)$ to give an error of a similar nature to the
Central Limit Limit. We give a version of Theorem 1.3(i,ii) in Theorems 4.1
and 4.2 with explicit rates of convergence. These results can be used to
produce probability $1$ statements even in the case that
$\liminf\left\lvert\int_{C}f_{n}dM\right\rvert=0$, although they will no
longer be of the form

$N(\mathscr{G},f_{n})\sim\int_{C}f_{n}dM.$

##  3 Well-defined counting functions

A sequence of such $L^{1}$-functions $f_{n}$ is called an $L^{1}$-ordering as
in Definition 1.2. As in Definition 1.1, we define the counting function

$N(\mathscr{G},f_{n})=\sum_{G\in C/\cong}f_{n}(G)\\#{\rm Epi}(\mathscr{G},G)$

for each $\mathscr{G}\in\mathcal{P}$. By allowing $f_{n}$ to have possibly
infinite support, we need to make sure that the counting function is well-
defined and the linearity of expectation behaves as we expect. In the case of
$L^{1}$-orderings, we prove this below:

######  Lemma 3.1.

If $f_{n}$ is an $L^{1}$-ordering, then

  * (a)

$N(\mathscr{G},f_{n})$ is well-defined as a function on the positive integers
$n$ almost surely (i.e. with probability $1$) with respect to $\mu$, and

  * (b)

$\displaystyle\int_{\mathcal{P}}N(\mathscr{G},f_{n})\
d\mu(\mathscr{G})=\int_{C}f_{n}\ dM.$

###### Proof.

This follows from the Fubini-Tonelli theorem. Indeed, $f_{n}$ being $L^{1}$
means

$\sum_{C}\int_{\mathcal{P}}|f_{n}(G)|\\#{\rm Epi}(\mathscr{G},G)\
d\mu(\mathscr{G})<\infty.$

All probability spaces are $\sigma$-finite, so it follows from the Fubinit-
Tonelli theorem that

$\displaystyle\int_{\mathcal{P}}\left\lvert\sum_{G\in
C/\cong}|f_{n}(G)|\\#{\rm Epi}(\mathscr{G},G)\right\rvert\ d\mu(\mathscr{G})$
$\displaystyle=\int_{\mathcal{P}}\sum_{G\in C/\cong}|f_{n}(G)|\\#{\rm
Epi}(\mathscr{G},G)\ d\mu(\mathscr{G})$ $\displaystyle=\sum_{G\in
C/\cong}\int_{\mathcal{P}}|f_{n}(G)|\\#{\rm Epi}(\mathscr{G},G)\
d\mu(\mathscr{G})$ $\displaystyle=\sum_{G\in C/\cong}|f_{n}(G)|M_{G}<\infty.$

Certainly $\mathbb{E}[|X|]<\infty$ implies $X<\infty$ almost surely, so it
must be that for fixed $n$ the counting function

$N(\mathscr{G},f_{n})=\sum_{G\in C/\cong}f_{n}(G)\\#{\rm Epi}(\mathscr{G},G)$

converges absolutely for almost all $\mathscr{G}$. As there are countably many
$n$, countable additivity implies $N(\mathscr{G},f_{n})$ is well-defined
almost surely as a function on the positive integers.

The fact that the integrals of $|f_{n}|$ are finite also implies

$\int_{\mathcal{P}}\sum_{C}f_{n}(G)\\#{\rm Epi}(\mathscr{G},G)\
d\mu(\mathscr{G})=\sum_{C}\int_{\mathcal{P}}f_{n}(G)\\#{\rm
Epi}(\mathscr{G},G)\ d\mu(\mathscr{G})$

by the Fubini-Tonelli Theorem. Evaluating the inner sum/integral gives

$\int_{\mathcal{P}}N(\mathscr{G},f_{n})\
d\mu=\sum_{C}f_{n}(G)M_{G}=\int_{C}f_{n}\ dM\leq\int_{C}|f_{n}|\ dM<\infty,$

proving (b). ∎

##  4 Law of Large Numbers for categories

We prove Theorem 1.3 and Corollary 1.6 in this section. The probabilistic
content of each proof is essentially Chebyshev’s bound, the Borel-Cantelli
lemma, and a few tricks for nonegative orderings previously applied in [KP10].
We prove Theorem 1.3(i,ii) with explicit control on the rate of convergence,
in part to make the proof of Corollary 1.6 more straight forward.

###  4.1 The Weak Law of Large Numbers

We prove the following result, which is Theorem 1.3(i) together with an
explicit rate of convergence.

######  Theorem 4.1.

Let $f_{n}:C/\cong\to\mathbb{R}$ be a real-valued $L^{1}$-ordering. Suppose
there exists an integer $k$, a non-decreasing function
$\gamma:\mathbb{N}\to\mathbb{R}^{+}$ for which
$\lim_{t\to\infty}\gamma(t)=\infty$, and a function
$E:\mathbb{N}\to\mathbb{R}^{+}$ for which

$\displaystyle\int_{\mathscr{G}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\
dM\right\rvert^{k}d\mu(\mathscr{G})=O\left(\frac{E(n)^{k}}{\gamma(n)}\right).$
(4)

Then

$\frac{N(\mathscr{G},f_{n})-\displaystyle\int_{C}f_{n}\
dM}{E(n)}\overset{p.}{\longrightarrow}0$

as $n\to\infty$, where the “p.” stands for converges in probability with
respect to $\mu$.

Theorem 1.3(i) follows by taking $E(n)=\max\left\\{\left\lvert\int_{C}f_{n}\
dM\right\rvert,\delta\right\\}$ for some small $\delta>0$.

###### Proof.

If $Y$ is a random variable with $\mathbb{E}[Y]=0$, then Chebyshev’s bound
states that for each positive integer $k$,

${\rm Prob}\left(|Y|>\lambda\right)\leq\lambda^{-k}\mathbb{E}[|Y|^{k}].$

(See, for instance, [MU17].) In the context of this proof, we let $\mathbb{E}$
denote the expected value with respect to $\mu(\mathscr{G})$ for the sake of
convenience. Set

$Y_{n}=N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM.$

It then follows that

$\displaystyle\mathbb{E}[|Y_{n}|^{k}]$
$\displaystyle=\int_{\mathcal{P}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM\right\rvert^{k}\ d\mu(\mathscr{G})$
$\displaystyle=O\left(\frac{E(n)^{k}}{\gamma(n)}\right).$

Taking $\lambda=\epsilon E(n)$, it follows that

$\displaystyle{\rm Prob}\left(|Y_{n}|>\epsilon
E(n)\right)\ll\frac{1}{\epsilon^{k}\gamma(n)}$

for each $\epsilon>0$. Thus,

$\displaystyle\limsup_{n\to\infty}{\rm
Prob}\left(\frac{|Y_{n}|}{E(n)}>\epsilon\right)=0.$

This is the definition of $Y_{n}/E(n)$ converging in probability to $0$, and
so concludes the proof. ∎

###  4.2 The Strong Law of Large Numbers

We prove the following result, which is Theorem 1.3(ii) together with an
explicit rate of convergence.

######  Theorem 4.2.

Let $f_{n}:C/\cong\to\mathbb{R}$ be a real-valued $L^{1}$-ordering. Suppose
there exists an integer $k$, a non-decreasing function
$\gamma:\mathbb{N}\to\mathbb{R}^{+}$ for which
$\lim_{t\to\infty}\gamma(t)=\infty$, and a function
$E:\mathbb{N}\to\mathbb{R}^{+}$ for which

$\displaystyle\int_{\mathscr{G}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\
dM\right\rvert^{k}d\mu(\mathscr{G})=O\left(\frac{E(n)^{k}}{\gamma(n)}\right).$
(5)

Additionally, assume that $\sum_{n=1}^{\infty}\frac{1}{\gamma(n)}<\infty$.
Then

$\frac{N(\mathscr{G},f_{n})-\displaystyle\int_{C}f_{n}\
dM}{E(n)}\overset{a.s.}{\longrightarrow}0$

as $n\to\infty$, where the “a.s.” stands for converges almost surely with
respect to $\mu$.

Theorem 1.3(ii) follows by taking $E(n)=\max\left\\{\left\lvert\int_{C}f_{n}\
dM\right\rvert,\delta\right\\}$ for some small $\delta>0$.

###### Proof.

We proceed in a similar way to the proof of Theorem 4.1. Set

$Y_{n}=N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM$

so that

$\displaystyle\mathbb{E}[|Y_{n}|^{k}]$
$\displaystyle=\int_{\mathcal{P}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM\right\rvert^{k}\ d\mu(\mathscr{G})$
$\displaystyle=O\left(\frac{E(n)^{k}}{\gamma(n)}\right).$

It then follows from Chebyshev’s bound that

${\rm
Prob}\left(|Y_{n}|>\lambda\right)\ll\frac{E(n)^{k}}{\lambda^{k}\gamma(n)}.$

Taking $\lambda=\epsilon E(n)$, it follows that

$\displaystyle\sum_{n=1}^{\infty}{\rm Prob}\left(|Y_{n}|>\epsilon
E(n)\right)\ll\sum_{n=1}^{\infty}\frac{1}{\epsilon^{k}\gamma(n)}<\infty.$

for each $\epsilon>0$. Thus, the Borel-Cantelli lemma implies

${\rm Prob}\left(|Y_{n}|>\epsilon E(n)\text{ infinitely often }\right)=0,$

so that

$\displaystyle{\rm
Prob}\left(\limsup_{n\to\infty}\frac{|Y_{n}|}{E(n)}>\epsilon\right)=0.$

By countable additivity and taking $\epsilon=1/m$ for $m\in\mathbb{Z}$ tending
towards $\infty$, this implies $Y_{n}/E(n)$ converges to $0$ almost surely,
concluding the proof. ∎

###  4.3 Almost Sure Upper Bounds

Corollary 1.6 follows almost immediately from Theorem 4.2.

###### Proof of Corollary 1.6.

We bound

$\displaystyle\int_{\mathscr{G}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM\right\rvert^{k}d\mu(\mathscr{G})$
$\displaystyle\leq\int_{\mathscr{G}}\left(|N(\mathscr{G},f_{n})|+\int_{C}|f_{n}|\
dM\right)^{k}d\mu(\mathscr{G})$
$\displaystyle\leq\int_{\mathscr{G}}\left(\sum_{G\in
C/\cong}|f_{n}(G)|(\\#{\rm
Epi}(\mathscr{G},G)+M_{G})\right)^{k}d\mu(\mathscr{G})$
$\displaystyle\leq\sum_{(G_{1},...,G_{k})\in
C^{k}/\cong}|f_{n}(G_{1},...,G_{k})|\sum_{j=0}^{2k}\sum_{\sigma\in
S_{k}}M_{(G_{\sigma(1)},...,G_{\sigma(j)})}^{(j)}M_{G_{\sigma(j+1)}}\cdots
M_{G_{\sigma(k)}},$

where the last line follows from $f_{n}$ being an $L^{1}$ function and moving
the integral all the way to the inside. We remark that the sum over
permutations $\sigma\in S_{k}$ is a bit larger than the truth, but this bound
will be sufficient for our purposes. This is exactly a sum of mixed moments

$\displaystyle\leq\sum_{j=0}^{2k}\sum_{\sigma\in
S_{k}}\int_{\sigma(C^{k})}|f_{n}|\ dM^{(j)}(dM)^{k-j}$
$\displaystyle\ll_{k}\max_{j\in\\{0,1,...,k\\}}\left\\{\int_{C^{k}}|f_{n}|\
dM^{(j)}(dM)^{k-j}\right\\},$

noting that $f_{n}$ is invariant under the permuting the coordinates. Taking

$E(n)=n^{\frac{1+\epsilon}{k}}\max_{j\in\\{0,1,...,k\\}}\left\\{\int_{C^{k}}|f_{n}|\
dM^{(j)}(dM)^{k-j}\right\\}^{1/k}$

and $\gamma(n)=n^{1+\epsilon}$ in Theorem 4.2 is sufficient to conclude the
proof. ∎

###  4.4 The Strong Law of Large Numbers for nonnegative counting functions

Theorem 1.3(iii) takes advantage of some tricks employed by [KP10] for
nonegative functions. The following proof is in essense the same as their main
result, however the functions $f_{n}(G)$ we allow are slightly more general
than the sequence of weights $w_{k}$ utilized in [KP10, Theorem 1].

These tricks are not compatible with controlling the rate of convergence, so
we do not state a more general result for this part. It is certainly possible
that more can be proven for this case but we do not do so here.

###### Proof of Theorem 1.3(iii).

We remark that $n\mapsto N(\mathscr{G},f_{n})$ being nondecreasing almost
everywhere implies

$n\mapsto\int_{C}f_{n}\ dM=\int_{\mathcal{P}}N(\mathscr{G},f_{n})\
d\mu(\mathscr{G})$

is also nondecreasing.

The primary trick is to prove convergence along a subsequence of indices
first, then interpolate to the remaining indices.

Fix $b>1$. Define the subsequence of natrual numbers $(n_{i})$ by

$n_{i}=\inf\left\\{n:\int_{C}f_{n}\ dM\geq b^{i}\right\\}.$

These necessarily exist by $\int_{C}f_{n}dM\to\infty$ with $n$. We apply
Chebyshev’s inequality to

$Y_{n}=N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM$

with $\lambda=\epsilon\int_{C}f_{n}\ dM$ to prove that

$\displaystyle\sum_{i=1}^{\infty}{\rm
Prob}\left(|Y_{n_{i}}|>\epsilon\int_{C}f_{n_{i}}\ dM\right)$
$\displaystyle\ll\sum_{i=1}^{\infty}\frac{1}{\epsilon^{k}\psi\left(\int_{C}f_{n}dM\right)}$
$\displaystyle\leq\sum_{i=1}^{\infty}\frac{1}{\epsilon^{k}\psi\left(b^{i}\right)},$

where the second comparison follows from $\psi$ being nondecreasing. A
straight foward exercise in the convergence of infinite series shows that if
$\sum_{n=1}^{\infty}\frac{1}{n\psi(n)}$ converges, then so does
$\sum_{i=1}^{\infty}\frac{1}{\psi(b^{i})}$ for any $b>1$ (for a reference, see
[Pet09a, Lemma 1]). Similar to the proof of Theorem 4.2, the Borel-Cantelli
Lemma then implies

${\rm
Prob}\left(\lim_{i\to\infty}\frac{|Y_{n_{i}}|}{\displaystyle\int_{C}f_{n_{i}}\
dM}=0\right)=1.$

Next, we interpolate to other indices $n$ not belonging to the subsequence
$(n_{i})$. There exists an $i$ for which $n_{i}\leq n\leq n_{i+1}$. We expand

$\displaystyle\frac{N(\mathscr{G},f_{n})-\int_{C}f_{n}dM}{\int_{C}f_{n}dM}$
$\displaystyle=\frac{N(\mathscr{G},f_{n})-\int_{C}f_{n_{i+1}}dM}{\int_{C}f_{n}dM}+\frac{\int_{C}f_{n_{i+1}}dM-\int_{C}f_{n}dM}{\int_{C}f_{n}dM}.$

Given that the counting function and its moments are nondecreasing almost
everywhere, this produces an almost everywhere upper bound

$\displaystyle\leq\frac{\left\lvert
N(\mathscr{G},f_{n_{i+1}})-\int_{C}f_{n_{i+1}}dM\right\rvert}{\int_{C}f_{n_{i+1}}dM}\frac{\int_{C}f_{n_{i+1}}dM}{\int_{C}f_{n_{i}}dM}+\frac{\int_{C}f_{n_{i+1}}dM-\int_{C}f_{n_{i}}dM}{\int_{C}f_{n_{i}}dM}.$

The subsequence $(n_{i})$ is nondecreasing, and if $n_{i}<n_{i+1}$ it follows
that

$b^{i}\leq\int_{C}f_{n_{i}}dM\leq\int_{C}f_{n_{i+1}-1}dM<b^{i+1}\leq\int_{C}f_{n_{i+1}}dM.$

We can simplify the upper bound to

$\displaystyle\leq\frac{\left\lvert
N(\mathscr{G},f_{n_{i+1}})-\int_{C}f_{n_{i+1}}dM\right\rvert}{\int_{C}f_{n_{i+1}}dM}b^{2}+(b^{2}-1).$

Taking the limit as $i\to\infty$ implies

$\limsup_{n\to\infty}\frac{N(\mathscr{G},f_{n})-\int_{C}f_{n}dM}{\int_{C}f_{n}dM}\leq
b^{2}-1.$

Taking $b>1$ sufficiently close to $1$ gives the desired $\limsup$. The
$\liminf$ is calculated similarly using the lower bound

$\displaystyle\frac{N(\mathscr{G},f_{n})-\int_{C}f_{n}dM}{\int_{C}f_{n}dM}$
$\displaystyle\geq\frac{N(\mathscr{G},f_{n_{i}})-\int_{C}f_{n_{i}}dM}{\int_{C}f_{n_{i}}dM}\frac{\int_{C}f_{n_{i}}dM}{\int_{C}f_{n_{i+1}}dM}-\frac{\int_{C}f_{n}dM-\int_{C}f_{n_{i}}dM}{\int_{C}f_{n}dM}$
$\displaystyle\geq\frac{N(\mathscr{G},f_{n_{i}})-\int_{C}f_{n_{i}}dM}{\int_{C}f_{n_{i}}dM}b^{-2}-1+\frac{\int_{C}f_{n_{i}}dM}{\int_{C}f_{n}dM}$
$\displaystyle\geq\frac{N(\mathscr{G},f_{n_{i}})-\int_{C}f_{n_{i}}dM}{\int_{C}f_{n_{i}}dM}b^{-2}-1+b^{-2}.$

The limit as $i\to\infty$ can again be made arbitrarily close to $0$ by taking
$b$ close to $1$. ∎

##  5 Epi-Products

We let $C$ be a diamond category as defined in [SW22, Definition 1.3] with
$\mathcal{P}$ the corresponding category of pro-objects. We fix throughout a
probability measure $\mu$ on $\mathcal{P}$ with finite moments $M_{G}$.

Theorem 1.4 and Theorem 1.5 rely on the concept of an epi-product, which is a
categorical construction capturing the lack of correlation between $\\#{\rm
Epi}(\mathscr{G},G)$ and $\\#{\rm Epi}(\mathscr{G},H)$ as $\mathscr{G}$ varies
according to $\mu$.

###  5.1 The Definition of an Epi-Product

######  Definition 5.1.

We define the epi-product of $G_{1}$ and $G_{2}$ to be an object
$G_{1}\times_{\rm Epi}G_{2}$ with epimorphisms to $G_{1}$ and $G_{2}$ which
satisfies the universal property

${H}$${G_{2}}$${G_{1}}$${G_{1}\times_{\rm Epi}G_{2}}$

where all morphisms in the diagram (including the universal morphism) are
required to be epimorphisms.

This is similar to the definition of a product, except we require all the
morphisms (including the universal morphism) to be epimorphisms. Notice that
we do not ask that $C$ has epi-products in general. This flexibility will
allow us to choose a wider variety of categories to work over. We remark that
if both the usual product $G_{1}\times G_{2}$ and $G_{1}\times_{\rm Epi}G_{2}$
exist then there is necessarily a unique isomorphism between them. However,
the existence of one does not imply the existence of the other. We will
discuss some basic examples at the end of the paper.

######  Lemma 5.2.

Let $\mu$ be a probability measure on the pro-objects of $C$ with finite
moments $M_{G}$, and let $\mathscr{G}$ vary with respect to $\mu$. Then the
random variables $\\#{\rm Epi}(\mathscr{G},G)$ and $\\#{\rm
Epi}(\mathscr{G},H)$ for $G,H\in C/\cong$ are uncorrelated if $G\times_{\rm
Epi}H$ exists and $M_{G\times_{\rm Epi}H}=M_{G}M_{H}$.

###### Proof.

By the definition of uncorrelated, it suffices to prove that

$\mathbb{E}[\\#{\rm Epi}(\mathscr{G},G)\\#{\rm
Epi}(\mathscr{G},H)]=\mathbb{E}[\\#{\rm Epi}(\mathscr{G},G)]\mathbb{E}[\\#{\rm
Epi}(\mathscr{G},H)].$

The universal property of epi-products implies that

$\\#{\rm Epi}(\mathscr{G},G)\\#{\rm Epi}(\mathscr{G},H)=\\#{\rm
Epi}(\mathscr{G},G\times_{\rm Epi}H).$

Therefore

$\displaystyle\mathbb{E}[\\#{\rm Epi}(\mathscr{G},G)\\#{\rm
Epi}(\mathscr{G},H)]$ $\displaystyle=\mathbb{E}[\\#{\rm
Epi}(\mathscr{G},G\times_{\rm Epi}H)]$ $\displaystyle=M_{G\times_{\rm Epi}H}$
$\displaystyle=M_{G}M_{H}$ $\displaystyle=\mathbb{E}[\\#{\rm
Epi}(\mathscr{G},G)]\mathbb{E}[\\#{\rm Epi}(\mathscr{G},H)].$

∎

###  5.2 The Proof of Theorem 1.4

Theorem 1.4 follows almost immediately from Lemma 5.2.

###### Proof of Theorem 1.4.

For simplicity in this proof, we write $\mathbb{E}$ for the expected value
with respect to $\mu$. When $f_{n}$ is real-valued, the square is always
positive and we can ignore the absolute value. Thus

$\displaystyle\int_{\mathscr{G}}\left(N(\mathscr{G},f_{n})-\int_{C}f_{n}\
dM\right)^{2}d\mu(\mathscr{G})$
$\displaystyle=\mathbb{E}\left[\left(\sum_{G\in C/\cong}f_{n}(G)(\\#{\rm
Epi}(\mathscr{G},G)-M_{G})\right)^{2}\right]$
$\displaystyle=\sum_{(G_{1},G_{2})\in
C^{2}/\cong}f_{n}(G_{1},G_{2})\mathbb{E}\left[\prod_{i=1}^{2}\left(\\#{\rm
Epi}(\mathscr{G},G_{i})-M_{G_{i}}\right)\right]$
$\displaystyle=\sum_{(G_{1},G_{2})\in
C^{2}/\cong}f_{n}(G_{1},G_{2})\left(M^{(2)}_{(G_{1},G_{2})}-M_{G_{1}}M_{G_{2}}\right).$

By Lemma 5.2, $(G_{1},G_{2})\in E(2,M)$ implies $\\#{\rm
Epi}(\mathscr{G},G_{1})$ and $\\#{\rm Epi}(\mathscr{G},G_{2})$ are
uncorrelated. By definition, this is equivalent to
$M^{(2)}_{(G_{1},G_{2})}=M_{G_{1}}M_{G_{2}}$. Therefore the sum simplifies to

$\displaystyle=\sum_{(G_{1},G_{2})\in C^{2}\setminus
E(2,M)/\cong}f_{n}(G_{1},G_{2})\left(M^{(2)}_{(G_{1},G_{2})}-M_{G_{1}}M_{G_{2}}\right)$
$\displaystyle=\int_{C^{2}\setminus E(2,M)}f_{n}\
dM^{(2)}-\int_{C^{2}\setminus E(2,M)}f_{n}\ (dM)^{2}.$

The result then follows from the triangle inequality. ∎

###  5.3 The Proof of Theorem 1.5

Theorem 1.5 is proven similarly to Theorem 1.4, although the higher power is
trickier to keep track of. In particular, some notion of “mutually
uncorrelated” random variables is needed to compare the moments of three or
more of the $\\#{\rm Epi}(\mathscr{G},G_{i})$ terms.

The extra categorical requirements in Theorem 1.5 are used to reframe the
mixed moments as first moments, which allows us to more easily capture the
required “mutually uncorrelated”ness using subobjects of product objects. This
reframing is also useful for computing the mixed moments, so we state and
prove it separately:

######  Lemma 5.3.

Suppose $C$ is a category with for which every morphism factors uniquely (up
to isomorphism) as a composition of an epimorphism with a monomorphism. Let
$G_{1},...,G_{j}\in C$ be objects for which the product
$G_{1}\times\cdots\times G_{j}$ exists in $C$ and has finitely many subobjects
up to isomorphism. Then

$M^{(j)}_{(G_{1},...,G_{j})}=\sum_{\begin{subarray}{c}\iota:H\hookrightarrow\prod_{i}G_{i}\\\
\rho_{m}\iota\text{ is an epimorphism}\end{subarray}}M_{H},$

where the sum is over subobjects $\iota:H\hookrightarrow\prod_{i}G_{i}$ up to
isomorphism for which $\rho_{m}\iota$ is also an epimorphism for each
projection morphism $\rho_{m}:\prod_{i}G_{i}\to G_{m}$.

###### Proof.

By the universal property of the product, there is an embedding

$\prod_{i}{\rm
Epi}(\mathscr{G},G_{i})\hookrightarrow\textnormal{Hom}\left(\mathscr{G},\prod_{i}G_{i}\right).$

Partitioning this embedding based on the possible images gives a bijection to
a disjoint union

$\prod_{i}{\rm
Epi}(\mathscr{G},G_{i})\longleftrightarrow\coprod_{\begin{subarray}{c}\iota:H\hookrightarrow\prod_{i}G_{i}\\\
\rho_{m}\iota\text{ is an epimorphism}\end{subarray}}{\rm
Epi}(\mathscr{G},H).$

Taking cardinalities and integrating with respect to $\mu$ concludes the
proof. ∎

We can now prove Theorem 1.5.

###### Proof of Theorem 1.5.

For simplicity in this proof, we write $\mathbb{E}$ for the expected value
with respect to $\mu$. When $f_{n}$ is real-valued and we take the $2k^{\rm
th}$ moment, we can essentially ignore the absolute values. We then compute

$\displaystyle\int_{\mathscr{G}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM\right\rvert^{2k}d\mu(\mathscr{G})$
$\displaystyle=\mathbb{E}\left[\left(\sum_{G\in C/\cong}f_{n}(G)(\\#{\rm
Epi}(\mathscr{G},G)-M_{G})\right)^{2k}\right]$
$\displaystyle=\sum_{(G_{1},...,G_{2k})\in
C^{2k}/\cong}f_{n}(G_{1},...,G_{k})\mathbb{E}\left[\prod_{i=1}^{2k}\left(\\#{\rm
Epi}(\mathscr{G},G_{i})-M_{G_{i}}\right)\right],$

where we can switch the order of the integral defining $\mathbb{E}$ and the
countable sum because $f_{n}$ is an $L^{1}$-ordering.

We multiply out the product of random variables to produce

$\displaystyle\mathbb{E}\left[\prod_{i=1}^{2k}\left(\\#{\rm
Epi}(\mathscr{G},G_{i})-M_{G_{i}}\right)\right]$
$\displaystyle=\sum_{A\subseteq\\{1,2,...,k\\}}(-1)^{|A|}M^{(|A|)}_{(G_{m})_{m\in
A}}\prod_{m\not\in A}M_{G_{m}}.$

It suffices to show this quantity is $0$ when $(G_{1},...,G_{2k})\in E(2k,M)$,
as the fact that $f_{n}$, $M^{(j)}$, and $E(2k,M)$ are invariant under
permutation implies

$\displaystyle\sum_{(G_{1},...,G_{k})\in C^{2k}\setminus
E(2k,M)/\cong}f_{n}(G_{1},...,G_{2k})\sum_{A\subseteq\\{1,2,...,k\\}}(-1)^{|A|}M^{(|A|)}_{(G_{m})_{m\in
A}}\prod_{m\not\in A}M_{G_{m}}$
$\displaystyle\ll_{k}\max_{j\in\\{1,2,...,2k\\}}\sum_{\begin{subarray}{c}A\subseteq\\{1,2,...,k\\}\\\
|A|=j\end{subarray}}\sum_{(G_{1},...,G_{k})\in C^{2k}\setminus
E(2k,M)/\cong}|f_{n}(G_{1},...,G_{2k})|M^{(j)}_{(G_{m})_{m\in
A}}\prod_{m\not\in A}M_{G_{m}}$
$\displaystyle\ll_{k}\max_{j\in\\{1,...,2k\\}}\int_{C^{2k}\setminus
E(2k,M)}|f_{n}|\ dM^{(j)}(dM)^{2k-j}.$

Now, suppose that $(G_{1},...,G_{2k})\in E(2k,M)$, with distinguished index
$i$. We separate the $i^{\rm th}$ term of the product out, and compute

$\displaystyle\mathbb{E}\left[\prod_{m=1}^{2k}\left(\\#{\rm
Epi}(\mathscr{G},G_{m})-M_{G_{m}}\right)\right]$
$\displaystyle=\mathbb{E}\left[\\#{\rm Epi}(\mathscr{G},G_{i})\prod_{m\neq
i}\left(\\#{\rm
Epi}(\mathscr{G},G_{m})-M_{G_{m}}\right)\right]-M_{G_{i}}\mathbb{E}\left[\prod_{m\neq
i}\left(\\#{\rm Epi}(\mathscr{G},G_{m})-M_{G_{m}}\right)\right]$
$\displaystyle=\sum_{\begin{subarray}{c}A\subset\\{1,...,2k\\}\\\ i\not\in
A\end{subarray}}\sum_{\begin{subarray}{c}\iota:H\hookrightarrow G_{A}\\\
\rho_{m}\iota\text{ is an epi.}\end{subarray}}\left(\mathbb{E}\left[\\#{\rm
Epi}(\mathscr{G},G_{i})\\#{\rm
Epi}(\mathscr{G},H)\right]-M_{G_{i}}M_{H}\right)\prod_{m\not\in A}M_{G_{m}}.$

By Lemma 5.2 and the properties of the distinguished $i^{\rm th}$ index, it is
necessarily the case that

$\mathbb{E}\left[\\#{\rm Epi}(\mathscr{G},G_{i})\\#{\rm
Epi}(\mathscr{G},H)\right]=M_{G_{i}}M_{H}$

for each $H$. Therefore this is a sum of zeros, and cancels out as claimed. ∎

##  6 Examples

Determining the bound in Theorem 1.4 and Theorem 1.5 is fairly specific to the
category, and the motivating examples of number field counting and the
Batyrev-Manin conjecture take a fair bit of set up to define the corresponding
diamond category $C$, construct a measure modeling arithmetic behavior,
translate classical orderings into a sequence $f_{n}$, and compute the moments
$\int f_{n}\ dM$. All this before even considering the question of bounding
the higher moments. The plus side is the immense return value of Theorem 1.3
and Corollary 1.6. If you understand when epi-products exist enough to apply
these results for one ordering, then it is often the case that the same
reasoning will apply to numerous other orderings on the same category.

The author will construct a model for Malle’s counting function and determine
the reasonableness of a large collection of orderings in forthcoming work
[Alb23]. For the purposes of this paper, we include some more basic examples
where these steps are either easier or already done for us in works like
[SW22].

###  6.1 Random subsets with independent elements

We prove a typical tool used to give random models for sets of integers as a
special case of our methods:

######  Theorem 6.1.

Let $\mathscr{G}$ be a random subset of $\mathbb{N}$ where we let the events
$(n\in\mathscr{G})$ be pairwise independent with probability $r_{n}\in[0,1]$.
This forms a true probability measure on $2^{\mathbb{N}}$, and for any any
$\epsilon>0$

$\frac{\displaystyle\\#(\mathscr{G}\cap\\{1,...,n\\})-\sum_{j\leq
n}r_{j}}{\displaystyle n^{\epsilon}\sqrt{\sum_{j\leq
n}r_{j}}}\overset{a.s.}{\longrightarrow}0$

as $n\to\infty$.

If we let $r_{n}=\frac{1}{\log(n)}$ be the probability that $n$ is prime (with
$r_{1}=0$ and $r_{2}=1$ in order to make sense) this is precisely Cramér’s
original random model for the set of primes [Cra94]. We reproduce Cramér’s
main term with this asymptotic:

$\\#(\mathscr{G}\cap\\{1,...,n\\})=\sum_{j=3}^{n}\frac{1}{\log(j)}+o\left(n^{\epsilon}\sqrt{\sum_{j=3}^{n}\frac{1}{\log(j)}}\right).$

with probability $1$. One easily checks using difference calculus that the
main term is of the same order of magnitude as ${\rm Li}(n)$ in agreement with
the prime number theorem. The error we produce, while not as strong as
Cramér’s, is still $o(n^{1/2+\epsilon})$ suggesting the truth of the Riemann
hypothesis.

###### Proof.

Let $C$ be the category gotten by letting $C^{\rm op}$ be the category of
finite subsets of $\mathbb{N}$ whose morphisms are inclusions. The pro-objects
are all the subsets of $\mathbb{N}$. This category is incredibly nice for
numerous reasons:

  * (a)

$\textnormal{Hom}(A,B)={\rm Epi}(A,B)$ contains only the inclusion map
$A\hookleftarrow B$ if $B\subseteq A$ and is empty otherwise.

  * (b)

The product of $A$ and $B$ is given by the union $A\cup B$, and the fact that
all morphisms are epimorphisms implies this is an epi-product too.

  * (c)

Any sequence of finite moments $M_{A}$ on this category is well-behaved in the
sense of [SW22], because the well-behavedness sum is always finite. A level is
the power set $2^{D}$ for some finite set $D$, while the elements with
epimorphisms from $A$ are precisely the subsets of $A$.

  * (d)

The Möbius function on the lattice of isomorphism classes ordered by
epimorphisms is given by $\mu(B,A)=(-1)^{|A\setminus B|}$.

The function $\\#{\rm Epi}(B,A)$ is then the characteristic function of the
event $(A\subseteq B)$, which among a level $2^{D}$ has expected value
precisely $\prod_{a\in A}r_{a}$, so we define this to be $M_{A}$. Certainly
$M_{A\cup B}=M_{A}M_{B}$ whenever $A\cap B=\emptyset$. We check that $M_{A}$
corresponds to a measure on the pro-objects using [SW22, Theorem 1.7]. Indeed,

$\displaystyle v_{2^{D},B}$ $\displaystyle=\sum_{A\subseteq
D}\frac{\hat{\mu}(B,A)}{|\textnormal{Aut}(A)|}M_{A}$
$\displaystyle=\sum_{B\subset A\subseteq D}\frac{(-1)^{|A\setminus
B|}}{1}\prod_{a\in A}r_{a}$ $\displaystyle=\prod_{b\in B}r_{b}\prod_{d\in
D\setminus b}(1-r_{d})$ $\displaystyle\geq 0$

so the measure $\mu$ exists.

We also consider that $\\#{\rm Epi}(\mathscr{G},A)$ is the characteristic
function of the event $(A\subseteq\mathscr{G})$. Thus, we conclude that

$\\#{\rm Epi}(\mathscr{G},A)\\#{\rm Epi}(\mathscr{G},B)=\\#{\rm
Epi}(\mathscr{G},A\cup B),$

so that $M_{(A_{1},A_{2},...,A_{2k})}=M_{A_{1}\cup A_{2}\cup\cdots\cup
A_{2k}}$. In this example, we have a very convenient bound for the finite
moments of unions

$M_{A\cup B}=\prod_{a\in A\cup B}r_{a}\geq\prod_{a\in A}r_{a}\prod_{b\in
B}r_{b}=M_{A}M_{B}$

which follows from the assumption that $r_{n}\in[0,1]$. In particular, this
implies $M^{(j)}\leq M^{(2k)}$ for each $j\leq 2k$.

Any ordering $f_{n}$ supported on a family of pairwise disjoint sets
necessarily satisfies that $C^{2k}\setminus E(2k,M)$ intersected with the
support of $f_{n}$ is precisely the collection of tuples $(A_{1},...,A_{2k})$
in the support of $f_{n}$ which has at most $k$ distinct coordinates. We
choose $f_{n}$ to be the characteristic function of singleton sets $\\{m\\}$
for which $m\leq n$. Up to the number of ways to choose matching coordinates,
it suffices to consider objects in $C^{k}$. Thus, we can bound

$\displaystyle\int_{C^{2k}\setminus E(2k,M)}|f_{n}|\ dM^{(j)}(dM)^{2k-j}$
$\displaystyle\leq\int_{C^{2k}\setminus E(2k,M)}|f_{n}|\ dM^{(2k)}$
$\displaystyle\ll_{k}\sum_{(A_{1},...,A_{k})\in C^{k}}f_{n}(A_{1})\cdots
f_{n}(A_{k})M_{A_{1}\cup\cdots\cup A_{k}}$

Noting that $f_{n}$ is supported on singleton sets, this is equivalent to

$\displaystyle=\sum_{|A|\leq k}\\#\\{a_{1},...,a_{k}\in
A:A=\\{a_{1},...,a_{k}\\}\\}\cdot\prod_{a\in A}f_{n}(\\{a\\})M_{\\{a\\}}$
$\displaystyle\ll_{k}\sum_{1\leq|A|\leq k}\prod_{a\in
A}f_{n}(\\{a\\})M_{\\{a\\}}$ $\displaystyle=\left(1+\int_{C}f_{n}\
dM\right)^{k}-1.$

If $\int_{C}f_{n}\ dM\geq 1$, we can bound this by a constant times the
leading term $(\int_{C}f_{n}\ dM)^{k}$, and otherwise this is $O(1)$. Thus we
conclude via Theorem 1.5 that

$\displaystyle\int_{\mathcal{P}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM\right\rvert^{2k}d\mu(\mathscr{G})$
$\displaystyle\ll_{k}\left(\int_{C}f_{n}\ dM\right)^{k}+O(1).$

Taking $E(n)=\left(\int_{C}f_{n}\ dM\right)^{1/2}$ and $\gamma(n)=n^{2}$ in
Theorem 4.2 then implies

$\frac{N(\mathscr{G},f_{n})-\displaystyle\int_{C}f_{n}\
dM}{n^{1/k}\sqrt{\displaystyle\int_{C}f_{n}\
dM}}\overset{a.s.}{\longrightarrow}0.$

The counting function is given by

$\displaystyle N(\mathscr{G},f_{n})$ $\displaystyle=\sum_{m\leq n}\\#{\rm
Epi}(\mathscr{G},\\{m\\})$ $\displaystyle=\\#(\mathscr{G}\cap\\{1,...,n\\}),$

while the moments of the ordering are given by

$\displaystyle\int_{C}f_{n}\ dM$ $\displaystyle=\sum f_{n}(A)M_{A}$
$\displaystyle=\sum_{m\leq n}r_{m}.$

Taking $k$ sufficiently large concludes the proof. ∎

Keeping in mind the connection with random models for prime numbers, we also
prove the following:

######  Corollary 6.2.

Let $\mathscr{G}$ be a random subset of $\mathbb{N}$ where we let the events
$(n\in\mathscr{G})$ be pairwise independent with probability $r_{n}\in[0,1]$.
Let $\mathbf{1}_{S}$ be a characteristic function of some subset
$S\subseteq\mathbb{N}$, and define the corresponding ordering $f_{n}$
supported on singleton sets by $f_{n}=\mathbf{1}_{S\cap\\{1,...,n\\}}$. Then

$\frac{\displaystyle\\#\\{m\leq n:m\in\mathscr{G}\cap
S\\}-\sum_{\begin{subarray}{c}j\leq n\\\ j\in
S\end{subarray}}r_{j}}{\displaystyle
n^{\epsilon}\sqrt{\sum_{\begin{subarray}{c}j\leq n\\\ j\in
S\end{subarray}}r_{j}}}\overset{a.s.}{\longrightarrow}0$

as $n\to\infty$.

If we let $r_{n}$ be the probabilities of Cramér’s model (or of the
modifications improving the model), Heuristic 1.7 makes predictions for sets
of prime numbers belonging to any subset $S\subseteq\mathbb{N}$ obeying the
incredibly mild density condition

$\sum_{\begin{subarray}{c}j\leq n\\\ j\in S\end{subarray}}r_{j}\gg n^{\delta}$

for some $\delta>0$. This prediction is extremely broad, where the currently
“known issues” amount to divisibility by small primes which are accounted for
in corrections to Cramér’s random model (see [Gra95] for a nice summary). This
example is not “novel”, such measure $1$ statements exist throughout the
literature on random models for the primes generalizing Cramér’s work as a
means to justify a number of the most famous conjectures on the distribution
of primes. This includes the likes of the Hardy-Littlewood conjecture (where
$f$ is the characteristic function on admissible constellations starting from
$j$) and primes of the form $x^{2}+1$ (where $f$ is the characteristic
function of integers of the form $x^{2}+1$).

In addition to being a short example with nice properties, this is meant to
demonstrate two things. Firstly this example demonstrates that important
existing random models (and corresponding results for those models) for the
distribution of prime numbers arise as special cases of Heuristic 1.7
(respectively Theorem 1.3) for the category of subsets of $\mathbb{N}$.
Secondly, this example demonstrates the power of working at this level of
generality. We have given a concrete description for when such probability $1$
results will exist, which allow us to tackle numerous orderings at a time.

###  6.2 Random groups

Any sequence $M_{G}=O(|G|^{n})$ on the category of finite groups is well-
behaved in the sense of [SW22], so their results can be used to determine when
such a moment sequence gives rise to a probability distribution. We will focus
on the example $M_{G}=1$ discussed in [SW22]. We can use Theorem 1.3 to make
predictions for (asymptotically) how many quotients of a “random” profinite
group have a given behavior. While questions of this nature are easy enough to
formulate, the moments of the most natural orderings $f_{n}$ can be much
harder to compute in practice. Utilizing the results of the preceeding
subsection, we give the following example:

######  Theorem 6.3.

Let $M_{G}=1$ on the category of finite _abelian_ groups with associated
probability measure $\mu$ on the profinite abelian groups. Then the average
number of maximal subgroups in a random pro-abelian group of index bounded
above by $n$ tends to $\log\log n$ in almost surely. More specifically,

$\frac{\\#\\{N\leq\mathscr{G}\text{ maximal}:[\mathscr{G}:N]\leq
n\\}}{\log\log n}\overset{a.s.}{\longrightarrow}1$

as $n\to\infty$.

###### Proof.

We take the ordering

$f_{n}(G)=\begin{cases}\frac{1}{p-1}&G\cong C_{p},\ p\leq n\\\
0&\text{else}.\end{cases}$

All maximal subgroups of an abelian group are normal with quotient isomorphic
to $C_{p}$ for some prime $p$, so we can compute

$\displaystyle\\#\\{N\leq\mathscr{G}\text{ maximal}:[\mathscr{G}:N]\leq n\\}$
$\displaystyle=\sum_{p\leq n}\frac{\\#{\rm Epi}(\mathscr{G},C_{p})}{p-1}$
$\displaystyle=\sum_{G\in C/\cong}f_{n}(G)\\#{\rm Epi}(\mathscr{G},C_{p})$
$\displaystyle=N(\mathscr{G},f_{n}).$

We note that $f_{n}$ is nonegative and that $n\mapsto N(\mathscr{G},f_{n})$ is
certainly increasing in $n$. Thus, we can apply Theorem 1.3(iii). We first
compute

$\displaystyle\int_{C}f_{n}\ dM$ $\displaystyle=\sum_{p\leq
n}\frac{1}{p-1}\sim\log\log n.$

We next apply Theorem 1.4 to bound (3). The ordering $f_{n}$ is supported on
the finite simple abelian groups $C_{p}$, which are of pairwise coprime order.
Thus, the intersection of $C^{2}\setminus E(2,M)$ with the support of $f_{n}$
is precisely the diagonal objects $(C_{p},C_{p})$ with $p\leq n$. We compute

$\displaystyle\int_{C^{2}\setminus{\rm Epi}_{M}^{2}C^{2}}f_{n}\ dM^{(2)}$
$\displaystyle=\sum_{p\leq n}\frac{1}{(p-1)^{2}}M_{(C_{p},C_{p})}$
$\displaystyle=\sum_{p\leq n}\frac{1}{(p-1)^{2}}\left(M_{C_{p}\times
C_{p}}-(p-1)M_{C_{p}}\right)$ $\displaystyle=\sum_{p\leq
n}\frac{p}{(p-1)^{2}}$ $\displaystyle\sim\log\log n,$

where the second equality follows from Lemma 5.3, noting that the subgroups of
$C_{p}\times C_{p}$ that surject onto each coordinate are the whole group, and
$p-1$ subgroups isomorphic to $C_{p}$. The $j=1$ integral actually converges
as $n\to\infty$ by a similar computation, so this is the maximum. All together
Theorem 1.4 implies

$\displaystyle\int_{\mathcal{P}}\left\lvert
N(\mathscr{G},f_{n})-\int_{C}f_{n}\ dM\right\rvert^{2}d\mu(\mathscr{G})$
$\displaystyle=O(\log\log n)$ $\displaystyle=O\left(\frac{(\log\log
n)^{2}}{(\log\log n)^{1/2}}\right)$
$\displaystyle=O\left(\frac{\left(\int_{C}f_{n}dM\right)^{2}}{\psi\left(\int_{C}f_{n}dM\right)}\right),$

where $\psi(t)=t^{1/2}$ is non decreasing with
$\lim_{t\to\infty}\psi(t)=\infty$. We confirm that
$\sum\frac{1}{n\psi(n)}=\sum n^{-3/2}<\infty$ converges, so the result follows
from Theorem 1.3(iii). ∎

More general questions on random groups can be asked, but much about the
asymptotic behavior of large families of groups is difficult to compute. This
makes asymptotic growth rates and error terms also difficult to compute in
practice. For instance, it would be nice to study the ordering
$f_{n}(G)=\frac{1}{|\textnormal{Aut}(G)|}$ if $|G|\leq n$ and $0$ otherwise.
The corresponding counting function is then

$N(\mathscr{G},f_{n})=\\#\\{N\trianglelefteq\mathscr{G}\mid[\mathscr{G}:N]\leq
n\\}.$

This is a naturally interesting function to ask about. However, the
corresponding moment

$\int_{C}f_{n}\ dM=\sum_{|G|\leq n}\frac{1}{|\textnormal{Aut}(G)|}$

is not easy to determine. For instance, there are reasons to suspect that 100%
of groups ordered by cardinality are $2$-groups, but the author is not aware
of a proof of this statement. Moreover, the bound in Theorem 1.4 requires some
more serious group theory to translate the results of Lemma 5.3 into bounds on
the error.

A number of interesting statistical questions that are simple to state arise
in this way. The language of Theorem 1.3 is useful to create a framework for
determining asymptotic growth rates, even if the moments of the ordering
requires additional study to prove any results.

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# Topological susceptibility and axial symmetry at finite temperature

JLQCD Collaboration: a, Sinya Aokib, Shoji Hashimotoa,c, Takashi Kanekoa,c,
Hideo Matsufurua, Jun-ichi Noakia, Eigo Shintanid

aTheory Center, IPNS, High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801, Japan
bGraduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba
305-8571, Japan
cSchool of High Energy Accelerator Science, The Graduate University for
Advanced Studies (Sokendai), Tsukuba 305-0801, Japan
dRIKEN-BNL Research Center, Upton, NY 11973-5000, USA E-mail:
<EMAIL_ADDRESS>
###### Abstract:

We consider the simulation of finite temperature QCD with two flavors of
dynamical overlap fermions in order to study the suppression of the axial
$U(1)$ symmetry breaking at the chiral phase transition point. As a
preliminary study, pure gauge simulations are performed to investigate how
fixing the topology affects physical quantities like the topological
susceptibility, $\chi_{t}$, at finite temperature, showing that it is possible
to reconstruct known results from the fixed topology sector. First results on
the degeneracy of meson correlators in the high temperature QGP sector are
shown.

##  1 Introduction

It is an interesting and long standing problem to understand whether the
flavour-singlet axial $U(1)$ symmetry is restored or not above the finite
temperature transition in the chirally symmetric phase of QCD. At low
temperature it is well known that the chiral symmetry is spontaneously broken
while the axial $U(1)$ symmetry is broken at quantum level by the presence of
configurations with non-trivial topological structure. The semi-classical
configurations that contribute to the axial charge are called instantons and
it can be shown that zero modes of the Dirac operator are related to the
presence of these and other configurations that have non-trivial topological
charge. Hence, topology plays a major role in this subject. One clear example
is the Witten-Veneziano relation [1, 2] that connects the topological
susceptibility of pure gauge theory to the heavy mass of the
$\eta^{\prime}(958)$ particle, the candidate to be the would-be Nambu-
Goldstone boson of the axial $U(1)$ symmetry.

Our purpose is to study the behavior of meson correlators at high temperature,
looking for signals of (approximate) degeneracy in all the singlet and triplet
channels ($\sigma,\delta,\pi,\eta^{\prime}$) toward the chiral limit. This
would be an evidence of effective restoration of both chiral and axial
symmetries, several intermediate steps are to be considered to control the
systematic errors. The best available discretization of the Dirac operator is
the Overlap fermions [3]. Dynamical simulations with Overlap fermions are
possible with current machines and were performed by JLQCD collaboration in
the past years [4]. The price payed in order to be able to have a very precise
fulfillment of the Ginsparg-Wilson relation [5] on the lattice is to fix the
topology. Changing topology in a HMC trajectory requires generating rough
configurations associated with near-zero modes of the hermitian Wilson
operator ($H_{W}$). A discontinuity arises in the HMC when one of these modes
crosses zero. Treating this discontinuity is a costly task, although it could
be achieved using the reflection/refraction methods [6]. One solution to this
problem is to suppress near zero modes of $H_{W}$ and to avoid the zero
crossing, with the disadvantage of preventing topology change during the
simulation. The simulation is no more ergodic, and some technique has to be
developed to obtain the physical results at $\theta=0$. This program was
indeed developed by JLQCD collaboration (see [7] for details) and it works
nicely at zero temperature. In order to accomplish our main target, we need to
be sure that the same methods can be applied even in the finite temperature
case. The first step towards the understanding and controlling the effect of
fixing topology at finite temperature is to reproduce the known results of
pure gauge theory. In particular we focus on the most relevant quantity, i.e.
the topological susceptibility $\chi_{t}$, and see if it can be reproduced
with the same methods adopted at zero temperature.

The rest of the paper consists of two main sections. The first is devoted to
the discussion of our study of topological susceptibility in pure gauge theory
at finite temperature. The second is the measurements of meson correlators, in
flavor singlet and triplet channels, and the spectral density of the Overlap
Dirac operator, where we show some evidences that the axial $U(1)$ symmetry
breaking is suppressed after the chiral phase transition. In the last section
we will draw some conclusions.

##  2 Pure gauge simulations

In a previous paper [8], the JLQCD-TWQCD collaboration calculated the
topological susceptibility at zero temperature in the case of two flavors of
overlap fermions. They demonstrated that it is possible to obtain physical
quantities even with simulations at fixed topological sector. They checked the
prediction of $\chi$PT, i.e. $\chi_{t}=(m_{q}\Sigma)/N_{f}+O(m_{q}^{2})$,
finding the expected linear behavior of $\chi_{t}$ versus the sea quark mass.
The methods described in [7] were applied in that case, i.e. the topological
susceptibility is measured by the long distance value of disconnected
correlators of pseudoscalar operators (they are non-zero because in a fixed
topology environment clustering property of field theory is violated). We will
investigate numerically whether we can apply the same methodology at finite
temperature too, with a preliminary test in pure gauge simulations, in order
to be trained in dealing with this systematic error in the most interesting
full QCD case.

The correlators are calculated assuming that the major contribution comes
solely from the lowest 50 eigenmodes. We tested for one value of $\beta$ that
this is really the case by changing the number of eigenmodes to 30 and 40. The
result is that the long distance behavior saturates at 40-50 eigenvalues, the
short distance correlator has a worse approximation, as expected.

In the pure gauge theory,there is an anomalous contribution to the pseudo-
scalar meson correlator in the flavor singlet channel, which is called hairpin
diagram [9, 10]:

$H(p)=f_{P}\frac{1}{p^{2}+m_{\pi}^{2}}m_{0}^{2}\frac{1}{p^{2}+m_{\pi}^{2}}f_{P}.$
(1)

This is a zero temperature result, and we assume that it is valid at finite
temperature with all parameters depending on temperature. We actually find
that this fits our data very well. Also, the results are independent of the
valence quark mass, as it should be in the case of pure gauge theory. For
statistical purposes, results are obtained with a joint fit of connected and
disconnected part. The joint fit assumes an identical decay rate for the two
correlators, long distance of connected part going to zero, to constrain the
rest of parameters. Fits also discard the first and last three points of the
spatial separation range.

Simulations were performed using an Iwasaki lattice gauge action, on a lattice
of dimensions $24^{3}\times 6$ and $\beta\in[2.35,\dots,2.55]$ in order to be
in the range of temperatures $[249,\dots,347]$ MeV. The phase transition was
estimated by the Polyakov loop behavior to be at $T=288$ MeV. Except for one
run at $Q=1$ all configurations were generated at $Q=0$.

The outcome for the topological susceptibility measurements is shown in figure
1. Our results are shown as black dots. They follow with a good accuracy the
reference results by the Regensburg group [11], obtained with direct zero
eigenmode counting. In order to cross-check the methodology we also
accumulated configurations without fixing topology, by using the same HMC
algorithm. The parameters for this run were chosen so that it corresponds to
the same temperature as the $\beta=2.50$ run, $T/T_{C}\simeq 1.1$. We then
extracted two subsets of configurations with $Q=0$ and $Q=1$ and applied the
correlator methodology to extract topological susceptibility. The final
results agree among two sectors and moreover are in accordance with an
estimate using the Atiyah-Singer’s index theorem for all configurations
(points indicated by cross $Q=$all, blue dot $Q=0$ and star $Q=1$ in the
figure). This result indicates that the term in the action that forces fixed
topology is not preventing appearance of local topological fluctuations, like
pairs of caloron-anticalorons (so called topological molecules) which are
associated to very low eigenmodes. These topological fluctuations are
distributed in the same way as the ones generated without fixing topology.

Figure 1: (Topological susceptibility on the $24^{3}\times 6$ lattice.
Statistical errors only. Reference data points are diamonds and squares. The
black dots are our results from measurements at Q=0. Figure 2: Lowest
eigenvalues at several temperatures. On abscissas the HMC trajectory number,
one point every 100 trajectories except for RUN-II where we saved one
configuration every 50.

So far we demonstrated that our method works very well even in the finite
temperature case. However we discovered and reported at the Lattice conference
that topology fixing poses potential problems in the high temperature regime,
at finer lattice spacing. The lowest eigenmodes (main source of the
topological susceptibility signal) take longer time to thermalize than usual
thermodynamical quantities. This is a known problem but in our case the effect
appeared more strongly. The points at $\beta=2.50$ and $\beta=2.55$ were the
most problematic, exhibiting some dependence on the initial configuration, see
Figure 2. In particular the run tagged RUN-II, obtained by starting from a
thermalized configuration at $\beta=2.50$ and heated up to $\beta=2.55$, was
giving different results from a similar run starting from unit configuration.
This was a severe problem, mining the reliability of higher temperature runs.
We thus further investigated the issue by accumulating more configurations,
especially for the highest temperature point. Final result is shown in Figure
2. We recall that all runs started from a unit configuration, high
temperature, and then thermalized (except RUN-II). Thermalization is monitored
by plaquette and Polyakov Loop as standard observables. By looking at the
lowest eigenvalue history, the first three panels do not show any particular
problem in thermalization, except the expected increasing autocorrelation time
going to finer lattices. With newest data no anomalous behavior is detected
also for $\beta=2.55$. At Lattice 2011 the history was much shorter, less than
10k trajectories, and no jumps were present. Transition between two different
values of the lowest eigenmode is sharp but the tunneling rate is non-zero,
and no dependence on the starting configuration is observed. The issue was
just statistical and is not expected to affect full QCD results.

We thus conclude that physical quantities can be obtained from fixed topology
simulations at finite temperature.

##  3 Dynamical overlap simulations

In this section we will discuss some of the results from simulations of two
flavors of dynamical overlap fermions in the high temperature region.

We accumulated O(200-300) thermalized configurations per $\beta$ with several
masses and temperatures, see table 1. Accumulating the corresponding zero
temperature configurations would have been a really demanding task so we do
not have currently a measurement of the lattice spacing and pion mass for
every $\beta$. At $\beta=2.30$ and bare quark mass $am=0.015$ was 286 MeV, 360
MeV at $am=0.025$ [12]. By looking at the Dirac operator spectral density we
observe that a gap is already opened at a temperature of around 192 MeV
($\beta=2.25$) near the chiral limit, $am=0.01$, confirmed by higher
temperature simulations. This could be a signal of strong suppression of axial
symmetry breaking, because the splitting between flavor-singlet and non-
singlet correlators dominantly comes from the near-zero modes, as we
demonstrate below. We do not observe this behavior with current data at the
temperature of 177 MeV ($\beta=2.20$, $am>0.025$).

We report also our first observation of the meson correlators in this range of
temperatures (figure 4). We plot the singlet and non-singlet pseudo-scalars
($\eta^{\prime}$ and $\pi$) and singlet and non-singlet scalars ($\sigma$ and
$\delta$).

Data at $\beta=2.20$ do not show any degeneracy among the correlators of the
lightest mesons. This could be an effect of the high mass ($am=0.025$) but it
is also possible that this temperature is still in the chirally broken region
(see the spectral density plot). The number of configurations is not
sufficient to precisely determine the transition temperature, so we cannot
settle this question. The meson correlators at the highest temperature,
starting from $\beta=2.25$, show a clear tendency toward degeneracy in the
chiral limit, a signal that $U(1)_{A}$ axial symmetry is effectively restored.

We need to accumulate more data in order to be conclusive, but certainly we
observe robust signals of axial symmetry restoration. Further study around the
phase transition is on the way.

$\beta$ |  $a$(fm) |  $T$(MeV) | Masses ($am$)
---|---|---|---
2.18 | 0.144 | 171 | 0.05
2.20 | 0.139 | 177 | 0.05, 0.025
2.25 | 0.128 | 192 | 0.01
2.30 | 0.118 | 208 | 0.05, 0.025, 0.01
Table 1: Parameters of some of the collected data for two flavors of dynamical
overlap fermions, lattice dimensions $16^{3}\times 8$. The reported lattice
spacings are the extrapolations to the chiral limit. Figure 3: Spectral
density of the overlap Dirac operator for $N_{f}=2$. The several $\beta$s were
isolated to emphasize the mass dependence of the density. Zero counting of
eigenvalues is intended on the left when line stops.

Figure 4: Meson correlators at several temperatures and masses.

##  4 Conclusions and perspectives

We have shown that it is possible to perform finite temperature lattice
simulations with overlap fermions at fixed topology. It is tested in the case
of pure gauge theory where measurements of topological susceptibility can be
compared with known results.

Having under control the systematics of topology fixing we performed
simulations of two flavors of overlap fermions at several temperatures in the
range $[170\sim 208]$ MeV. We measured the correlators of meson operators in
all the (pseudo)scalar channels looking for their degeneracy in the chiral
limit, signal of effective restoration of both chiral and axial symmetry. We
found evidence of this restoration, corroborated also by the spectral density
analysis that exhibits a gap in the chiral limit at temperatures above $>192$
MeV, in the current data set.

We can currently fairly say that we have clear evidence of $U(1)_{A}$
effective restoration in a region just above the chiral phase transition in
two flavors QCD. The next step is narrowing the region of uncertainty about
the temperature when the gap starts opening. In a forthcoming paper in
preparation a complete analysis and the newly collected data will be
presented.

This work is supported in part by the HPCI Strategic Program of Ministry of
Education and in part by the Grant-in-Aid for Scientific Research on
Innovative Areas (No. 2004: 20105001, 20105003, 20105005, 21674002, 21684013).

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