# Poisson-Dirichlet asymptotics in
condensing particle systems

Paul Chleboun  ,  Simon Gabriel  and  Stefan Grosskinsky  P. Chleboun,
Department of Statistics, University of Warwick, Coventry, CV4 7AL, United
Kingdom<EMAIL_ADDRESS>S. Gabriel, Mathematics Institute,
University of Warwick, Coventry CV4 7AL, United Kingdom
<EMAIL_ADDRESS>S. Grosskinsky, Department of Applied Mathematics,
Delft University of Technology, 2600 AA Delft, The Netherlands
<EMAIL_ADDRESS>
###### Abstract.

We study measures on random partitions, arising from condensing stochastic
particle systems with stationary product distributions. We provide fairly
general conditions on the stationary weights, which lead to Poisson-Dirichlet
statistics of the condensed phase in the thermodynamic limit. The Poisson-
Dirichlet distribution is known to be the unique reversible measure of split-
merge dynamics for random partitions, which we use to characterize the limit
law. We also establish concentration results for the macroscopic phase, using
size-biased sampling techniques and the equivalence of ensembles to
characterize the bulk distribution of the system.

###### Key words and phrases:

Poisson-Dirichlet distribution, split-merge dynamics, random partitions,
equivalence of ensembles, interacting particle systems, condensation

S. Gabriel acknowledges financial support from EPSRC through grant
EP/R513374/1

###### Contents

  1. 1 Introduction and results
  2. 2 Background on partitions
  3. 3 Proof of Theorem 1.3
  4. 4 Proof of Theorem 1.6
  5. 5 Application to interacting particle systems and conclusion
  6. A Equivalence of ensembles

##  1\. Introduction and results

###  1.1. Mathematical setting and motivation

The results presented in this paper are motivated by the study of interacting
particle systems. We consider finite systems consisting of $N$ particles on
$L$ sites indexed by the set $\Lambda$, where $|\Lambda|=L$. For simplicity we
take $\Lambda=\\{1,\ldots,L\\}$ in the remainder of this paper. The space of
such particle configurations $\eta=(\eta_{1},\ldots,\eta_{L})$ with
$\eta_{x}\in\mathbb{N}_{0}$ is given by

$\Omega_{L,N}:=\left\\{\eta\in\mathbb{N}_{0}^{L}:\sum_{x=1}^{L}\eta_{x}=N\right\\}\,,$

which we equip with the discrete topology. The dynamics of systems we consider
are assumed to be irreducible Markov processes on $\Omega_{L,N}$, conserving
only the quantities $N$ and $L$. Thus, there exists a unique invariant
distribution $\pi_{L,N}$ of the system on $\Omega_{L,N}$, which is called the
_canonical distribution_.

We will focus on models where the canonical distributions are of product form

$\pi_{L,N}[d\eta]=\begin{cases}\displaystyle\frac{1}{Z_{L,N}}\prod_{x=1}^{L}w_{L}(\eta_{x})\,d\eta\
&\text{if }\sum_{x=1}^{L}\eta_{x}=N\,,\\\ 0&\text{otherwise}.\end{cases}$ (1)

Here $d\eta$ denotes the counting measure and
$(w_{L}(n))_{n\in\mathbb{N}_{0}}$ is a sequence of positive weights, possibly
depending on the system size $L$. The normalising constant, called _canonical
partition function_ , is given as

$Z_{L,N}=\sum_{\eta\in\Omega_{L,N}}\prod_{x=1}^{L}w_{L}(\eta_{x})\,.$

Note that the weights $(w_{L}(n))_{n\in\mathbb{N}_{0}}$ are independent of the
site $x$, thus the $\pi_{L,N}$ are permutation invariant and in particular
spatially homogeneous, so that single-site marginals
$\pi_{L,N}[\eta_{x}\in\cdot]$ do not depend on $x$.

We are primarily interested in the limiting behaviour of $\pi_{L,N}$ in the
thermodynamic limit $N,L\to\infty$ such that $\tfrac{N}{L}$ converges to
$\rho\;\geqslant\;0$, which we will subsequently abbreviate by
$N/L\rightarrow\rho$. Assume for now that the weak limit of the single-site
marginals exists for all $\rho\;\geqslant\;0$,

$\displaystyle\pi_{L,N}[\eta_{x}\in\cdot\,]\to\nu_{\rho}[\,\cdot\,]\quad\text{as
}N/L\rightarrow\rho\,,$ (2)

and the limit is a probability measure on $\mathbb{N}_{0}$. This implies in
particular convergence of the expectations for bounded functions
$f:\mathbb{N}_{0}\mapsto\mathbb{R}$, i.e.

$\displaystyle\sum_{n=0}^{\infty}f(n)\pi_{L,N}[\eta_{x}=n]\to\nu_{\rho}(f)\,,$

where we write $\nu_{\rho}(f)$ for the expectation of $f$ under $\nu_{\rho}$.
Sometimes we will also write $\nu_{\rho}(f(\eta_{x}))$ for the corresponding
expectation if needed for clarity. Looking at a single site’s expected
occupation number under $\pi_{L,N}$, we see that due to spatial homogeneity we
have

$\pi_{L,N}(\eta_{x})=\sum_{n=0}^{N}n\,\pi_{L,N}[\eta_{x}=n]=\frac{N}{L}\to\rho\,,$

when taking the thermodynamic limit. However, because the identity $f(n)=n$ is
an unbounded function, we cannot guarantee that the particle density of the
system is conserved in the limit $\nu_{\rho}$ and $\nu_{\rho}(\eta_{x})$ may
be strictly smaller than $\rho$. This phenomenon is known as condensation.

######  Definition 1.1 (Condensation).

A system characterised by spatially homogeneous canonical distributions
$(\pi_{L,N})_{L,N}$ exhibits condensation in the thermodynamic limit
$N/L\rightarrow\rho$ if $\nu_{\rho}$ in (2) exists and

$\nu_{\rho}(\eta_{x})<\rho=\lim_{N/L\rightarrow\rho}\pi_{L,N}(\eta_{x})\,.$

Furthermore, we say that the system has a condensation transition with
critical density $\rho_{c}\;\geqslant\;0$ if

$\nu_{\rho}(\eta_{x})\begin{cases}=\rho&\quad\text{ if }\rho<\rho_{c}\,,\\\
<\rho&\quad\text{ if }\rho>\rho_{c}\,.\end{cases}$

In a nutshell, condensation means that a positive fraction of the total
density $\rho$ is not observed in the thermodynamic limit, since it
concentrates on sites with diverging occupation numbers called the _condensed
phase_. Clearly, the number of such sites has a vanishing volume fraction and
does not contribute to the weak limit $\nu_{\rho}$, which describes the
distribution of the _background_ or _bulk phase_.

Condensation in homogeneous systems has been studied previously in great
generality, partially reviewed e.g. in [4, 13, 20]. Early results are
formulated in the context of zero-range processes in [7, 12, 19] and in [28,
24, 2, 1] on a rigorous level, where the condensed phase concentrates on a
single lattice site. Our goal here is to understand details of the condensed
phase when it extends over more than one site and exhibits a non-trivial
structure. Such structures have previously been observed as a result of
spatial correlations [38, 39, 37] and as a result of $L$-dependent stationary
weights, with a soft cut-off for site occupation numbers under zero-range
dynamics [36] or in the inclusion process [27].

On the level of particle configurations the condensed phase disappears in the
thermodynamic limit due to its vanishing volume fraction. To study its
structure, it is more useful to interpret a configuration as an ordered
partition of the total mass. For models of type (1) partitions and particle
configurations are equivalent, since $\pi_{L,N}$ is permutation invariant and
the underlying lattice structure is irrelevant.

We will represent particle configurations rescaled by the total mass $N$ as
ordered partitions of the unit interval $[0,1]$ on the set

$\displaystyle\overline{\nabla}:=\left\\{p=(p_{i})_{i}\in[0,1]^{\mathbb{N}}:\sum_{i=1}^{\infty}p_{i}\;\leqslant\;1\quad\text{and}\quad
p_{1}\;\geqslant\;p_{2}\;\geqslant\;\cdots\right\\}\,.$ (3)

We use the map $T=T(L,N):\Omega_{L,N}\to\overline{\nabla}$ with

$T(\eta):=\frac{1}{N}\left(\widehat{\eta}_{1},\ldots,\widehat{\eta}_{L},0,\ldots\right)\,,$
(4)

where $\widehat{\eta}=(\widehat{\eta}_{1},\ldots,\widehat{\eta}_{L})$ denotes
the entries in $\eta$ in decreasing order with
$\widehat{\eta}_{1}\;\geqslant\;\widehat{\eta}_{2}\;\geqslant\;\ldots\;\geqslant\;\widehat{\eta}_{L}$.
Since any permutation of entries in $\eta$ yields the same partition in
$\overline{\nabla}$, the map $T$ is not injective. Thus, the push-forward
measure of $\pi_{L,N}$ under $T$ on $\overline{\nabla}$ is given by

$\mu_{L,N}[dp]:=\pi_{L,N}\circ
T^{-1}[dp]=\pi_{L,N}[d(Np)]\,|T^{-1}(\\{p\\})|\,,$ (5)

with $Np$ denoting the configuration in $\Omega_{L,N}$ induced by the finite
ordered partition $p\in\overline{\nabla}$. Note that $\mu_{L,N}$ concentrates
on finite partitions with at most $L$ non-zero entries and
$T^{-1}(\\{p\\})=\emptyset$ otherwise. In fact, the $\mu_{L,N}$ further
concentrate on the subset where $\sum_{i=1}^{\infty}p_{i}=1$. However, this
space is not compact, unlike $\overline{\nabla}$ which is compact w.r.t. the
product topology by Tychonoff’s theorem, ensuring existence of subsequential
weak limits of $\mu_{L,N}$ in the thermodynamic limit $N/L\to\rho$. The
objective of this article is to identify general assumptions on the weights
$(w_{L})_{L}$, such that for $\rho>0$ large enough $\mu_{L,N}$ converges
weakly to a Poisson-Dirichlet distribution as $N/L\rightarrow\rho$. Details on
this distribution are introduced in Section 2.

The starting point of our analysis is the recent paper [27] in which weights
of the form

$w_{L}(n)=\frac{\Gamma(n+d)}{n!\Gamma(d)}\,,$ (6)

with $d=d(L)\in\mathbb{R}$ such that
$\lim_{L\to\infty}dL=\theta\in(0,\infty)$, are considered. Such weights emerge
for example from the dynamics of the inclusion process introduced in [16, 5],
which can also be applied in population genetics as a multi-species Moran
model [33], with the above scaling of the parameter $d$ corresponding to a
small mutation rate. With weights (6) the system (1) exhibits a condensation
transition with critical density $\rho_{c}=0$, leaving an empty bulk behind,
i.e. $\nu_{\rho}(\eta_{x})=0$. For the condensed phase in this model we have

$\displaystyle\mu_{L,N}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\text{PD}(\theta)\,,\quad\text{ as
}\,N/L\rightarrow\rho\,\text{ for all }\rho>0\,,$ (7)

where PD$(\theta)$ denotes the Poisson-Dirichlet distribution with parameter
$\theta$ [27, Theorem 1]. The proof uses the fact that $\pi_{L,N}$ with
weights (6) is a Dirichlet multinomial distribution which permits an exact,
simple expression for the corresponding partition function $Z_{L,N}$. This
leads to exact expressions for the distribution of size-biased marginals which
characterize the Poisson-Dirichlet limit (cf. Section 2.1).

Our main result provides a generalization to models with more general weights
that do not lead to exact expressions for $Z_{L,N}$, and with non-trivial bulk
distribution where $0<\nu_{\rho}(\eta_{x})<\rho$. In our proof, we not only
make use of the Poisson-Dirichlet distribution’s characterisation via size-
biased sampling, which was essential for the arguments in [27], but also use
the characterisation as the unique reversible distribution under split-merge
dynamics as explained in Section 2.1. This allows us to avoid explicit
expressions or approximations of the partition function $Z_{L,N}$ which are
not always at hand. Our approach is motivated by a recent paper by Ioffe and
Tóth [26], where the embedding of integer configurations into partitions of
$[0,1]$ was used to show convergence of cycle-length processes of stationary
random stirring to the split-merge dynamics.

###  1.2. Main results

We recall from (1) that the $\pi_{L,N}$ are probability measures on
$\Omega_{L,N}$ given by

$\displaystyle\pi_{L,N}[d\eta]=\frac{1}{Z_{L,N}}\prod_{x=1}^{L}w_{L}(\eta_{x})\,d\eta\,.$

For our first result we fix a density $\rho>0$ and choose $N,L\to\infty$ such
that $N/L\rightarrow\rho$. We assume that

  1. (A1)

$w_{L}(n)\;\geqslant\;0$ for all $n\in\mathbb{N}$ and the limit

$\lim_{L\to\infty}w_{L}(n)=:w(n)\;\geqslant\;0\quad\mbox{exists for all fixed
}n\in\mathbb{N}_{0}\,,$

such that $w$ is summable and non-trivial,

and a weak form of the equivalence of ensembles:

  1. (A2)

The limiting probability distribution (2) exists and is of the form

$\pi_{L,N}[\eta_{1}=n]\to\nu_{\rho}[n]=\frac{w(n)\phi^{n}}{\mathcal{Z}}\qquad\forall\,n\;\geqslant\;0\,,$

for some $\phi>0$, and $\mathcal{Z}=\sum_{n}w(n)\phi^{n}\in(0,\infty)$ is the
corresponding normalising constant.

######  Remark 1.2.

The equivalence of ensembles is the main mathematical framework to understand
the large scale behaviour of statistical mechanics models, and in particular
to show condensation as in Definition 1.1 (see Section 4 for details).
Assumption (A2) has therefore been established for all homogeneous particle
systems that are known to exhibit condensation (see citations above). We will
see in our second result Theorem 1.6, that equivalence of ensembles and
Assumption (A2) can be shown for a large class of models under slightly
stronger assumptions on convergence of the stationary weights.

To understand the role of the parameters $\phi$ and $\mathcal{Z}$, assume that
the limit in (2) and the measure $w$ defined by the limiting weights in (A1)
are equivalent and $w(0),\,w(1)>0$. Then we can define

$\phi:=\frac{\nu_{\rho}[1]}{w(1)}\frac{w(0)}{\nu_{\rho}[0]}>0\quad\text{and}\quad\mathcal{Z}=\frac{w(0)}{\nu_{\rho}[0]}\in(0,\infty)$

and use a telescopic product argument for the ratio of partition functions to
see that

$\displaystyle\frac{Z_{L-1,N-n}}{Z_{L,N}}=\frac{Z_{L-1,N}}{Z_{L,N}}\prod_{k=1}^{n}\frac{Z_{L-1,N-k}}{Z_{L-1,N-k+1}}\to\frac{1}{\mathcal{Z}}\phi^{n}\,.$

Thus, we recover Assumption (A2) in this case, and a similar argument works in
the degenerate case $w(0)>0$ and $w(k)=0$ for all $k\;\geqslant\;1$, which is
the case for the inclusion process with weights (6).

In order to formulate our main result with simple notation, we assume without
loss of generality that $\phi=\mathcal{Z}=1$, since we can absorb $\phi$ and
$\mathcal{Z}$ into the weights by

$\widetilde{w}_{L}(n):=\frac{w_{L}(n)\phi^{n}}{\mathcal{Z}}\quad\text{and
}\quad\widetilde{w}(n):=\frac{w(n)\phi^{n}}{\mathcal{Z}}\,.$

So $w$ can be assumed to be the probability mass function of $\nu_{\rho}$,
i.e.

$\nu_{\rho}[\eta_{x}=n]=w(n)\quad\mbox{for all }n\;\geqslant\;0\,.$ (8)

We are interested in the macroscopic part of the condensed phase, i.e. the
distribution of occupation numbers $\eta_{x}$ that scale linearly with the
total mass $N$ in the system when taking the thermodynamic limit. The
structure of this macroscopic phase will depend on the asymptotic behaviour of
the stationary weights. In order to see Poisson-Dirichlet statistics in this
phase, the weights $w_{L}(n)$ must scale like $(nL)^{-1}$, for at least all
$n$ which are visible under macroscopic rescaling:

  1. (A3)

there exists $\theta>0$ such that for all $\varepsilon\in(0,1)$ we have

$\sup_{\varepsilon
N\;\leqslant\;n\;\leqslant\;N}\left|nw_{L}(n)L-\theta\right|\to
0\qquad\text{as }N/L\rightarrow\rho\,.$

We also impose a second moment condition:

  1. (A4)

The limit

$\alpha^{2}:=\frac{1+\theta}{\rho}\lim_{N/L\rightarrow\rho}\frac{\pi_{L,N}(\eta_{x}^{2})}{N}\in[0,1]\quad\text{exists}\,.$
(9)

In Section 3 we will see that

$\alpha=\mu(\|p\|_{1})\,,\quad\mbox{with}\quad\|p\|_{1}=\sum_{j=1}^{\infty}p_{j}\,,$

i.e. $\alpha$ coincides with the expected total mass fraction for each
accumulation point $\mu$ of the measures $(\mu_{L,N})_{L,N}$, and that the
variance of $\|p\|_{1}$ vanishes. Therefore, Assumption (A4) guarantees that
the macroscopic phase is well defined in the thermodynamic limit, excluding
fluctuations of mass towards other scales, and plays an important role when
identifying the accumulation points $\mu$. This leads to our first main
result.

######  Theorem 1.3.

Let $\rho>0$ and $(w_{L})_{L}$ be a sequence of weights satisfying (A1) \-
(A4) for some $\theta\in(0,1]$. Then the laws of mass partitions (5) converge
as

$(\mu_{L,N})_{L,N}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\text{PD}_{[0,\alpha]}(\theta)\,,\quad\text{ as }\quad
N/L\rightarrow\rho\,,$

where PD${}_{[0,\alpha]}(\theta)$ denotes the Poisson-Dirichlet distribution
with parameter $\theta$, concentrating on partitions of the interval
$[0,\alpha]$.

######  Remark 1.4.

(a) If $\theta=0$, assumption (A3) does not specify the leading order limiting
behaviour of the weights. Our proof can cover this case, if we assume in
addition that $n\mapsto nw_{L}(n)$ is a regularly varying function111i.e.
$nw_{L}(n)/\big{(}\lambda nw_{L}([\lambda n])\big{)}\to C\in(0,\infty)$ for
all $\lambda>0$ as $n\to\infty$ for large enough $L$ (see (28) in the proof).
This is consistent with choosing fixed weights of the form
$w_{L}(n)=w(n)=n^{-b}/\mathcal{Z}$ for $b>2$ whenever $n>0$ and $w_{L}(0)=1$.
Indeed, for this choice, as part of a larger class of sub-exponential weights,
it is a well known result that the condensed phase consists of a single
cluster [12, 24, 2], which can be interpreted as a degenerate Poisson-
Dirichlet distribution with $\theta=0$.

(b) The result covers also the case $\alpha=0$ where the macroscopic phase is
empty and its limiting distribution is trivial. This includes models that do
not condense at all, or where the condensed phase concentrates on sub-
macroscopic scales such as for models with spatial correlations [38, 39, 37].
Of course our result does not say anything interesting in this case, since
there is no mass on the macroscopic scale.

(c) All results in this paper extend to arbitrary $\theta>0$ under the
assumption that PD$(\theta)$ is the unique reversible distribution for the
split-merge dynamics introduced in Section 2. It seems widely accepted that
this is indeed the case, though to the authors’ best knowledge no proof exists
for $\theta>1$.

Assumption (A3) is the core premise which guarantees the Poisson-Dirichlet
limit of the macroscopic phase, and is consistent with the scaling of weights
(6) for the inclusion process

$w_{L}(n)=\frac{\Gamma(n+d)}{n!\Gamma(d)}\simeq\frac{\theta}{n\,L}\quad\text{if
we set}\quad d=\theta/L\,.$

This scaling implies that a size-biased sample of a macroscopic cluster has
the stationary weights $nw_{L}(n)\simeq\theta/L$ which are independent of $n$.
That means, picking a particle uniformly at random, the size of its cluster is
uniformly distributed. This is the trademark of Poisson-Dirichlet statistics,
and the distribution can only be normalized in the scaling limit due to the
factor $1/L$. Thus, to get a non-trivial macroscopic phase with $\theta>0$, it
is necessary that the weights $w_{L}$ do depend on the system size $L$. The
role of $\theta$ and more details on size-biased sampling will be given in
Section 2.

######  Remark 1.5.

In the case where only (A1) \- (A3) are satisfied, there is not enough control
on the limiting mean behaviour of intermediate scales to yield uniqueness of
the limit in the macroscopic phase. However, following the proof of Theorem
1.3, we can conclude that every accumulation point $\mu$ of the sequence
$(\mu_{L,N})_{L,N}$ is a Poisson-Dirichlet distribution of parameter $\theta$
on an interval $[0,\alpha_{\mu}]$. Here $\alpha_{\mu}$ denotes the respective
subsequential limit of (9).

Recall that in Theorem 1.3 we have fixed the density $\rho>0$ and it provides
a very general result that also includes trivial cases without condensation.
But we had to assume the equivalence of ensembles in (A2) and regularity of
the macroscopic phase in (A4), which are not easy to check in general (if not
established already for particular models). Strengthening the requirements
(A1) and (A3) on the stationary weights $(w_{L})_{L}$, we can use Theorem 1.3
to show a stronger but more specialized result, including the equivalence of
ensembles and regularity of the macroscopic phase in the conclusion.

######  Theorem 1.6.

Assume $(w_{L})_{L}$ is sequence of non-negative weights satisfying the
following two conditions:

  1. (B1)

$(w_{L})_{L}$ converges in the sup-norm, $\|\cdot\|_{\infty}$, to a sequence
$w$ with

$\sum_{n=0}^{\infty}w(n)=1\quad\text{and}\quad\sum_{n=0}^{\infty}n^{2}w(n)<\infty\,,$
(10)

and either $w(0)=1$ or

$\displaystyle w(0)>0\quad\mbox{and}\quad\sup_{n}[w(n-1)\wedge w(n)]>0\,.$
(11)

  2. (B2)

There exists some $\theta\in(0,1]$ such that

$\lim_{J\to\infty}\lim_{L\to\infty}\sup_{n>J}\left|nw_{L}(n)L-\theta\right|=0\,.$

Then the system exhibits a condensation transition according to Definition 1.1
with critical density

$\displaystyle\rho_{c}:=\sum_{n=0}^{\infty}nw(n)\in[0,\infty)\,.$

Furthermore, we have bulk density $\nu_{\rho}(\eta_{x})=\rho_{c}$ for all
$\rho\;\geqslant\;\rho_{c}$ and

$\displaystyle(\mu_{L,N})_{L,N}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\text{PD}_{[0,\alpha]}(\theta)\,,\quad\text{ as }\quad
N/L\rightarrow\rho\;\geqslant\;\rho_{c}\,,$

with $\alpha=\alpha(\rho)=1-\frac{\rho_{c}}{\rho}$.

######  Remark 1.7.

By assumption (B2), there exists an $A\in\mathbb{N}_{0}$ such that

$\lim_{L\to\infty}\sup_{n>A}\left|nw_{L}(n)L-\theta\right|\;\leqslant\;C\quad\text{with
an arbitrary constant }C>0\,.$

So for each $n\;\geqslant\;A$ we have $w_{L}(n)\to 0$ as $L\to\infty$, and
hence by assumption (B1)

$w(n)=0\quad\text{for all }n>A\,,$ (12)

and the limiting distribution $w$ can only have finite support.

Clearly, the restriction in (10) that $w$ can be interpreted as a probability
mass function is for notational convenience, we could just assume summability.
Furthermore, with $\theta>0$ all models covered by this result have a
macroscopic phase with non-trivial structure, excluding systems where the
latter concentrates on a single site which have been studied previously (see
citations above). Condition (11) is necessary to avoid lattice effects and
establish the equivalence of ensembles for the bulk part of the distribution
(see Proposition A.1). In the special case that $w(0)=1$ there is a simpler
proof of the equivalence of ensembles result.

We want to stress that single-site-condensation in models is typically due to
a strong enough attraction between particles, whereas for systems covered in
Theorem 1.6 particle attraction alone is too weak, and condensation only
occurs in combination with particle expulsion from the bulk as represented by
condition (12) on the limiting weights. This strict exclusion condition for
occupation numbers larger than $A$ in the limiting weights prevents clustering
of particles on sub-macroscopic scales. It should be possible to weaken this,
but some form of bulk exclusion is essential for condensation with non-trivial
macroscopic phase in models with stationary product measures. In Section 5 we
provide an intuitive explanation of this in terms of dynamics of generic
particle systems covered by our result.

###  1.3. Key steps of the proofs

The essential steps of the proof of Theorem 1.3 may be summarised as follows.
By compactness we know that $(\mu_{L,N})_{L,N}$ has weak accumulation points.
In order to determine the limit points’ distributions, we prove that
$(\mu_{L,N})_{L,N}$ is approximately reversible w.r.t. a discrete split-merge
dynamics. These discrete dynamics converge to the generator of the
coagulation-fragmentation process with split-merge dynamics, which we will
introduce in Section 2, see (13). Lastly, we use size-biased sampling together
with a disintegration argument to prove that the corresponding limit points
concentrate and therefore have a Poisson-Dirichlet law. This is due to the
fact that the Poisson-Dirichlet distribution is the unique distribution which
concentrates and is reversible w.r.t. the limiting split-merge dynamics
mentioned above [40, 35]. Here, we say that $\mu$ concentrates if
$\|p\|_{1}=\alpha$ $\mu$-a.s. for some $\alpha\in[0,1]$.

Essentially, Theorem 1.6 is a direct application of Theorem 1.3. Additionally,
the stronger assumptions allow us to establish the equivalence of ensembles in
Appendix A, which in our case implies the condensation transition. The proof
is based on the application of a local central limit theorem (LCLT) which,
together with a relative entropy bound, shows convergence of single-site
marginals to a distribution independent of the particle density
$\rho\;\geqslant\;\rho_{c}$.

The remainder of the paper will be structured as follows: in Section 2 we will
give a short review on Poisson-Dirichlet distributions and size-biased
sampling. Section 3 will focus on the proof of Theorem 1.3 whereas in Section
4 we state the proof of Theorem 1.6. Lastly, we discuss possible applications
of Theorem 1.6 to a family of zero-range and generalized inclusion processes
in Section 5. In Appendix A we give a brief introduction to grand-canonical
ensembles before proving equivalence of ensembles for size-dependent weights
under a sub-exponential growth condition.

##  2\. Background on partitions

###  2.1. The Poisson-Dirichlet distribution

The Poisson-Dirichlet (PD) distribution is a one-parameter family of
probability measures on the space of ordered partitions
$\nabla=\nabla_{[0,1]}$ of the unit interval, where for any $\alpha>0$ we
denote

${\nabla}_{[0,\alpha]}:=\left\\{p=(p_{i})_{i}\in[0,1]^{\mathbb{N}}:\|p\|_{1}=\alpha\quad\text{and}\quad
p_{1}\;\geqslant\;p_{2}\;\geqslant\;\cdots\right\\}\,.$

Note that elements $p$ in $\nabla$ are not partitions themselves but induce
partitions of the form $\\{[0,p_{1}),[p_{1},p_{1}+p_{2}),\ldots\\}$. The
family of measures was first introduced by Kingman [30] in the study of random
distributions on countably infinite sets, motivated by Bayesian inference and
decision theory. Apart from the original construction as a limit of Dirichlet
distributions, the PD distribution can be more intuitively constructed via a
stick-breaking procedure. Let $U_{1},U_{2},\ldots$ be independent
Beta$(1,\theta)$-distributed random variables and define

$V_{1}:=U_{1}\,,\;\;V_{2}:=(1-U_{1})U_{2}\,,\;\;V_{3}:=(1-U_{1})(1-U_{2})U_{3}\,,\ldots,$

i.e. we start with a stick of unit length and continue by breaking a random
fraction of $U_{1}$ apart. Then we do the same with the remaining part of the
stick and iterate. The resulting random vector $V=(V_{i})_{i\;\geqslant\;1}$
is said to be GEM($\theta$)-distributed, named after Griffiths [22, 23] Engen
[11] and McCloskey [32]. Reordering the entries of $V$ in decreasing order
yields $\widehat{V}$ which is known to be PD($\theta$)-distributed (see e.g.
[14]).

Note the two special cases, $\theta=1$ where the $U_{i}$’s are uniformly
distributed on the interval $[0,1]$, and $\theta=0$ where the $U_{i}$’s are
degenerated point-measures on one and hence $\widehat{V}=V=(1,0,0,\ldots)$.
Clearly, the choice of the interval $[0,1]$ is arbitrary and one can construct
PD and GEM distributions on intervals $[0,\alpha]$ for arbitrary $\alpha>0$
just by rescaling

$p\sim PD_{[0,\alpha]}(\theta)\quad\Leftrightarrow\quad
p/\alpha:=(p_{1}/\alpha,p_{2}/\alpha,\ldots)\sim PD_{[0,1]}(\theta)\,,$

and analogously for $GEM_{[0,\alpha]}(\theta)$. Since its introduction in [30]
the PD distribution emerged first in population biology [30, 10], before
appearing in statistical mechanics [31, 25, 3, 17, 26] and interacting
particle systems [27].

Besides its characterisation via the GEM-construction, the PD distribution was
furthermore found to be the unique invariant (in fact reversible) measure on
$\nabla$ of the coagulation-fragmentation process with split-merge dynamics
for $\theta\in(0,1]$. This is a Markov process on the state space
$\overline{\nabla}$ with infinitesimal generator given by

$\mathcal{G}_{\theta}f(p)=\sum_{i\neq
j}p_{i}p_{j}\left[f(\widehat{M}_{ij}p)-f(p)\right]+\theta\sum_{i}p_{i}^{2}\left[\int_{0}^{1}f(\widehat{S}_{i}^{u}p)du-f(p)\right]\,.$
(13)

Here $\widehat{M}_{ij}p$ denotes the operator that merges the parts $p_{i}$
and $p_{j}$ to a single block of size $p_{i}+p_{j}$ and then reorders the
partition to maintain the decreasing order. On the other hand,
$\widehat{S}_{i}^{u}p$ defines the operation of splitting $p_{i}$ into two
blocks of size $up_{i}$ and $(1-u)p_{i}$ before reordering the resulting
partition.

######  Proposition 2.1 ([40, 35]).

For $\theta\in[0,1]$, the Poisson-Dirichlet distribution PD($\theta$) is the
unique invariant measure on $\nabla$ with respect to split-merge dynamics
defined by $\mathcal{G}_{\theta}$, and it is also reversible.

Since the generator in (13) conserves the total mass of partitions, it is
clear that there exist stationary distributions for split-merge dynamics on
$\nabla_{[0,\alpha]}$ for all $\alpha>0$, which are unique and equal to
$PD_{[0,\alpha]}(\theta)$ with the above result. In general, the set
$C_{b}(\overline{\nabla})$ of bounded continuous functions is the natural
domain for the (Feller) Markov semigroup associated to split-merge dynamics.
Under the product topology on $\overline{\nabla}$ these include in particular
bounded cylinder functions, which depend only on finitely many entries of a
partition, and for all such functions the generator (13) is well defined (see
also [34, Lemma 4]).

Originally, the split-merge process was constructed in discrete time, see
[34], the extension to continuous time can be found in [25, Section 7.4]. The
uniqueness of the invariant measure when $\theta=1$ was proven in [40] by
Zerner, Zeitouni, Mayer-Wolf and Diaconis. An alternative technique allowed
Schramm to extend this result to $\theta\in(0,1]$, see [35] and Theorem 7.1 in
[25]. Lastly, consider the case $\theta=0$, then clearly
$\delta_{(1,0,\ldots)}$ is invariant because $\mathcal{G}_{0}$ only consists
of the merge term and there cannot exist another invariant measure on
$\nabla$.

###  2.2. Size-biased sampling

Partitions in $\nabla$ can be interpreted as probability mass functions
themselves, which allows for a natural size-biased resampling of its elements.
Given $p=(p_{i})_{i\in\mathbb{N}}\in\nabla$ we sample an index
$i\in\mathbb{N}$ at random according to $(p_{i})_{i\in\mathbb{N}}$. Continuing
this procedure, while renormalising the remaining partition to a total mass of
one in each round, we construct a so-called _size-biased sample_
$\widetilde{p}$ of $p$. For given $p\in\nabla$, $\widetilde{p}$ is a random
element of the unordered set

${\Delta}:=\left\\{p=(p_{i})_{i}\in[0,1]^{\mathbb{N}}:\|p\|_{1}=1\right\\}\,,$

and we denote its distribution by $\sigma_{p}$. Of course $\widetilde{p}$ can
also be defined in the same way for $p\in\Delta$.

The above procedure can be generalised to partitions in the compact space

$\overline{\Delta}:=\left\\{p=(p_{i})_{i}\in[0,1]^{\mathbb{N}}:\|p\|_{1}\;\leqslant\;1\right\\}\,,$

which includes in particular
$\overline{\nabla}=\bigcup_{\alpha\in[0,1]}\nabla_{[0,\alpha]}$ that we
already introduced in (3). More precisely, fix an element
$p\in\overline{\Delta}$. In the following, $q=\widetilde{p}$ will denote the
size-biased sample with distribution $\sigma_{p}$ on $\overline{\Delta}$ which
is defined recursively:

  * •

the first entry of $q$ is assigned the value222 We refrain from using an equal
sign instead of ’$\leftarrow$’, since this could lead to mathematically wrong
statements. For example, if $p=(\tfrac{1}{2},\tfrac{1}{2},0,\ldots)$ then
$q_{1}=\tfrac{1}{2}$ with probability $\frac{1}{2}+\frac{1}{2}=1$ which is
reflected by our notation in (14), but would read $q_{1}=\tfrac{1}{2}$ with
probability $\tfrac{1}{2}$ when replacing ’$\leftarrow$’ with ’$=$’. In a
fully rigorous construction of size-biased samples we actually sample the
index $j$ at random and not the value $p_{j}$, see [18] for the full
construction and more details. Because our analysis does not differentiate
between entries of the same size, we omit this step to significantly simplify
notation and assign the value $p_{j}$ directly.

$q_{1}\leftarrow\begin{cases}p_{j}&\quad\text{ w.p. }p_{j}\text{ for all
}j\in\mathbb{N}\,,\\\ 0&\quad\text{ w.p. }1-\|p\|_{1}\,,\end{cases}$ (14)

  * •

for $i>1$, let $V$ be the set of indices of $p$ assigned to $q_{k}$ for $k<i$,
then

$q_{i}\leftarrow\begin{cases}p_{j}&\quad\text{ w.p.
}\displaystyle\frac{p_{j}}{1-\sum_{k=1}^{i-1}q_{k}}\text{ for all
}j\in\mathbb{N}\setminus V\,,\\\ &\\\ 0&\quad\text{ w.p.
}\displaystyle\frac{1-\|p\|_{1}}{1-\sum_{k=1}^{i-1}q_{k}}\,.\end{cases}$ (15)

In the case where $p$ consists of finitely many non-zero components only, we
sample zeros in each iteration after exhausting all non-zero components. In
contrast to size-biased sampling on $\Delta$ with $\|p\|_{1}=1$, which is
usually defined in terms of shuffling indices of the original sequence, size-
biased sampling of $p\in\overline{\Delta}$ assumes a non-exhaustive reservoir
of zeros from which we pick with probability proportional to $1-\|p\|_{1}$ in
each round. For given $p\in\overline{\Delta}$, the distribution
$\sigma_{p}[dq]$ then denotes the law on $\overline{\Delta}$ of $q$ defined
above, and concentrates on partitions with $\|q\|_{1}=\|p\|_{1}$.

For an arbitrary probability measure
$\mu\in\mathcal{M}_{1}(\overline{\nabla})$ we then define its size-biased
distribution $\sigma(\mu)$ as the law

$\sigma(\mu)[dq]:=\int_{\overline{\nabla}}\sigma_{p}[dq]\mu[dp]\,.$ (16)

One interesting result regarding size-biased distributions with reservoirs of
zeros, is that weak convergence of measures on $\overline{\nabla}$ implies
weak convergence of the corresponding size-biased distributions.

######  Lemma 2.2.

If a sequence $(\mu_{n})_{n\in\mathbb{N}}$ of probability measures on
$\overline{\nabla}$ converges weakly to a measure $\mu$, then also
$\sigma(\mu_{n})\stackrel{{\scriptstyle d}}{{\rightarrow}}\sigma(\mu)$ on
$\overline{\Delta}$.

Originally this result was stated in [8, Theorem 1] with a flawed construction
and proof. A correct proof of the lemma can be found in [18, Theorem 1] along
with a nice exposition on size-biased sampling.

From the stick-breaking construction of the PD distribution it is easy to see
(e.g. in [14]) that the size-biased distribution of PD$(\theta)$ on
$\nabla_{[0,1]}$ is precisely the GEM($\theta$) distribution. Considering on
the other hand PD${}_{[0,\alpha]}(\theta)$ on $\nabla_{[0,\alpha]}$, its size-
biased distribution, as defined in (14) and (15), contains $0$-elements and
does not coincide with GEM${}_{[0,\alpha]}(\theta)$ whenever $\alpha<1$.
However, this connection still holds for a modified (positive) size-biasing
$\tilde{p}^{\prime}$ without $0$-elements, defined again via scaling. For
$p\in\overline{\Delta}\setminus\\{0\\}$ (i.e. we have $\|p\|_{1}>0$), the
positive size-biased sample is defined as

$\tilde{p}^{\prime}:=\|p\|_{1}q\quad\mbox{where}\quad
q\sim\sigma_{p/\|p\|_{1}}\,.$ (17)

Therefore the law of $\tilde{p}^{\prime}$ on
$\overline{\Delta}\setminus\\{0\\}$ is
$\sigma^{\prime}_{p}[dq]=\sigma_{p/\|p\|_{1}}\circ\frac{1}{\|p\|_{1}}[dq]$,
the pushforward measure under rescaling of the size-biasing
$\sigma_{p/\|p\|_{1}}$ on $\Delta$, which does not contain any $0$-elements.
As for $\sigma_{p}$, $\sigma^{\prime}_{p}[dq]$ concentrates on partitions with
$\|q\|_{1}=\|p\|_{1}$, and for a distribution $\mu$ on $\overline{\Delta}$ we
define $\sigma^{\prime}(\mu)$ analogously to (16).

Note that $\sigma(\mu)=\sigma^{\prime}(\mu)$ on $\mathcal{M}_{1}(\nabla)$
since $\|p\|_{1}=1$ $\mu$-a.s., and in this case we will write
${\mu}[\widetilde{p}\in\cdot]$ instead to ease notation. Furthermore, we can
recover the finite dimensional marginals of $\sigma^{\prime}(\mu)$ from
$\sigma(\mu)$ by conditioning on positivity of the corresponding entries:

$\sigma^{\prime}(\mu)[({q}_{1},\ldots,{q}_{n})\in\cdot\,]=\sigma(\mu)\left[({q}_{1},\ldots,{q}_{n})\in\cdot\;\middle|\;q_{k}>0,\
1\;\leqslant\;k\;\leqslant\;n\right]\,.$ (18)

Returning to the case of the PD distribution $\mu\sim
PD_{[0,\alpha]}(\theta)$, we can retrieve the first marginal of the
corresponding GEM distribution on $[0,\alpha]$:

$\sigma^{\prime}(\mu)[{q}_{1}\in\cdot]=\sigma(\mu)\left[q_{1}\in\cdot\;\middle|\;q_{1}>0\right]=\mathrm{Beta}_{[0,\alpha]}(1,\theta)\,.$

Note also that for positive size-biased distributions the equivalent statement
of Lemma 2.2 does not hold, because loss of mass of the corresponding
sequences $p\sim\mu_{n}$ may occur. This would correspond to a positive
probability of sampling a block size of zero in the limit, which is precisely
the case for distributions $\mu_{L,N}$ we study in this paper.

We will work with size-biased sampling not only on $\overline{\Delta}$ but
also on $\Omega_{L,N}$, which corresponds to uniformly picking a particle and
sampling the occupation number on its site. Using our definition from above
(cf. [27, Section 2.3] for a more detailed construction), the size-biased
distribution of $\pi_{L,N}$ is given by

$\sigma(\pi_{L,N})[d\eta]:=\sigma(\mu_{L,N})\left[d\left(\frac{\eta}{N}\right)\right]\,,$

where $\mu_{L,N}$ (5) is the distribution of the ordered partition
corresponding to a particle configuration $\eta$. Because
$\mu_{L,N}\in\mathcal{M}_{1}(\nabla)$, we will write
$\pi_{L,N}[\widetilde{\eta}\in\cdot]$ instead of $\sigma(\pi_{L,N})$, and note
that due to the product structure of (1) we have e.g. for the first marginal

$\pi_{L,N}[\widetilde{\eta}_{1}=n]=\frac{L}{N}nw_{L}(n)\frac{Z_{L-1,N-n}}{Z_{L,N}}\quad\mbox{for
all }n=0,\ldots,N\,.$ (19)

##  3\. Proof of Theorem 1.3

Recall the canonical distributions $\pi_{L,N}$ on $\Omega_{L,N}$ given by

$\pi_{L,N}[d\eta]=\frac{1}{Z_{L,N}}\prod_{z=1}^{L}w_{L}(\eta_{z})d\eta$

and their macroscopic counterparts $\mu_{L,N}$ on
$\nabla\subset\overline{\nabla}$, see (5). The proof of Theorem 1.3 can be
broken down into two main steps. We begin by showing that every accumulation
point of $(\mu_{L,N})_{L,N}$ is reversible with respect to the infinitesimal
generator $\mathcal{G}_{\theta}$ of the split-merge process (13). The second
step consists of proving that each limiting measure on $\overline{\nabla}$ in
fact concentrates on $\nabla_{[0,\alpha]}$ for some $\alpha$. Together with
Proposition 2.1 this implies that the limiting measure must be a Poisson-
Dirichlet distribution on the interval $[0,\alpha]$.

###  3.1. Reversibility of weak accumulation points

The space $\overline{\nabla}$ of ordered (sub-)partitions is compact w.r.t.
the product topology on $[0,1]^{\mathbb{N}}$, which implies compactness of the
space $\mathcal{M}_{1}(\overline{\nabla})$ w.r.t. the topology induced by weak
convergence. Therefore, every subsequence of $(\mu_{L,N})_{L,N}$ has a weakly
convergent subsequence and $(\mu_{L,N})_{L,N}$ has at least one accumulation
point.

The first step in the proof of Theorem 1.3 is the following intermediate
result.

######  Proposition 3.1.

Consider weights $(w_{L})_{L}$ satisfying assumptions (A1) \- (A3) with
$\theta>0$. All weak accumulation points of $(\mu_{L,N})_{L,N}$ are reversible
w.r.t. $\mathcal{G}_{\theta}$.

Because $\mu_{L,N}$ concentrates on discrete partitions and by Assumption (A3)
we only control the weights on the macroscopic scale, it will be convenient to
consider a discrete version
$\mathcal{G}^{(N,\varepsilon)}_{\theta}:C_{b}(\overline{\nabla})\to
C_{b}(\overline{\nabla})$ of the split-merge process corresponding to
$\mathcal{G}_{\theta}$ (13), which only acts on the parts of the partition
exceeding a fixed size of $\varepsilon\in(0,1)$,

$\begin{split}\mathcal{G}^{(N,\varepsilon)}_{\theta}f(p)&:=\frac{N}{N-1}\sum_{i\neq
j}p_{i}p_{j}\mathds{1}\\{p_{i},p_{j}\;\geqslant\;\varepsilon\\}\left[f(\widehat{M}_{ij}p)-f(p)\right]\\\
&\qquad+\frac{\theta}{N-1}\sum_{i}p_{i}\mathds{1}\\{p_{i}\;\geqslant\;2\varepsilon\\}\sum_{k=\varepsilon
N}^{N(p_{i}-\varepsilon)}\left[f(\widehat{S}_{i}^{k/Np_{i}}p)-f(p)\right]\,.\end{split}$

As we will see Lemma 3.3, it suffices to control the weights on macroscopic
scales larger than $\varepsilon$ in order to observe a PD limit.

Recall that under $\mu_{L,N}$ partitions consist of (at most) $L$ blocks.
Since we will lift the split-merge dynamics to the space of particle
configurations $\Omega_{L,N}$, the resulting partitions should not exceed
length $L$ either, and we have to adapt the split-merge dynamics to achieve
that. Recall from (8) that without loss of generality we can assume that
$\nu_{\rho}[\eta_{x}=n]=w(n)$. Then, Assumption (A2) guarantees that there
exists $m\in\mathbb{N}$ such that $w(m)=\lim_{L\to\infty}w_{L}(m)>0$. This
implies that under $\mu_{L,N}$ a positive fraction of blocks has size $m/N$,
see Lemma 3.2 below. Therefore, instead of leaving empty sites behind when
merging and splitting onto empty sites, we can impose to leave a fraction of
$m/N$ behind when merging and only split onto blocks of size $m/N$. For
example, consider the case $m=1$ where $\lim_{L\to\infty}w_{L}(0)=w(0)=0$ but
$w(1)>0$. This will not affect the statistics on a macroscopic scale, where
microscopic blocks are indistinguishable; this is why henceforward we assume
$m=0$ for notational convenience.

######  Lemma 3.2.

Let $\\#_{0}(\eta)$ be the number of sites with zero occupation in the
configuration $\eta$. We have that

$\pi_{L,N}\left(\left(\frac{\\#_{0}(\eta)}{L}-w(0)\right)^{2}\right)\to
0\,,\quad\text{as }N/L\rightarrow\rho\,.$

In particular, we have a weak law of large numbers and for every
$\varepsilon>0$

$\displaystyle\pi_{L,N}\left[\left|\frac{\\#_{0}(\eta)}{L}-w(0)\right|>\varepsilon\right]\to
0\,,\quad\text{as }N/L\rightarrow\rho\,.$

###### Proof.

By direct calculation of the second moment, it suffices to show that (using
the product structure of $\pi_{L,N}$)

$\pi_{L,N}[\eta_{x}=0,\eta_{y}=0]=\pi_{L-1,N}[\eta_{x}=0]\,\pi_{L,N}[\eta_{y}=0]$

converges to $w(0)^{2}$ for all $1\;\leqslant\;x,y\;\leqslant\;L$, $x\neq y$,
which holds due to (A2). The second statement follows immediately by
Chebyshev’s inequality. ∎

The operator $\mathcal{G}_{\theta}^{(N,\varepsilon)}$ approximates the
corresponding continuous process acting on blocks of size larger than
$\varepsilon$, which is characterised by the generator

$\begin{split}\mathcal{G}^{(\varepsilon)}_{\theta}f(p)&:=\sum_{\begin{subarray}{c}i,j=1\\\
i\neq
j\end{subarray}}^{\infty}p_{i}p_{j}\mathds{1}\\{p_{i},p_{j}\;\geqslant\;\varepsilon\\}\left[f(\widehat{M}_{ij}p)-f(p)\right]\\\
&\qquad+\theta\sum_{i=1}^{\infty}p_{i}^{2}\mathds{1}\\{p_{i}\;\geqslant\;2\varepsilon\\}\left[\int_{\varepsilon}^{1-\varepsilon}f(\widehat{S}_{i}^{u}p)du-f(p)\right]\,.\end{split}$

We summarise the corresponding convergence behaviour of the generators in the
following lemma:

######  Lemma 3.3.

We have

$\mathcal{G}_{\theta}^{(N,\varepsilon)}\to\mathcal{G}_{\theta}^{(\varepsilon)}\,,\quad\text{as
}N\to\infty\,,\qquad\text{and}\qquad\mathcal{G}_{\theta}^{(\varepsilon)}\to\mathcal{G}_{\theta}\,,\quad\text{as
}\varepsilon\to 0\,,$

in the strong operator topology on bounded continuous functions
$C_{b}(\overline{\nabla})$.

###### Proof.

Let $g\in C_{b}(\overline{\nabla})$. To prove the first part of the statement,
we need to show that
$\|(\mathcal{G}_{\theta}^{(N,\varepsilon)}-\mathcal{G}_{\theta}^{(\varepsilon)})g\|_{\infty}$
vanishes in the limit $N\to\infty$. We will compare the two parts of each
operator separately. We start with the merge term:

$\displaystyle\left|\sum_{i\neq
j}^{L}p_{i}p_{j}\mathds{1}\\{p_{i},p_{j}\;\geqslant\;\varepsilon\\}\left[g(\widehat{M}_{ij}p)-g(p)\right]\right.$
$\displaystyle\qquad\qquad\qquad-\left.\frac{N}{N-1}\sum_{i\neq
j}^{L}p_{i}p_{j}\mathds{1}\\{p_{i},p_{j}\;\geqslant\;\varepsilon\\}\left[g(\widehat{M}_{ij}p)-g(p)\right]\right|$
$\displaystyle\qquad\qquad\;\leqslant\;\frac{2\|g\|_{\infty}}{N-1}\sum_{i\neq
j}^{L}p_{i}p_{j}\mathds{1}\\{p_{i},p_{j}\;\geqslant\;\varepsilon\\}\;\leqslant\;\frac{2\|g\|_{\infty}}{N-1}\,.$

And similarly for the split term:

$\displaystyle\left|\sum_{i=1}^{L}p_{i}^{2}\mathds{1}\\{p_{i}\;\geqslant\;2\varepsilon\\}\left[\int_{0}^{1}g(\widehat{S}_{i}^{u}p)du-g(p)\right]\right.$
$\displaystyle\qquad\qquad-\left.\frac{1}{N-1}\sum_{i=1}^{L}p_{i}\mathds{1}\\{p_{i}\;\geqslant\;2\varepsilon\\}\sum_{k=1}^{Np_{i}-1}\left[g(\widehat{S}_{i}^{k/Np_{i}}p)-g(p)\right]\right|$
$\displaystyle\quad\;\leqslant\;\sum_{i=1}^{L}p_{i}^{2}\left|\int_{0}^{1}g(\widehat{S}_{i}^{u}p)du-\frac{1}{(N-1)p_{i}}\sum_{k=1}^{Np_{i}-1}g(\widehat{S}_{i}^{k/Np_{i}}p)\right|+\frac{\|g\|_{\infty}}{N-1}\sum_{i=1}^{L}p_{i}(1-p_{i})$
$\displaystyle\qquad\;\leqslant\;\frac{2\|g\|_{\infty}}{N-1}+\frac{\|g\|_{\infty}}{N-1}\,,$

where the last inequality holds since the sum inside the absolute value
approximates the Riemann-sum which in turn converges to the given integral.
Since the sum of the first estimate and second estimate multiplied by $\theta$
bounds
$\|(\mathcal{G}_{\theta}^{(N,\varepsilon)}-\mathcal{G}_{\theta}^{(\varepsilon)})g\|_{\infty}$
from above, we take the thermodynamic limit $N/L\to\rho$ and conclude the
first part of the lemma.

The second statement requires us to show that
$\|(\mathcal{G}_{\theta}-\mathcal{G}_{\theta}^{(\varepsilon)})g\|_{\infty}\to
0$ as $\epsilon\to 0$. First, note that

$\displaystyle(\mathcal{G}_{\theta}-\mathcal{G}_{\theta}^{(\varepsilon)})g(p)$
$\displaystyle=\sum_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{\infty}p_{i}p_{j}\mathds{1}\\{p_{i},p_{j}<\varepsilon\\}\left[g(\widehat{M}_{ij}p)-g(p)\right]$
$\displaystyle\qquad+\theta\sum_{i=1}^{\infty}p_{i}^{2}\left[\int_{0}^{1}g(\widehat{S}_{i}^{u}p)du-g(p)\right]$
$\displaystyle\qquad-\theta\sum_{i=1}^{\infty}p_{i}^{2}\mathds{1}\\{p_{i}\;\geqslant\;2\varepsilon\\}\left[\int_{\varepsilon}^{1-\varepsilon}g(\widehat{S}_{i}^{u}p)du-g(p)\right]$
$\displaystyle=\sum_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{\infty}p_{i}p_{j}\mathds{1}\\{p_{i},p_{j}<\varepsilon\\}\left[g(\widehat{M}_{ij}p)-g(p)\right]$
$\displaystyle\qquad+\theta\sum_{i=1}^{\infty}p_{i}^{2}\mathds{1}\\{p_{i}<2\varepsilon\\}\left[\int_{0}^{1}g(\widehat{S}_{i}^{u}p)du-g(p)\right]$
$\displaystyle\qquad+\theta\sum_{i=1}^{\infty}p_{i}^{2}\mathds{1}\\{p_{i}\;\geqslant\;2\varepsilon\\}\left[\int_{0}^{1}g(\widehat{S}_{i}^{u}p)du-\int_{\varepsilon}^{1-\varepsilon}g(\widehat{S}_{i}^{u}p)du\right]\,.$

The first two terms vanish by dominated convergence and for the last term we
have the estimate

$\left|\sum_{i=1}^{\infty}p_{i}^{2}\mathds{1}\\{p_{i}{\;\geqslant\;}2\varepsilon\\}\left[\int_{0}^{1}g(\widehat{S}_{i}^{u}p)du-\int_{\varepsilon}^{1-\varepsilon}\\!\\!\\!\\!\\!g(\widehat{S}_{i}^{u}p)du\right]\right|\;\leqslant\;2\varepsilon\|g\|_{\infty}\sum_{i=1}^{\infty}p_{i}^{2}\;\leqslant\;2\varepsilon\|g\|_{\infty}\,.$

This concludes the proof. ∎

The proof of Proposition 3.1, requires the following key observation which
states that $\mu_{L,N}$ is approximately reversible w.r.t. the dynamics
corresponding to $\mathcal{G}_{\theta}^{(N,\varepsilon)}$.

######  Lemma 3.4.

For every $f,g\in C_{b}(\overline{\nabla})$ and $\varepsilon\in(0,1)$ we have

$\left|\mu_{L,N}(f\,\mathcal{G}_{\theta}^{(N,\varepsilon)}g)-\mu_{L,N}(g\,\mathcal{G}_{\theta}^{(N,\varepsilon)}f)\right|\to
0$ (20)

in the thermodynamic limit $N/L\to\rho\;\geqslant\;0$.

We postpone the proof of Lemma 3.4 until after the one of Proposition 3.1. Now
we have everything to state the proof of this section’s main finding.

###### Proof of Proposition 3.1:.

Due to compactness of the space $\mathcal{M}_{1}(\overline{\nabla})$ w.r.t.
the topology induced by weak convergence, the sequence $(\mu_{L,N})$ has weak
accumulation points. Let $\mu$ be such an accumulation point and
$(\mu_{L_{j},N_{j}})_{j}$ a subsequence converging to it. Then for each
$\epsilon\in(0,1)$ we have

$\begin{split}\left|\mu(f\mathcal{G}_{\theta}g)-\mu(g\mathcal{G}_{\theta}f)\right|&\;\leqslant\;\left|\mu(f\mathcal{G}_{\theta}g)-\mu_{L_{j},N_{j}}(f\mathcal{G}_{\theta}^{(N_{j},\varepsilon)}g)\right|\\\
&\qquad+\left|\mu_{L_{j},N_{j}}(f\mathcal{G}_{\theta}^{(N_{j},\varepsilon)}g)-\mu_{L_{j},N_{j}}(g\mathcal{G}_{\theta}^{(N_{j},\varepsilon)}f)\right|\\\
&\qquad+\left|\mu_{L_{j},N_{j}}(g\mathcal{G}_{\theta}^{(N_{j},\varepsilon)}f)-\mu(g\mathcal{G}_{\theta}f)\right|\,.\end{split}$

The middle term on the r.h.s. vanishes due to Lemma 3.4, whereas the first
term can be estimated by

$\begin{split}\left|\mu(f\mathcal{G}_{\theta}g)-\mu_{L_{j},N_{j}}(f\mathcal{G}_{\theta}^{(N_{j},\varepsilon)}g)\right|&\;\leqslant\;\left|\mu(f\mathcal{G}_{\theta}g)-\mu_{L_{j},N_{j}}(f\mathcal{G}_{\theta}g)\right|\\\
&\qquad+\left|\mu_{L_{j},N_{j}}(f\mathcal{G}_{\theta}g)-\mu_{L_{j},N_{j}}(f\mathcal{G}_{\theta}^{(N_{j},\varepsilon)}g)\right|\,.\end{split}$
(21)

Since $(\mu_{L_{j},N_{j}})_{j}$ converges in distribution to $\mu$ and
$\mathcal{G}_{\theta}g$ is bounded and continuous, see [34, Lemma 4], the
first term on the r.h.s. of (21) vanishes as $L$ diverges. Also, by Lemma 3.3,
the second term on the right vanishes after taking the limit $L,N\to\infty$
before $\varepsilon\to 0$, since

$\displaystyle\left|\mu_{L_{j},N_{j}}(f(\mathcal{G}_{\theta}-\mathcal{G}_{\theta}^{(N_{j},\varepsilon)})g)\right|\;\leqslant\;\|f\|_{\infty}\|(\mathcal{G}_{\theta}-\mathcal{G}_{\theta}^{(N_{j},\varepsilon)})g\|_{\infty}\,.$

The same steps hold when applied to
$\left|\mu_{L_{j},N_{j}}(g\mathcal{G}_{\theta}^{(N_{j},\varepsilon)}f)-\mu(g\mathcal{G}_{\theta}f)\right|$.
Overall, this yields $\mu(f\mathcal{G}_{\theta}g)=\mu(g\mathcal{G}_{\theta}f)$
which finishes the proof. ∎

It only remains to state the proof of Lemma 3.4.

###### Proof of Lemma 3.4:.

First, we note that we can write $\mu_{L,N}(f\mathcal{G}^{(N,\varepsilon)}g)$
in terms of the canonical distribution $\pi_{L,N}$:

$\displaystyle N(N-1)\mu_{L,N}(f\mathcal{G}^{(N,\varepsilon)}g)$ (22)
$\displaystyle=\sum_{\eta\in\Omega_{L,N}}f\left(\frac{\widehat{\eta}}{N}\right)\sum_{i\neq
j}^{L}\widehat{\eta}_{i}\widehat{\eta}_{j}\mathds{1}\\{\widehat{\eta}_{i},\widehat{\eta}_{j}\;\geqslant\;\varepsilon
N\\}\left[g\left(\widehat{M}_{ij}\left(\frac{\widehat{\eta}}{N}\right)\right)-g\left(\frac{\widehat{\eta}}{N}\right)\right]\pi_{L,N}(\eta)$
$\displaystyle+\theta\sum_{\eta\in\Omega_{L,N}}f\left(\frac{\widehat{\eta}}{N}\right)\sum_{i=1}^{L}\widehat{\eta}_{i}\mathds{1}\\{\widehat{\eta}_{i}\;\geqslant\;2\varepsilon
N\\}\sum_{k=\varepsilon N}^{\widehat{\eta}_{i}-\varepsilon
N}\left[g\left(\widehat{S}_{i}^{k/\widehat{\eta}_{i}}\left(\frac{\widehat{\eta}}{N}\right)\right)-g\left(\frac{\widehat{\eta}}{N}\right)\right]\pi_{L,N}(\eta)\,,$

where we multiplied with $N(N-1)$ for convenience.

In the following it will be easier not to work with discrete partitions, but
with corresponding particle configurations without worrying about the order of
the corresponding sites. Hence, we require a new notation to lift split and
merge operations to the space of particle configurations: for
$\eta\in\Omega_{L,N}$ we define

$\begin{split}M_{xy}\eta:=\eta+\eta_{y}(e^{(x)}-e^{(y)})\,,\\\
S_{xy}^{k}\eta:=\eta+k(e^{(y)}-e^{(x)})\,,\\\
S_{x}^{k}\eta:=\eta+k(e^{(L+1)}-e^{(x)})\,,\end{split}$

where $e^{(x)}\in\Omega_{L,1}$ denotes the configuration with a single
particle at site $x$, i.e. $(e^{(x)})_{z}=\delta_{x,z}$. The operator
$S_{x}^{k}$ is only necessary for the case of a full particle configuration
$\eta$, i.e. $\\#_{0}(\eta)=0$. Then, we simply append the additional block of
particles at the end of the configuration, which then has length $(L+1)$. This
arbitrary but convenient choice of course does not change the projection of
the dynamics on the level of partitions. As such we can rewrite (22) as

$\displaystyle N(N-1)$
$\displaystyle\mu_{L,N}(f\mathcal{G}^{(N,\varepsilon)}g)=\pi_{L,N}\left(\widehat{f}\left(\eta\right)\sum_{x\neq
y}^{L}\eta_{x}\eta_{y}\mathds{1}\\{\eta_{x},\eta_{y}\;\geqslant\;\varepsilon
N\\}\left[\widehat{g}\left(M_{xy}\eta\right)-\widehat{g}\left(\eta\right)\right]\right)$
$\displaystyle\quad+\theta\,\pi_{L,N}\left(\mathds{1}\\{\\#_{0}(\eta)>0\\}\widehat{f}\left(\eta\right)\sum_{x=1}^{L}\eta_{x}\mathds{1}\\{\eta_{x}\;\geqslant\;2\varepsilon
N\\}\right.$ $\displaystyle\qquad\quad\times\left.\sum_{k=\varepsilon
N}^{\eta_{x}-\varepsilon
N}\sum_{y=1}^{L}\frac{\mathds{1}\\{\eta_{y}=0\\}}{\\#_{0}(\eta)}\left[\widehat{g}\left(S_{xy}^{k}\eta\right)-\widehat{g}\left(\eta\right)\right]\right)$
(23)
$\displaystyle\quad+\theta\,\pi_{L,N}\left(\mathds{1}\\{\\#_{0}(\eta)=0\\}\widehat{f}\left(\eta\right)\sum_{x=1}^{L}\eta_{x}\mathds{1}\\{\eta_{x}\;\geqslant\;2\varepsilon
N\\}\right.$ $\displaystyle\qquad\quad\times\left.\sum_{k=\varepsilon
N}^{\eta_{x}-\varepsilon
N}\left[\widehat{g}\left(S_{x}^{k}\eta\right)-\widehat{g}\left(\eta\right)\right]\right)\,,$

where $\widehat{f}:=f\circ T$ and $\widehat{g}:=g\circ T$ with the ordering
map $T$ given in (4). Furthermore, we can see that the terms that depend on
$\widehat{f}$ and $\widehat{g}$ only through the product
$\widehat{f}(\eta)\widehat{g}(\eta)$ cancel when taking the difference between
$\mu_{L,N}(f\mathcal{G}^{(N,\varepsilon)}g)$ and
$\mu_{L,N}(g\mathcal{G}^{(N,\varepsilon)}f)$ in (20). Additionally, the very
last term in (3.1) is negligible since, using
$(\eta_{x}-1)/(N-1)\;\leqslant\;1$ and $\sum_{x=1}^{L}\eta_{x}/N=1$
$\pi_{L,N}$-almost surely, we have

$\begin{split}\frac{1}{N(N-1)}\pi_{L,N}&\left(\mathds{1}\\{\\#_{0}(\eta)=0\\}\sum_{x=1}^{L}\eta_{x}(\eta_{x}-1)\right)\;\leqslant\;\pi_{L,N}\left[\\#_{0}(\eta)=0\right]\,,\end{split}$

which vanishes due to Lemma 3.2 because $w(0)>0$.

To simplify notation we introduce

$V_{f,g}^{(L,N,\varepsilon)}:=\pi_{L,N}\left(\widehat{f}\left(\eta\right)\sum_{x\neq
y}^{L}\eta_{x}\eta_{y}\mathds{1}\\{\eta_{x},\eta_{y}\;\geqslant\;\varepsilon
N\\}\widehat{g}\left(M_{xy}\eta\right)\right)$ (24)

and

$\displaystyle U_{f,g}^{(L,N,\varepsilon)}$
$\displaystyle:=\theta\,\pi_{L,N}\left(\mathds{1}\\{\\#_{0}(\eta)>0\\}\widehat{f}\left(\eta\right)\sum_{x=1}^{L}\eta_{x}\mathds{1}\\{\eta_{x}\;\geqslant\;2\varepsilon
N\\}\right.$ $\displaystyle\qquad\qquad\times\left.\sum_{k=\varepsilon
N}^{\eta_{x}-\varepsilon
N}\sum_{y=1}^{L}\frac{\mathds{1}\\{\eta_{y}=0\\}}{\\#_{0}(\eta)}\widehat{g}\left(S_{xy}^{k}\eta\right)\right)\,.$
(25)

We are then left to analyse

$\displaystyle N(N-1)$
$\displaystyle(\mu_{L,N}(f\mathcal{G}^{(N,\varepsilon)}g)-\mu_{L,N}(g\mathcal{G}^{(N,\varepsilon)}f))$
$\displaystyle=V_{f,g}^{(L,N,\varepsilon)}+U_{f,g}^{(L,N,\varepsilon)}-V_{g,f}^{(L,N,\varepsilon)}-U_{g,f}^{(L,N,\varepsilon)}+o(N^{2})\,.$

The goal is to compare
$V_{f,g}^{(L,N,\varepsilon)}-U_{g,f}^{(L,N,\varepsilon)}$ and
$U_{f,g}^{(L,N,\varepsilon)}-V_{g,f}^{(L,N,\varepsilon)}$, respectively, and
show that these differences vanish in the limit if divided by $N(N-1)$. In
order to prove this, we perform a change of measure, since both
$V^{(L,N,\varepsilon)}$ and $U^{(L,N,\varepsilon)}$ are expectations with
respect to $\pi_{L,N}$.

First, we note that the restriction of the merge map

$M_{xy}\Big{|}_{\\{\eta_{y}=k\\}}:\Omega_{L,N}\cap\\{\eta_{y}=k\\}\to\Omega_{L,N}$

is injective and therefore defines a bijection between the set
$\Omega_{L,N}\cap\\{\eta_{y}=k\\}$ and its image
$M_{xy}(\Omega_{L,N}\cap\\{\eta_{y}=k\\})=\Omega_{L,N}\cap\\{\eta_{y}=0,k\;\leqslant\;\eta_{x}\\}$
with inverse $S_{xy}^{k}$. Therefore, the change of measure of $\pi_{L,N}$ and
its pushforward measure under $M_{xy}$ for fixed $x,y,k$ on the set
$\Omega_{L,N}\cap\\{\eta_{y}=k\\}$ is given by:

$\begin{split}\frac{\pi_{L,N}}{\pi_{L,N}\circ
M_{xy}}[\eta]&=\frac{\prod_{z=1}^{L}w_{L}(\eta_{z})}{\prod_{z=1}^{L}w_{L}((M_{xy}\eta)_{z})}=\frac{w_{L}((M_{xy}\eta)_{x}-k)w_{L}(k)}{w_{L}((M_{xy}\eta)_{x})w_{L}(0)}\,.\end{split}$
(26)

This will allow us to perform a change of measure in the following sense. Fix
$x,y$ and $k$, furthermore let $h_{y,k}$ be a real valued function on
$\Omega_{L,N}$ with support in $\\{\eta_{y}=k\\}$. By definition of $M_{xy}$,
we can define $\widetilde{h}_{x,y,k}=h_{y,k}\circ M_{xy}^{-1}$ which is zero
outside of $\Omega_{L,N}\cap\\{\eta_{y}=0,k\;\leqslant\;\eta_{x}\\}$. We then
have

$\begin{split}\pi_{L,N}(h_{y,k}(\eta))&=\pi_{L,N}(\widetilde{h}_{x,y,k}(M_{xy}\eta))=\int_{\\{\eta_{y}=k\\}}\widetilde{h}_{x,y,k}(M_{xy}\eta)\;\pi_{L,N}[d\eta]\\\
&=\int_{\\{\xi_{y}=0,k\;\leqslant\;\xi_{x}\\}}\widetilde{h}_{x,y,k}(\xi)\;\pi_{L,N}\circ
M_{xy}^{-1}[d\xi]\\\
&=\int_{\\{\xi_{y}=0,k\;\leqslant\;\xi_{x}\\}}\widetilde{h}_{x,y,k}(\xi)\frac{w_{L}(\xi_{x}-k)w_{L}(k)}{w_{L}(\xi_{x})w_{L}(0)}\;\pi_{L,N}[d\xi]\,.\end{split}$

Before we apply the change of measure, we divide and multiply by
$(M_{xy}\eta)_{x}$, decompose over $\\{\eta_{y}=k\\}$, and interchange the
order of integration:

$\displaystyle V_{f,g}^{(L,N,\varepsilon)}$ $\displaystyle=\sum_{x\neq
y}^{L}\sum_{k=1}^{N}\pi_{L,N}\left((M_{xy}\eta)_{x}\widehat{f}\left(\eta\right)\widehat{g}\left(M_{xy}\eta\right)\frac{\eta_{x}\eta_{y}}{(M_{xy}\eta)_{x}}\mathds{1}\\{k,\eta_{x}\;\geqslant\;\varepsilon
N\\}\right.$
$\displaystyle\qquad\times\mathds{1}\\{\eta_{y}=k\\}\mathds{1}\\{(M_{xy}\eta)_{y}=0\\}\bigg{)}$
$\displaystyle=\sum_{x\neq
y}^{L}\sum_{k=1}^{N}\pi_{L,N}\left((M_{xy}\eta)_{x}\widehat{f}\left(S_{xy}^{k}M_{xy}\eta\right)\widehat{g}\left(M_{xy}\eta\right)\frac{((M_{xy}\eta)_{x}-k)k}{(M_{xy}\eta)_{x}}\right.\mathds{1}\\{k\;\geqslant\;\varepsilon
N\\}$
$\displaystyle\qquad\times\mathds{1}\\{k\;\leqslant\;(M_{xy}\eta)_{x}-\varepsilon
N\\}\mathds{1}\\{(S_{xy}^{k}M_{xy}\eta)_{y}=k\\}\mathds{1}\\{(M_{xy}\eta)_{y}=0\\}\bigg{)}\,,$

where in the last step we expressed $\eta$ in terms of $M_{xy}\eta$. Recall
now (26) and note that we can recover $\eta$ from $M_{xy}\eta$ and
$k=\eta_{y}$. Therefore, the change of measure yields

$\displaystyle V_{f,g}^{(L,N,\varepsilon)}$ $\displaystyle=\sum_{x\neq
y}^{L}\sum_{k=1}^{N}\pi_{L,N}\left(\mathds{1}\\{\\#_{0}(\eta)>0\\}\eta_{x}\widehat{f}\left(S_{xy}^{k}\eta\right)\widehat{g}\left(\eta\right)\frac{(\eta_{x}-k)k}{\eta_{x}}\frac{w_{L}(\eta_{x}-k)w_{L}(k)}{w_{L}(\eta_{x})w_{L}(0)}.\right.$
$\displaystyle\qquad\times\mathds{1}\\{k\;\leqslant\;\eta_{x}-\varepsilon
N\\}\mathds{1}\\{k\;\geqslant\;\varepsilon
N\\}\mathds{1}\\{(S_{xy}^{k}\eta)_{y}=k\\}\mathds{1}\\{\eta_{y}=0\\}\bigg{)}$
$\displaystyle=\pi_{L,N}\left(\mathds{1}\\{\\#_{0}(\eta)>0\\}\widehat{g}\left(\eta\right)\sum_{x=1}^{L}\eta_{x}\mathds{1}\\{\eta_{x}\;\geqslant\;2\varepsilon
N\\}\sum_{k=\varepsilon N}^{\eta_{x}-\varepsilon
N}\sum_{y=1}^{L}\frac{\mathds{1}\\{\eta_{y}=0\\}}{\\#_{0}(\eta)}\right.$
$\displaystyle\qquad\times\widehat{f}\left(S_{xy}^{k}\eta\right)\\#_{0}(\eta)\frac{(\eta_{x}-k)k}{\eta_{x}}\frac{w_{L}(\eta_{x}-k)w_{L}(k)}{w_{L}(\eta_{x})w_{L}(0)}\bigg{)}\,.$

Comparing now $V_{f,g}^{(L,N,\varepsilon)}$ to $U_{g,f}^{(L,N,\varepsilon)}$
in (25), the only discrepancy is the term

$\begin{split}\\#_{0}(\eta)\frac{(\eta_{x}-k)k}{\eta_{x}}\frac{w_{L}(\eta_{x}-k)w_{L}(k)}{w_{L}(\eta_{x})w_{L}(0)}=\frac{\\#_{0}(\eta)}{L}\frac{k\,w_{L}(k)L}{w_{L}(0)}\frac{(\eta_{x}-k)\,w_{L}(\eta_{x}-k)}{\eta_{x}\,w_{L}(\eta_{x})}\,.\end{split}$
(27)

First, note that by Assumption (A3)

$k\,w_{L}(k)L\frac{(\eta_{x}-k)w_{L}(\eta_{x}-k)}{\eta_{x}w_{L}(\eta_{x})}\to\theta\,,\quad\mbox{as
}L\to\infty\,,$ (28)

uniformly in $k$ and $\eta$, since $\varepsilon
N\;\leqslant\;k,\eta_{x}-k,\eta_{x}$. Therefore, we have

$\displaystyle\left|\mu_{L,N}(f\mathcal{G}^{(N,\varepsilon)}g)-\mu_{L,N}(g\mathcal{G}^{(N,\varepsilon)}f)\right|$
$\displaystyle\qquad=\frac{1}{N(N-1)}\left|\left(V_{f,g}^{(L,N,\varepsilon)}-U_{g,f}^{(L,N,\varepsilon)}\right)+\left(U_{f,g}^{(L,N,\varepsilon)}-V_{g,f}^{(L,N,\varepsilon)}\right)\right|+o(1)$
$\displaystyle\qquad\;\leqslant\;\frac{2\|f\|_{\infty}\|g\|_{\infty}\theta}{N(N-1)}\pi_{L,N}\left(\sum_{x=1}^{L}\eta_{x}\right.\left.\sum_{k=\varepsilon
N}^{\eta_{x}-\varepsilon
N}\left|\frac{\\#_{0}(\eta)}{L}\frac{1}{w_{L}(0)}-1\right|\right)+o(1)$
$\displaystyle\qquad\;\leqslant\;2\|f\|_{\infty}\|g\|_{\infty}\theta\,\pi_{L,N}\left(\left|\frac{\\#_{0}(\eta)}{L}\frac{1}{w_{L}(0)}-1\right|\right)+o(1)\,,$

where we used (28) before dropping all indicator functions in the first
inequality. Lastly, by Lemma 3.2 we know that

$\displaystyle\lim_{N/L\rightarrow\rho}\pi_{L,N}\left(\left|\frac{\\#_{0}(\eta)}{L}\frac{1}{w_{L}(0)}-1\right|\right)=0\,,$

which completes the proof.

∎

###  3.2. Concentration and uniqueness of the limit

As already mentioned in the introduction, the only invariant distribution
w.r.t. $\mathcal{G}_{\theta}$ which concentrates on full partitions
$\nabla_{[0,\alpha]}$, for some $\alpha\in[0,1]$, is the Poisson-Dirichlet
distribution PD${}_{[0,\alpha]}(\theta)$. Hence, in order to prove that weak
accumulation points of measures $(\mu_{L,N})_{L,N}$ coincide, it is enough to
show that each such weak limit $\mu$ from Proposition 3.1 satisfies

$\operatorname{Var}_{\mu}\left(\sum_{i=1}^{\infty}p_{i}\right)=0$

and there exists $\alpha\in[0,1]$ such that
$\mu\left(\sum_{i=1}^{\infty}p_{i}\right)=\alpha$ for every limit $\mu$.

In the following we consider not only the $\ell^{1}$-norm but also the
$\ell^{2}$-norm as a function on $\overline{\nabla}$ and write

$\|p\|_{1}=\sum_{i=1}^{\infty}p_{i}\quad\text{ and
}\quad\|p\|_{2}^{2}=\sum_{i=1}^{\infty}|p_{i}|^{2}\,,$

respectively. Note that the latter has the advantage, in contrast to
$\|\cdot\|_{1}$, of being a continuous function on $\overline{\nabla}$ with
respect to the product topology.

First, we recall the first part of Lemma 5 in [34] where Mayer-Wolf et al.
proved the following relationship between $\mu(\|p\|_{1}^{2})$ and
$\mu(\|p\|_{2}^{2})$, which is a natural consequence of the balance between
expected split and merge rates for the stationary distribution $\mu$.

######  Lemma 3.5 (Mayer-Wolf et al.).

If $\mu\in\mathcal{M}_{1}(\overline{\nabla})$ is invariant w.r.t.
$\mathcal{G}_{\theta}$, then

$\mu(\|p\|_{1}^{2})=(1+\theta)\mu(\|p\|_{2}^{2})\,.$

Together with its counterpart in the following lemma, this immediately proves
that each subsequential limit of $(\mu_{L,N})_{L,N}$ concentrates on some
$\nabla_{[0,\alpha]}$.

######  Lemma 3.6.

Let $\mu\in\mathcal{M}_{1}(\overline{\nabla})$ be an accumulation point of
$(\mu_{L,N})_{L,N}$. Then

$\mu(\|p\|_{1})^{2}=(1+\theta)\mu(\|p\|_{2}^{2})\,.$

###### Proof.

Since the statement is trivial for $\mu=\delta_{(0,0,\ldots)}$, we assume
without loss of generality $\mu\neq\delta_{(0,0,\ldots)}$. Let
$(\mu_{L_{j},N_{j}})_{j\in\mathbb{N}}$ be a sequence converging weakly to
$\mu$. For such $\mu$ we first note that

$\displaystyle\mu(\|p\|_{2}^{2})$
$\displaystyle=\lim_{j\to\infty}\mu_{L_{j},N_{j}}(\|p\|_{2}^{2})=\lim_{j\to\infty}\frac{L_{j}}{N_{j}^{2}}\pi_{L_{j},N_{j}}(\eta_{1}^{2})$
(29)
$\displaystyle=\lim_{j\to\infty}\frac{1}{N_{j}}\pi_{L_{j},N_{j}}(\widetilde{\eta}_{1})=\lim_{j\to\infty}\mu_{L_{j},N_{j}}(\widetilde{p}_{1})=\lim_{j\to\infty}\sigma(\mu_{L_{j},N_{j}})(q_{1})\,,$

where we used that $\mu_{L_{j},N_{j}}$ converges weakly to $\mu$ together with
(19), and in the second to last step that the size-biased distribution is
invariant under reordering of the configuration. Furthermore, weak convergence
of $\mu_{L_{j},N_{j}}$ implies weak convergence of the size biased
distributions by Lemma 2.2, i.e.

$\sigma(\mu_{L_{j},N_{j}})\to\sigma(\mu)$

in the topology induced by weak convergence. Recall that under $\sigma(\mu)$,
the first component in $q$ is zero with probability

$\sigma(\mu)[q_{1}=0]=\mu(1-\|p\|_{1})\,.$ (30)

Since the projection map on the first component is continuous w.r.t. the
product topology, we have $\sigma(\mu_{L,N})(q_{1})\to\sigma(\mu)(q_{1})$
which yields with (29)

$\mu(\|p\|_{2}^{2})=\sigma(\mu)(q_{1})=\sigma(\mu)(q_{1}\;|\;q_{1}>0)\sigma(\mu)[q_{1}>0]=\sigma^{\prime}(\mu)(q_{1})\mu(\|p\|_{1})\,,$
(31)

where $\sigma^{\prime}(\mu)[q_{1}\in\cdot]$ is the size-biased distribution
conditioned on positive components defined in (18). Now, using (17) and
disintegration of measures we write

$\displaystyle\sigma^{\prime}(\mu)(q_{1})$
$\displaystyle=\int_{\overline{\nabla}}\|p\|_{1}\int_{\nabla}q_{1}\,\sigma_{p/\|p\|_{1}}[dq]\,\mu[dp]$
$\displaystyle=\int_{0}^{1}\alpha\int_{\overline{\nabla}}\int_{\nabla}q_{1}\,\sigma_{p/\|p\|_{1}}[dq]\,\mu^{(\alpha)}[dp]\,\mu[\|p\|_{1}\in
d\alpha]\,,$

where $\mu^{(\alpha)}=\mu[\,\cdot\,|\|p\|_{1}=\alpha]$ denotes the measure
$\mu$ conditioned on $\nabla_{[0,\alpha]}$, which is well defined almost
surely w.r.t. $\mu[\|p\|_{1}\in d\alpha]$. Since $\mathcal{G}_{\theta}$ (13)
conserves $\|p\|_{1}$, $\mu^{(\alpha)}$ is also invariant for
$\mathcal{G}_{\theta}$ and thus equal to PD${}_{[0,\alpha]}(\theta)$ by
Proposition 2.1. Hence,
$\sigma(\mu^{(\alpha)})[dq]=\int_{\overline{\nabla}}\sigma_{p/\|p\|_{1}}[dq]\,\mu^{(\alpha)}[dp]$,
as in (16), is the GEM distribution and therefore

$\int_{\overline{\nabla}}\int_{\nabla}q_{1}\,\sigma_{p/\|p\|_{1}}[dq]\,\mu^{(\alpha)}[dp]=\frac{1}{1+\theta}\,,\quad\text{
for a.e. }\alpha\in(0,1]\,.$

Thus,

$\sigma^{\prime}(\mu)(q_{1})=\frac{1}{1+\theta}\int_{0}^{1}\alpha\;\mu[\|p\|_{1}\in
d\alpha]=\frac{1}{1+\theta}\mu(\|p\|_{1})$

which, with (31), yields with

$\mu(\|p\|_{2}^{2})=\frac{1}{1+\theta}\mu(\|p\|_{1})^{2}\,,$

concluding the proof. ∎

######  Remark 3.7.

It is interesting to note that Lemma 3.6 holds in greater generality, because
all $k$-norms are continuous whenever $k>1$. We have

$\displaystyle\alpha_{\mu}^{k}=\mu(\|p\|_{1})^{k}=\frac{1}{(k-1)!}\left[\prod_{j=1}^{k-1}(j+\theta)\right]\mu(\|p\|_{k}^{k})\qquad\text{for
all }k\;\geqslant\;1\,.$

This can be proven in exactly the same way as Lemma 3.6, with the only
difference that we replace $\mu(\|p\|_{2}^{2})$ with $\mu(\|p\|_{k}^{k})$ when
$k>1$. The case $k=1$ is trivial.

Lemma 3.5 and 3.6 imply $\mu(\|p\|_{1}^{2})=\mu(\|p\|_{1})^{2}$, which is
equivalent to

$\operatorname{Var}_{\mu}(\|p\|_{1})=\mu(\|p\|_{1}^{2})-\mu(\|p\|_{1})^{2}=0\,.$
(32)

Therefore, accumulation points $\mu$ concentrate on
$\nabla_{[0,\alpha_{\mu}]}$ with $\alpha_{\mu}:=\mu(\|p\|_{1})$. To conclude
the proof of Theorem 1.3, it is left to show that $\alpha_{\mu}$ is
independent of the choice of $\mu$, which is guaranteed by assumption (A4).

###### Proof of Theorem 1.3.

Let $\theta\in(0,1]$. By Proposition 3.1, we know that all weak accumulation
points of $(\mu_{L,N})_{L,N}$ are reversible w.r.t. $\mathcal{G}_{\theta}$.
Furthermore, every such limit $\mu$ of a subsequence
$(\mu_{L_{j},N_{j}})_{j\;\geqslant\;1}$ concentrates on
$\nabla_{[0,\alpha_{\mu}]}$ for some $\alpha_{\mu}\in[0,1]$ as we can see by
(32). Thus, Proposition 2.1 implies that $\mu$ must be the Poisson-Dirichlet
distribution on $[0,\alpha_{\mu}]$ with parameter $\theta$.

The only control we have on $\alpha_{\mu}$ from (A1) \- (A3) is that

$\alpha_{\mu}\;\leqslant\;1-\frac{\nu_{\rho}(\eta_{1})}{\rho}\,,\quad\text{where}\quad\nu_{\rho}(\eta_{1})=\sum_{n=0}^{\infty}nw(n)$

is the expected density in the bulk. However, Assumption (A4) together with
Lemma 3.6 implies uniqueness of the limit, since for every accumulation point
$\mu$ we have

$\mu(\|p\|_{1})=\sqrt{(1+\theta)\mu(\|p\|_{2}^{2})}=\sqrt{\frac{1+\theta}{\rho}\lim_{N/L\rightarrow\rho}\frac{\pi_{L,N}(\eta_{x}^{2})}{N}}=\alpha\,,$
(33)

where we used one of the identities deduced in (29) in the second equality.
This concludes the proof.

∎

##  4\. Proof of Theorem 1.6

In this section we prove Theorem 1.6, a specialized version of Theorem 1.3
which is better suited for application to condensation in particle systems. In
contrast to assumptions (A1) -(A3), we require uniform convergence of the
weights as well as stronger control on the weights in sub-$L$ scales in (B2).
However, this allows us to drop assumption (A4) that was needed to guarantee
the concentration of the macroscopic phase in Theorem 1.3. Thanks to the
stronger assumptions, we can explicitly calculate the form of limiting single-
site marginals, which will imply the equivalance of ensembles (A2) and the
condensation transition.

In Appendix A we prove the equivalence of ensembles and deduce that the system
defined by weights $(w_{L})_{L}$ satisfying (B1) (and a growth condition on
the weights which is weaker than (B2)) exhibits condensation for $\rho$ large
enough. We summarize this result here, a more general and detailed version is
given in Proposition A.1.

######  Proposition 4.1 (Equivalence of ensembles).

Consider weights $(w_{L})_{L}$ and $w$ satisfying (B1), (B2) with
corresponding canonical measures $(\pi_{L,N})_{L,N}$ as defined in (1). Then
the system exhibits a condensation transition in the thermodynamic limit
$N/L\rightarrow\rho\;\geqslant\;0$ (cf. Definition 1.1) with critical density
$\rho_{c}:=\sum_{n=1}^{\infty}nw(n)$. More precisely, we have convergence of
single-site marginals $\pi_{L,N}[\eta_{x}\in\cdot]\to\nu_{\rho}$ such that

$\nu_{\rho}(\eta_{x})=\left\\{\begin{array}[]{cl}\rho&\text{if
}\rho<\rho_{c}\,,\\\ \rho_{c}&\text{if
}\rho\;\geqslant\;\rho_{c}\,.\end{array}\right.$ (34)

and $\nu_{\rho}=w$ for $\rho\;\geqslant\;\rho_{c}$.

This establishes existence of the condensed phase for $\rho>\rho_{c}$, and the
following result guarantees that it agrees with the macroscopic phase, and
there is no mass on intermediate scales.

######  Proposition 4.2.

Consider weights $(w_{L})_{L}$ and $w$ satisfying (B1) and (B2) with
corresponding canonical measures $(\pi_{L,N})_{L,N}$. Then, for every
$\rho>\rho_{c}$

$\displaystyle\lim_{\varepsilon\to
0}\lim_{N/L\rightarrow\rho}\pi_{L,N}[\widetilde{\eta}_{1}>\varepsilon
N]=1-\frac{\rho_{c}}{\rho}\,.$

###### Proof.

We fix a density $\rho>\rho_{c}$. Now, by definition of the first size-biased
marginal, we have for every $\varepsilon\in(0,1]$

$\displaystyle\pi_{L,N}[J<\widetilde{\eta}_{1}\;\leqslant\;\varepsilon
N]=\frac{L}{N}\sum_{n=J+1}^{\varepsilon
N}n\,w_{L}(n)\frac{Z_{L-1,N-n}}{Z_{L-1,N}}\,,$

for $N$ large enough. By Assumption (B2), $n\,w_{L}(n)L$ is uniformly close to
$\theta$ for $J$ sufficiently large. Applying Lemma 4.3 below yields

$\displaystyle\limsup_{N/L\rightarrow\rho}\sup_{J<n\;\leqslant\;\varepsilon
N}\frac{Z_{L-1,N-n}}{Z_{L-1,N}}\;\leqslant\;\Big{(}1-\tfrac{\varepsilon}{1-\rho_{c}/\rho}\Big{)}^{-1}\,.$

Altogether,

$\displaystyle\lim_{N/L\rightarrow\rho}\pi_{L,N}[J<\widetilde{\eta}_{1}\;\leqslant\;\varepsilon
N]\;\leqslant\;\varepsilon\,(\theta+\varepsilon)\Big{(}1-\tfrac{\varepsilon}{1-\rho_{c}/\rho}\Big{)}^{-1}\,,$

which vanishes as we take the limit $\varepsilon\to 0$. This completes the
proof, since

$\lim_{N/L\rightarrow\rho}\pi_{L,N}[\widetilde{\eta}_{1}\;\leqslant\;J]=\lim_{N/L\rightarrow\rho}\frac{L}{N}\sum_{n=0}^{J}n\,\pi_{L,N}[\eta_{1}=n]=\frac{\rho_{c}}{\rho}\,,$

which is a direct implication of Proposition 4.1. ∎

We now prove the key estimate used in Proposition 4.2, which guarantees that
the ratio of partition functions $Z_{L-1,N-n}/Z_{L,N}$ does not blow up for
$n=o(N)$.

######  Lemma 4.3.

Assume that (B1) and (B2) are both satisfied. Then we have

$\displaystyle\limsup_{N/L\to\rho}\frac{Z_{L-1,(1-\kappa)N}}{Z_{L,N}}\;\leqslant\;\Big{(}1-\tfrac{\overline{\kappa}}{1-\rho_{c}/\rho}\Big{)}^{-1}\,,$

for every $\kappa=\kappa(L)=O(1)$ such that
$\kappa\;\leqslant\;1-\tfrac{\rho_{c}}{\rho}$, where
$\overline{\kappa}:=\limsup_{L\to\infty}\kappa(L)$.

###### Proof.

Fix $\varepsilon\in(0,1)$, by (B2) and Remark 1.7 there exists
$J\in\mathbb{N}_{0}$ such that

$\displaystyle\frac{m\,w_{L}(m)}{n\,w_{L}(n)}\;\geqslant\;1-\varepsilon\quad\textrm{for
all }m,n\;\geqslant\;J\,,$ (35)

for all $L$ sufficiently large (depending on $J$ and $\varepsilon$). Let
$K=\lceil\kappa N\rceil$, by definition of the canonical measures we have

$\displaystyle\pi_{L,N}(\eta_{1}\,;\,\eta_{1}>J)$
$\displaystyle\;\geqslant\;\pi_{L,N}(\eta_{1}\,;\,\eta_{1}>J+K)=\sum_{n>J+K}^{N}n\,w_{L}(n)\frac{Z_{L-1,N-n}}{Z_{L,N}}$
$\displaystyle=\sum_{n>J}^{N-K}(n+K)\,w_{L}(n+K)\frac{Z_{L-1,N-K-n}}{Z_{L,N}}$
$\displaystyle\;\geqslant\;\big{(}1-\varepsilon\big{)}\sum_{n>J}^{N-K}n\,w_{L}(n)\frac{Z_{L-1,N-K-n}}{Z_{L,N}}\,,$

where the last inequality follows from (35). The sums above are all non-empty
because $N-K>J$ for $L,N$ sufficiently large since $\kappa<1$. Multiplying and
dividing by $Z_{L,N-K}$ we have

$\displaystyle\pi_{L,N}(\eta_{1}\,;\,\eta_{1}>J)$
$\displaystyle\;\geqslant\;\big{(}1-\varepsilon\big{)}\frac{Z_{L,N-K}}{Z_{L,N}}\pi_{L,N-K}(\eta_{1}\,;\,\eta_{1}>J)\,.$
(36)

By equivalence of ensembles (Proposition 4.1), in the thermodynamic limit
$N/L\to\rho\;\geqslant\;\rho_{c}$ the single site marginals of $\pi_{L,N}$
converge weakly to $\nu_{\rho}$ which has mean $\rho_{c}$, so

$\pi_{L,N}(\eta_{1}\,;\,\eta_{1}>J)=\frac{N}{L}-\pi_{L,N}(\eta_{1}\,;\,\eta_{1}\;\leqslant\;J)\to\rho-\rho_{J}\,,\quad\textrm{as
}N/L\to\rho\,,$

where $\rho_{J}$ is given by
$\nu_{\rho}(\eta_{1}\,;\,\eta_{1}\;\leqslant\;J)$. Furthermore, by dominated
convergence, $\rho_{J}\to\rho_{c}$ as $J\to\infty$. Now taking the
thermodynamic limit in (36), followed by the limit $J\to\infty$, we have

$\displaystyle\limsup_{N/L\to\rho}\frac{Z_{L,(1-\kappa)N}}{Z_{L,N}}\;\leqslant\;\frac{\rho-\rho_{c}}{(1-\overline{\kappa})\rho-\rho_{c}}\frac{1}{1-\varepsilon}\,.$

From the equivalence of ensembles in Proposition 4.1, and the identity
$\pi_{L,N}[\eta_{1}=0]=w_{L}(0)Z_{L-1,N}/Z_{L,N}$, we observe that
$\lim_{N/L\to\rho}Z_{L-1,N}/Z_{L,N}=1$, which completes the proof, since we
can choose $\varepsilon$ arbitrarily small after taking $J\to\infty$. ∎

We are now ready to state the full proof of Theorem 1.6 which follows by
putting together the statements of Theorem 1.3, Proposition 4.1 and
Proposition 4.2.

###### Proof of Theorem 1.6.

The equivalence of ensembles implies the condensation transition with critical
density $\rho_{c}$, and with its formulation in Proposition A.1, also
Assumption (A2) is satisfied. Next, we recover assumptions (A1), (A3) and (A4)
for a fixed choice of $\rho>\rho_{c}$. Clearly, (A1) is a direct implication
of (B1), also (A3) follows immediately from (B2). This already yields that the
subsequential limits of the corresponding measures $\mu_{L,N}$ are Poisson-
Dirichlet distributions. It is only left to show that the macroscopic phase is
non-trivial and indeed agrees with the condensed phase. Recall from (30) that

$\displaystyle\mu(\|p\|_{1})=\sigma(\mu)[q_{1}>0]=\lim_{\varepsilon\to
0}\sigma(\mu)[q_{1}>\varepsilon]$

and, as $\varepsilon\to 0$,

$\displaystyle\sigma(\mu)[q_{1}>\varepsilon]$
$\displaystyle=\lim_{N/L\rightarrow\rho}\mu_{L,N}[\widetilde{p}_{1}>\varepsilon]$
$\displaystyle=\lim_{N/L\rightarrow\rho}\pi_{L,N}[\widetilde{\eta}_{1}>\varepsilon
N]\to 1-\frac{\rho_{c}}{\rho}\,,$

which follows from Proposition 4.2. Hence, (A4) holds with
$\alpha=\mu(\|p\|_{1})=1-\tfrac{\rho_{c}}{\rho}$. Altogether, we verified
(A1)-(A4) and so can apply Theorem 1.3. ∎

##  5\. Application to interacting particle systems and conclusion

As mentioned in the introduction, condensation transitions occur naturally and
have been studied extensively for interacting particle systems, more precisely
for stochastic lattice gases which model transport phenomena and conserve the
number of particles. A large class of such models with state space
$\Omega_{L,N}$ has been introduced in [6] with infinitesimal generator of the
form

$\mathcal{L}^{(L)}f(\eta)=\sum_{x,y=1}^{L}p(x,y)u(\eta_{x},\eta_{y})\left[f(\eta^{xy})-f(\eta)\right]\,,$
(37)

where $f\in C_{b}(\Omega_{L,N})$. Here $\eta^{xy}$ denotes the configuration
$\eta-e^{(x)}+e^{(y)}$ where one particle moved from site $x$ to site $y$,
with $e^{(x)}$ denoting the unit vector $e^{(x)}_{z}=\delta_{x,z}$. The jump
rate $u(n,m)\;\geqslant\;0$ is a non-negative function of the occupation
numbers $n$ on the departure and $m$ on the target site of a particle jump,
and to avoid degeneracies we assume that $u(n,m)=0$ if and only if $n=0$.
$p(x,y)$ denotes an irreducible probability kernel on $\\{1,\ldots,L\\}$ and
models the geometry of the underlying lattice. Systems have been studied, e.g.
on regular lattices in various dimensions and with different boundary
conditions, here we assume that the system is closed and conserves the total
number of particles $\sum_{x=1}^{L}\eta_{x}$.

For any fixed number of particles $N\in\mathbb{N}$, the operator
$\mathcal{L}^{(L)}$ defines an irreducible, continuous-time Markov process on
the finite state space $\Omega_{L,N}$, which therefore has a unique invariant
distribution $\pi_{L,N}$. It has been established in [6, 15] that this
distribution is indeed of product form and spatially homogeneous, cf. (1),
under the following conditions:

$\frac{u(n+1,m)}{u(m+1,n)}=\frac{u(n+1,0)}{u(1,n)}\frac{u(1,m)}{u(m+1,0)}\qquad\forall
n,m\;\geqslant\;0\,,$ (38)

and either $p(\cdot,\cdot)$ is symmetric, or it is doubly stochastic, i.e.
$\sum_{y}\big{(}p(x,y)-p(y,x)\big{)}=0$, and

$u(n,m)-u(m,n)=u(n,0)-u(m,0)\qquad\forall n,m\;\geqslant\;0\,.$ (39)

Then the stationary weights are

$w(n)=\prod_{k=1}^{n}\frac{u(1,k-1)}{u(k,0)}\qquad\forall n\;\geqslant\;0\,,$
(40)

which depend only on the jump rates $u$ but not on the kernel $p$.

For the special case of zero-range dynamics, (40) simplifies further since
$u(k,\cdot)=u(k)$, and (38) and (39) are fulfilled. In this case (40) leads to
the simple identification between stationary weights and rates

$\displaystyle u(n)=\frac{w(n-1)}{w(n)}\qquad\forall n\;\geqslant\;1\,.$ (41)

Due to this simple one-to-one correspondence between weights and transition
rates, zero-range processes provide a generic framework of studying
condensation transitions in interacting particle systems.

Another less restrictive simplification is to assume that
$u(n,m)=u_{1}(n)u_{2}(m)$ is of product form, which automatically satisfies
(38) and always leads to factorized stationary measures under symmetric
dynamics with weights from (40), now taking the form

$w(n)=\bigg{(}\frac{u_{1}(1)}{u_{2}(0)}\bigg{)}^{n}\prod_{k=1}^{n}\frac{u_{2}(k-1)}{u_{1}(k)}\qquad\forall
n\;\geqslant\;0\,.$ (42)

Note that due to the conservation law the exponential factor
$\Big{(}\frac{u_{1}(1)}{u_{2}(0)}\Big{)}^{n}$ is usually omitted, since it
cancels in the definition of $\pi_{L,N}$ (1). One particular example is the
inclusion process with rates

$u(n,m)=n\,(d+m)\quad\text{with }d>0\,,$ (43)

which leads to the stationary weights (6). If we set

$d=d(L)>0\quad\text{with}\quad dL\to\theta>0\text{ as }L\to\infty\,,$ (44)

the system exhibits a condensation transition with $\rho_{c}=0$ and a Poisson-
Dirichlet structure with PD${}_{[0,1]}(\theta)$, see [27, Theorem 1] or (7)
above. Theorem 1.6 recovers this result for $\theta\in(0,1]$. The restriction
of $\theta$ is solely due to the fact that the Poisson-Dirichlet distribution
is (so far) only proven to be the unique invariant distribution for the split-
merge process if $\theta\in(0,1]$.

Using the size-dependent parameter $d$ as above, we will provide a few
instructive examples of particle systems with size-dependent jump rates
$u_{L}$ of product form, where Theorem 1.6 applies with a non-trivial critical
density $\rho_{c}>0$. We start by fixing the weights

$w_{L}(n)=\begin{cases}w(n)&\quad\text{ if }n\;\leqslant\;A\,,\\\
\frac{d}{n}&\quad\text{ if }n>A\,,\end{cases}\quad\text{for some fixed
}A\in\mathbb{N}\,,$ (45)

where $d=d(L)$ as in (44), $\theta\in(0,1]$ and $w$ is a probability mass
function on the set $\\{0,\ldots,A\\}$. Let us fix for simplicity the uniform
distribution with $w(n)=(1+A)^{-1}\mathds{1}\\{n\;\leqslant\;A\\}$. Using
(41), the corresponding rates of a zero-range process are given by

$u_{L}(n)=\begin{cases}1&\quad\text{ if }1\;\leqslant\;n\;\leqslant\;A\,,\\\
d^{-1}\simeq L/\theta&\quad\text{ if }n=A+1\,,\\\ \frac{n}{n-1}&\quad\text{ if
}n>A+1\,.\end{cases}$ (46)

This underlines the mechanism that leads to condensation in such systems:
Sites with occupation numbers different from $A+1$ are stable and eject
particles at rates of order $1$. Sites with occupation number $A+1$ are
unstable and eject a particle at diverging rate of order $L$, creating a sharp
threshold between bulk sites ($\eta_{x}\;\leqslant\;A$) and cluster sites
($\eta_{x}>A+1$). If in the thermodynamic limit

$N/L\to\rho>\rho_{c}=\frac{A}{2}\,,$

i.e. the total density exceeds the expectation of the uniform bulk
distribution, the system exhibits a condensation transition with mass fraction
$1-\rho_{c}/\rho$ in the condensate. Heuristically, the excess mass is
expelled from the bulk and accumulates in stable clusters with occupation
numbers larger than $A+1$. The particular form of the rates for those clusters
leads to a macroscopic phase with Poisson-Dirichlet statistics
PD${}_{[0,1-\rho_{c}/\rho]}(\theta)$, which follows from Theorem 1.6 and is
illustrated in Figure 1. Clearly, the weights $(w_{L})_{L}$ converge uniformly
to $w$ and therefore satisfy both Assumption (B1) and (B2). The same process
with rates $du_{L}(n)$ corresponds to asymptotically vanishing exit rates from
stable sites, which is simply a time change and leads of course to the same
stationary behaviour.

Figure 1. Simulation results at stationarity for a zero-range process with
rates (46), parameters $\theta=A=\rho=1$ and $\rho_{c}=1/2$. The left boxplot
displays $\|p\|_{2}^{2}$ where $p=\tfrac{\widehat{\eta}}{N}$ was sampled
$200$-times for every system size, indicating convergence to $\tfrac{1}{8}$
which agrees with
$\mu_{L,N}(\|p\|_{2}^{2})\simeq(1+\theta)^{-1}(1-\tfrac{\rho_{c}}{\rho})^{2}$
from Lemma 3.6. The right plot shows three samples of accumulated
configurations in a system of size $N=L=1024$, where we clearly see occurance
of large clusters, with a background density of $\rho_{c}=1/2$ indicated by
dotted lines.

We can also generalize the inclusion process dynamics (43) to stationary
weights of the form (45). Consider a process with rates of product form
$u_{L}(n,m)=u_{1,L}(n)u_{2,L}(m)$ with

$u_{1,L}(n)=u_{2,L}(n)=\begin{cases}d\simeq\theta/L&\quad\text{ if
}1\;\leqslant\;n\;\leqslant\;A\,,\\\ n&\quad\text{ if }n>A\,,\end{cases}$ (47)

and $u_{2,L}(0)=d$. It is easy to see from (42) that the stationary weights
for this process are given by (45) and Theorem 1.6 applies. Heuristically,
sites with occupation number up to $A$ eject and attract particles at a slow
rate $d$, and particles on sites with higher occupation numbers become “free”
and leave independently at the same rate and also attract other particles, as
a simple generalization of the standard inclusion interaction. As a result,
the dynamics in the condensed phase happen at a much higher rate than in the
bulk. This separation of time scales leads to completely different dynamics
than in the zero-range example above, even though both models share the same
stationary distributions. We want to stress that due to the general nature of
Assumptions (B1) and (B2), Theorem 1.6 applies also to modifications of these
examples and the particular form of the weights (45) is not important.

Understanding the dynamics of the condensed phase in these models is a very
interesting question for future research, in particular the coarsening regime,
where macroscopic clusters emerge from homogeneous initial conditions and
approach stationarity by exchanging particles. Heuristically, their stationary
mass partition can be understood as a balance between aggregation and
fragmentation of macroscopic clusters. Note that these dynamics are not
described by split-merge processes, which we only use as an auxiliary tool to
characterize PD distributions, but are rather of a diffusive nature. A
diffusive model on partitions that has stationary PD distribution has been
introduced in [10], and it would be very interesting to study hydrodynamic
scaling limits in this context.

As we have seen, the Poisson-Dirichlet structure in the macroscopic phase
arises due to uniform stationary weights under size-biased sampling, which
leads to particular rates in the zero-range process (46) or the generalized
inclusion process (47) for large occupation numbers. In the context of the
dynamics of interacting particle systems this is only one particular case, and
it would be interesting to study the statistics of the condensed phase beyond
Poisson-Dirichlet under a different scaling behaviour of the weights.

##  Appendix A Equivalence of ensembles

In this section we prove that, under assumption (B1) and if the weights
$(w_{L})_{L}$ decay sub exponentially, i.e.

$\displaystyle\frac{1}{L}\log w_{L}(aL)\to 0\,,\qquad\forall a>0\,,$ (48)

we have equivalence of ensembles and condensation in the sense of weak
convergence of finite dimensional marginals. Note that this includes
condensation in the sense of Definition 1.1, and (48) is weaker than
Assumption (B2) in Theorem 1.6. The result is in the same spirit as previous
results on equivalence of ensembles and condensation in stochastic particle
systems with stationary product measures (see for example [4]). However, as
far as we know, this is the first general result in this direction for models
with size-dependent weights.

In order to state the result in more generality we first introduce some extra
notation. For a sequence of non-negative, non-trivial weights,
$(w_{L}(n))_{n\in\mathbb{N}_{0}}$, possibly depending on the system size $L$,
we define a family of probability measures on $\mathbb{N}_{0}$ by tilting the
weights by a non-negative _fugacity_ parameter $\phi\;\geqslant\;0$:

$\displaystyle\bar{\nu}_{\phi,L}[dn]:=\frac{1}{z_{L}(\phi)}w_{L}(n)\phi^{n}dn\quad\mbox{with}\quad
z_{L}(\phi):=\sum_{n=0}^{\infty}w_{L}(n)\phi^{n}\,,$

which is well defined for each $\phi\in
D_{L}:=\\{\phi\,:\,z_{L}(\phi)<\infty\\}$. The corresponding family of _grand-
canonical distributions_ is given by the product measures

$\bar{\nu}_{\phi,L}^{\otimes
L}[d\eta]=\frac{1}{z_{L}(\phi)^{L}}\prod_{x=1}^{L}w_{L}(\eta_{x})\phi^{\eta_{x}}d\eta\,,$
(49)

which are defined on the configuration space
$\Omega_{L}=\mathbb{N}_{0}^{L}=\bigcup_{N=0}^{\infty}\Omega_{L,N}$, where the
total number of particles is arbitrary. The expected number of particles per
site (density) $R_{L}:D_{L}\to[0,\infty)$ is a strictly increasing function of
$\phi$, with

$\displaystyle
R_{L}(\phi):=\bar{\nu}_{\phi,L}(\eta_{x})=\phi\,\partial_{\phi}\log
z_{L}(\phi)\,,\quad\textrm{for }\phi\in D_{L}\,.$ (50)

We denote the inverse of $R_{L}$ by $\Phi_{L}$. Furthermore, we define the
variance of $\bar{\nu}_{\phi,L}$

$\displaystyle\sigma_{L}^{2}(\phi)=\bar{\nu}_{\phi,L}(\eta_{x}^{2})-R_{L}(\phi)^{2}\,,$

which is finite for each $\phi$ in the interior of $D_{L}$. By construction,
the canonical measures (1) on $\Omega_{L,N}$ are given by conditioning any
grand-canonical measure on the total number of particles, i.e.

$\pi_{L,N}[d\eta]:=\bar{\nu}_{\phi,L}^{\otimes
L}\left[d\eta\;\middle|\;\sum_{x=1}^{L}\eta_{x}=N\right]=\frac{1}{Z_{L,N}}\prod_{x=1}^{L}w_{L}(\eta_{x})d\eta\,,$

which is independent of $\phi\in D_{L}$.

By Assumption (B1), the weights $(w_{L}(n))_{n,L}$ converge uniformly in $n$
as $L\to\infty$ to a probability measure $(w(n))_{n\in{\mathbb{N}}_{0}}$. We
define the limiting grand-canonical measures by

$\bar{\nu}_{\phi}[dn]:=\frac{1}{z(\phi)}w(n)\phi^{n}dn\quad\mbox{with}\quad
z(\phi):=\sum_{n=0}^{\infty}w(n)\phi^{n}\ .$

Since $(w(n))_{n\in{\mathbb{N}}_{0}}$ is normalised, these measures must exist
at least for each $\phi\in[0,1]$, and $\bar{\nu}_{1}$ corresponds to the
weights $w$. By analogy with (50), we define the function
$R(\phi)=\bar{\nu}_{\phi}(\eta_{1})$, which is a strictly increasing function
$R\colon[0,1]\to[0,\rho_{c}]$ with

$\rho_{c}=R(1)=\sum_{n=0}^{\infty}nw(n)\,,\quad\mbox{as given in Theorem
\ref{THEO_SPECIALISED_MAIN}}\ .$

We denote the inverse of $R$ by $\Phi$, so that the average particle density
under $\bar{\nu}^{\otimes{L}}_{\Phi(\rho)}$ is $\rho$ for all
$\rho\;\leqslant\;\rho_{c}$. Further, we denote the variance of
$\bar{\nu}_{\phi}$ by $\sigma^{2}(\phi)$ which is finite for $\phi\in[0,1]$ by
Assumption (B1). Note that it may be possible that $R(\phi)<\infty$ and
$\bar{\nu}_{\phi}$ is well defined also for $\phi>1$, but such measures are
not accessible as limits of $\bar{\nu}_{\phi,L}$ and do not play a role in the
following.

Under Assumption (B1) and sub-exponential weights (48), it turns out that
there is a condensation transition according to Definition 1.1 with critical
density $\rho_{c}=R(1)$. In this case, in the thermodynamic limit
$N/L\rightarrow\rho$, all finite dimensional marginals of the canonical
measures converge weakly to the limiting grand-canonical measures with density
$\rho$ if $\rho\;\leqslant\;\rho_{c}$, and with density $\rho_{c}$ if
$\rho\;\geqslant\;\rho_{c}$. This implies that for $\rho>\rho_{c}$ the excess
mass must condense on a vanishing volume fraction in the thermodynamic limit.

######  Proposition A.1 (Equivalence of ensembles).

Consider non-negative weights $(w_{L})_{L}$ satisfying (B1) with limit $w$.
Furthermore, assume that $w_{L}$’s have sub-exponential tails, in the sense of
(48). Then for each $M\subset\mathbb{N}_{0}$, with $|M|=m<\infty$, denoting
the marginal of $\pi_{L,N}$ on $M$ by $\pi_{L,N}^{M}$, we have

$\displaystyle\pi_{L,N}^{M}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\begin{cases}\bar{\nu}_{\Phi(\rho)}^{\otimes m}&\quad\text{
if }\rho<\rho_{c}\,,\\\ \bar{\nu}_{1}^{\otimes m}&\quad\text{ if
}\rho\;\geqslant\;\rho_{c}\,,\end{cases}$

where $\rho_{c}=R(1)<\infty$.

Throughout the proof we assume further that condition (11) is satisfied, i.e.

$\displaystyle w(0)>0\quad\mbox{and}\quad\sup_{n}[w(n-1)\wedge w(n)]>0\,,$

which implies that the variance, $\sigma^{2}(\phi)$, given by $\sum
n^{2}w(n)\phi^{n}$, is positive for each $\phi\in(0,1]$. The special case of
$w(0)=1$ is covered at the end of the proof.

We firstly observe that for each $\rho\in[0,\rho_{c}]$ we can construct a
sequence of size-dependent fugacities such that the mean of the size-dependent
grand-canonical measures converges to $\rho$ and the variance remains bounded.
The typical behaviour of $R_{L}(\phi)$ is illustrated in Figure 2.

Figure 2.  The dashed lines represent $R_{L}(\phi)$ for systems of sizes
$L\in\\{4,16,128\\}$ and weights (45) with $\theta=A=1$. The black line
denotes the limit $R(\phi)=\tfrac{\phi}{1+\phi}$.

######  Lemma A.2.

Under Assumption (B1) and if the weights $(w_{L})_{L}$ are sub-exponential in
the sense of (48), then for each $\varphi\in[0,1]$ there exists a sequence
$(\varphi_{L})_{L}$ in $[0,1)$, with limit point $\varphi$, such that

$\displaystyle\|w_{L}-w\|_{\infty}\frac{1}{(1-\varphi_{L})^{3}}\to
0\,,\quad\text{ as }L\to\infty\,,$ (51)

and

$\displaystyle R_{L}(\varphi_{L})\to
R(\varphi)\in[0,\rho_{c}]\,,\quad\sigma^{2}_{L}(\varphi_{L})\to\sigma^{2}(\varphi)>0\,,\
\text{ and }\ z_{L}(\varphi_{L})\to z(\varphi)\,.$ (52)

###### Proof.

For $\varphi\in[0,1)$ we may choose $\varphi_{L}=\varphi$ for each $L$. For
$\varphi=1$, we let $\varphi_{L}=1-\|w_{L}-w\|_{\infty}^{1/4}$. Then
$\varphi_{L}\to 1$ as $L\to\infty$ and condition (51) is satisfied since
$\|w_{L}-w\|_{\infty}\to 0$ by assumption (B1).

It is only left to prove the convergence in (52), which is equivalent to
showing that

$\lim_{L\to\infty}\sum_{n=0}^{\infty}n^{p}w_{L}(n)\varphi_{L}^{n}=\sum_{n=0}^{\infty}n^{p}w(n)\varphi^{n}\,,$

for $p\in\\{0,1,2\\}$. Since the weights converge uniformly, we have

$\displaystyle\bigg{|}\sum_{n=0}^{\infty}n^{p}(w_{L}(n)-w(n))\varphi_{L}^{n}\bigg{|}\;\leqslant\;\|w_{L}(n)-w(n)\|_{\infty}\sum_{n=0}^{\infty}n^{p}\varphi_{L}^{n}\,.$

Term-by-term differentiation of the geometric series yields for
$p\in\\{0,1,2\\}$

$\displaystyle\sum_{n=0}^{\infty}n^{p}\varphi_{L}^{n}=\frac{(p\,\vee
1)\varphi_{L}^{p}}{(1-\varphi_{L})^{p+1}}+\frac{\varphi_{L}}{(1-\varphi_{L})^{2}}\mathds{1}\\{p=2\\}\,.$

Hence, using again (51), we have

$\displaystyle\sum_{n=0}^{\infty}n^{p}w_{L}(n)\varphi_{L}^{n}\simeq\sum_{n=0}^{\infty}n^{p}w(n)\varphi_{L}^{n}\to\sum_{n=0}^{\infty}n^{p}w(n)\varphi^{n}\,,$

where we used dominated convergence in the last step. ∎

We will prove Proposition A.1 by showing that the relative entropy between the
single site marginal of $\pi_{L,N}$ and $\bar{\nu}_{\varphi_{L},L}$ vanishes,
where the limit density $R(\varphi)$ is equal to $\rho$ in the sub-critical
case, and $\rho_{c}$ in the super critical case. Optimally, in the super-
critical case, we would like to measure the relative entropy w.r.t. the
limiting measure directly, however this is not possible since
$\pi_{L,N}[\eta_{x}\in\cdot\,]\ll w$ is in general not satisfied and the
relative entropy would be ill defined.

The main tool we rely on in the proof of Proposition A.1 is a local central
limit theorem (see for example [9, Theorem 1.2]) which allows us to estimate
the decay of the relative entropy. For completeness we include the local limit
theorem here. To state it, we first introduce the Bernoulli part decomposition
$q$ of a probability measure $\mathds{P}_{x,L}$ on ${\mathbb{Z}}$ as

$\displaystyle
q(\mathds{P}_{x,L}):=\sum_{n\in{\mathbb{Z}}}(\mathds{P}_{x,L}[n]\wedge\mathds{P}_{x,L}[n+1])\,.$

Moreover, for a family of measures
$(\mathds{P}_{x,L})_{1\;\leqslant\;x\;\leqslant\;L}$ we define
$Q_{L}:=\sum_{x=1}^{L}q(\mathds{P}_{x,L})$.

######  Lemma A.3 ([9, Theorem 1.2] Local central limit theorem).

Consider a triangular array of independent integer valued random variables
$\eta_{x,L}$, for $1\;\leqslant\;x\;\leqslant\;L$, and $L\in\mathbb{N}$, where
$\eta_{x,L}$ has law $\mathds{P}_{x,L}$. Suppose there exist sequences $a_{L}$
and $b_{L}$, $L\;\geqslant\;1$, such that $b_{L}\to\infty$,
$\limsup_{L\to\infty}b_{L}^{2}/Q_{L}<\infty$ and

$\displaystyle\frac{1}{b_{L}}\sum_{x=1}^{L}(\eta_{x,L}-a_{L})\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,1)\,.$ (53)

Then

$\sup_{n\in\mathbb{Z}}\left|b_{L}\,\mathds{P}_{L}\Big{[}\sum_{x=1}^{L}\eta_{x,L}=n\Big{]}-g\Big{(}\frac{n-a_{L}}{b_{L}}\Big{)}\right|\to
0\,,$

where $\mathds{P}_{L}$ denotes the product measure
$\bigotimes_{x=1}^{L}\mathds{P}_{x,L}$ and $g$ the density of a standard
normal.

To apply Lemma A.3 we consider independent random variables

$\displaystyle(\eta_{x,L})_{1\;\leqslant\;x\;\leqslant\;L}\quad\text{ with law
}\quad\bar{\nu}_{\varphi_{L},L}^{\otimes L}\quad\mbox{for each
}L\;\geqslant\;1\,,$ (54)

where $(\varphi_{L})_{L\in\mathbb{N}}$ is a sequence in $[0,1)$ satisfying
(51). To apply Lemma A.3 in the proof of Proposition A.1, we first verify the
central limit theorem (53) for the $\eta_{x,L}$’s.

######  Lemma A.4.

Consider $(\varphi_{L})_{L}$ to be a sequence in $[0,1)$ with limit point
$\varphi\in[0,1]$ satisfying (51) and (52). Furthermore, let
$(\eta_{x,L})_{1\;\leqslant\;x\;\leqslant\;L,L\in\mathbb{N}}$ be as in (54)
and define

$\zeta_{x,L}:=\frac{\eta_{x,L}-R_{L}(\varphi_{L})}{\sqrt{L\,\sigma_{L}^{2}(\varphi_{L})}}\,.$

Then $\sum_{x=1}^{L}\zeta_{x,L}$ converges weakly to a standard normal, as $L$
tends to infinity.

###### Proof.

We want to apply the Lindeberg-Feller central limit theorem, see [29, Theorem
5.12]: because the $\zeta_{x,L}$’s are centered and normalised, it suffices
confirm that the following Lindeberg condition holds:

$\textrm{For every $\varepsilon>0$,
}\lim_{L\to\infty}\sum_{x=1}^{L}\bar{\nu}_{\varphi_{L},L}(\zeta_{x,L}^{2}\mathds{1}\\{|\zeta_{x,L}|>\varepsilon\\})=0\,.$

Since $R_{L}(\varphi_{L})$ and $\sigma^{2}_{L}(\varphi_{L})$ converge to
positive numbers, we have that for $L$ large enough

$\displaystyle\sum_{x=1}^{L}\bar{\nu}_{\varphi_{L},L}(\zeta_{x,L}^{2}\mathds{1}\\{|\zeta_{x,L}|>\varepsilon\\})$
$\displaystyle=\frac{1}{z_{L}(\varphi_{L})\,\sigma^{2}_{L}(\varphi_{L})}\sum_{n=0}^{\infty}(n-R_{L}(\varphi_{L}))^{2}w_{L}(n)\varphi_{L}^{n}\mathds{1}\Big{\\{}|n-R_{L}(\varphi_{L})|>\varepsilon\sqrt{L\,\sigma^{2}_{L}(\varphi_{L})}\Big{\\}}$
$\displaystyle\;\leqslant\;\frac{1}{z_{L}(\varphi_{L})\,\sigma^{2}_{L}(\varphi_{L})}\sum_{n=K_{\varepsilon,L}}^{\infty}n^{2}w_{L}(n)\varphi_{L}^{n}\,,$

where
$K_{\varepsilon,L}=\varepsilon\sqrt{L\,\sigma^{2}_{L}(\varphi_{L})}+R_{L}(\varphi_{L})$,
which diverges like $\sqrt{L}$. The denominator in the above expression
converges to a positive constant. For the numerator, we observe

$\displaystyle\bigg{|}\sum_{n=K_{\varepsilon,L}}^{\infty}n^{2}(w_{L}(n)-w(n))\varphi_{L}^{n}\bigg{|}\;\leqslant\;\|w_{L}(n)-w(n)\|_{\infty}\sum_{n=K_{\varepsilon,L}}^{\infty}n^{2}\varphi_{L}^{n}\to
0\,,$ (55)

where we used property (51) of the sequence $(\varphi_{L})_{L}$. Thus, using
the second-moment assumption on $w$ in (B1), we have, for each $\epsilon$

$\displaystyle\sum_{n=K_{\varepsilon,L}}^{\infty}n^{2}w_{L}(n)\varphi_{L}^{n}\simeq\sum_{n=K_{\varepsilon,L}}^{\infty}n^{2}w(n)\varphi_{L}^{n}\;\leqslant\;\sum_{n=K_{\varepsilon,L}}^{\infty}n^{2}w(n)\to
0\,,$

as $L\to\infty$, since $K_{\epsilon,L}\to\infty$. This concludes the Lindeberg
condition. ∎

###### Proof of Proposition A.1.

We first consider the sub-critical and critical case together, fix
$\rho\in(0,\rho_{c}]$. Let $(\varphi_{L})_{L}$ be a sequence converging to
$\varphi:=\Phi(\rho)\in(0,1]$ satisfying (51) and (52). We will measure the
relative entropy between single-site marginals of
$\pi_{L,N}[\eta_{x}\in\cdot\,]$ and $\bar{\nu}_{\varphi_{L},L}$.

We start with an expression for the relative entropy between $\pi_{L,N}$ and
$\bar{\nu}_{\varphi_{L},L}^{\otimes L}$ which is used frequently in the proof
of similar equivalence of ensembles results (see for example [4]),

$\displaystyle\begin{split}H\left(\pi_{L,N}\mid\bar{\nu}_{\varphi_{L},L}^{\otimes
L}\right)&=\sum_{\eta\in\Omega_{L,N}}\pi_{L,N}[\eta]\log\left(\frac{\pi_{L,N}[\eta]}{\bar{\nu}_{\varphi_{L},L}^{\otimes
L}[\eta]}\right)=\log\left(\frac{z_{L}(\varphi_{L})^{L}}{Z_{L,N}\varphi_{L}^{N}}\right)\\\
&=-\log\bar{\nu}_{\varphi_{L},L}^{\otimes
L}\left[\sum_{x=1}^{L}\eta_{x}=N\right]\,.\end{split}$

Then, by subadditivity of the relative entropy we have for marginals

$H\left(\pi_{L,N}[\eta_{x}\in\cdot\,]\mid\bar{\nu}_{\varphi_{L},L}\right)\;\leqslant\;-\frac{1}{L}\log\bar{\nu}_{\varphi_{L},L}^{\otimes
L}\left[\sum_{x=1}^{L}\eta_{x}=N\right]\,.$ (56)

We estimate the right-hand side using the local limit theorem in Lemma A.3
with the specific choices of

$\displaystyle a_{L}:=L\,R_{L}(\varphi_{L})\quad\text{and}\quad
b_{L}:=\sqrt{L\,\sigma_{L}^{2}(\varphi_{L})}\,.$

It follows from Lemma A.4 that

$\sum_{x=1}^{L}\zeta_{x,L}=\frac{\sum_{x=1}^{L}\eta_{x,L}-a_{L}}{b_{L}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,1)\,.$

Moreover, $b_{L}$ diverges in the large $L$ limit because
$\lim_{L\to\infty}\sigma_{L}^{2}(\varphi_{L})=\sigma^{2}(\varphi)>0$. Also,

$\displaystyle\limsup_{L\to\infty}\frac{b_{L}^{2}}{Q_{L}}=\limsup_{L\to\infty}\frac{\sigma_{L}^{2}(\varphi_{L})}{q(\bar{\nu}_{\varphi_{L},L})}\;\leqslant\;\frac{\sigma^{2}(\varphi)}{\sup_{n\in\mathbb{N}_{0}}[w(n)\wedge
w(n+1)]}\,,$

where the final inequality follows by dominated convergence. The right hand
side is finite by assumption (B1). Therefore, we may apply the local limit
theorem stated in Lemma A.3 which yields

$\bar{\nu}_{\varphi_{L},L}^{\otimes
L}\left[\sum_{x=1}^{L}\eta_{x}=N\right]=O\left(L^{-1/2}\right)\,,$

and coming back to (56)

$H\left(\pi_{L,N}[\eta_{x}\in\cdot\,]\mid\bar{\nu}_{\varphi_{L},L}\right)\sim\frac{\log
L}{L}\to 0\quad\mbox{as }L\to\infty\,.$

With Pinsker’s inequality (see e.g. [21, Lemma 6.2]) this implies for the
total variation distance

$\displaystyle
d_{TV}\left(\pi_{L,N}[\eta_{x}\in\cdot\,],\bar{\nu}_{\varphi_{L},L}\right)\to
0\,.$ (57)

Also, by (52) and uniform convergence of the weights,
$\bar{\nu}_{\varphi_{L},L}$ converges weakly to $\bar{\nu}_{\varphi}$, which
together with (57) implies
$\pi_{L,N}[\eta_{x}\in\cdot\,]\stackrel{{\scriptstyle
d}}{{\rightarrow}}\bar{\nu}_{\Phi(\rho)}$ as
$N/L\to\rho\;\leqslant\;\rho_{c}$.

Finally, we conclude the super-critical case $\rho>\rho_{c}$ using a large
deviation estimate. Now let $\varphi_{L}$ be a sequence converging to $1$ and
satisfying (51) and (52) so that $R_{L}(\varphi_{L})\to\rho_{c}$. Then

$\begin{split}-\frac{1}{L}\log\bar{\nu}_{\varphi_{L},L}^{\otimes
L}\left[\sum_{x=1}^{L}\eta_{x}=N\right]&\;\leqslant\;-\frac{1}{L}\log\bar{\nu}_{\varphi_{L},L}^{\otimes
L\setminus\\{1\\}}\left[\sum_{x=2}^{L}\eta_{x}=N-\lfloor(\rho-\rho_{c})L\rfloor\right]\\\
&\qquad-\frac{1}{L}\log\bar{\nu}_{\varphi_{L},L}\left[\eta_{1}=\lfloor(\rho-\rho_{c})L\rfloor\right]\,,\end{split}$

where the first term on the r.h.s. converges to zero by the local central
limit theorem, since $(N-\lfloor(\rho-\rho_{c})L\rfloor)/L\to\rho_{c}$. For
the second term,

$\displaystyle-\frac{1}{L}\log\bar{\nu}_{\varphi_{L},L}\left[\eta_{1}=\lfloor(\rho-\rho_{c})L\rfloor\right]$
$\displaystyle\qquad=-\frac{1}{L}\log
w_{L}(\lfloor(\rho-\rho_{c})L\rfloor)-(\rho-\rho_{c})\log\varphi_{L}+\frac{1}{L}\log
z_{L}(\varphi_{L})\to 0\,,$

where convergence follows from the sub-exponential assumption (48), and since
$\varphi_{L}\to 1$ and $z_{L}(\varphi_{L})\to z(1)\in(0,\infty)$. It follows
that
$H\left(\pi_{L,N}[\eta_{x}\in\cdot\,]\mid\bar{\nu}_{\varphi_{L},L}\right)$
vanishes and, for the same reason as in the sub-critical case,
$\pi_{L,N}[\eta_{x}\in\cdot\,]$ converges weakly to $w=\bar{\nu}_{1}$.

Finally, to establish weak convergence of finite dimensional marginals; fix
$n_{1},\ldots,n_{m}\in\mathbb{N}_{0}$ and $x_{1},\ldots,x_{m}$ distinct
indices, then

$\displaystyle\pi_{L,N}$
$\displaystyle[\eta_{x_{1}}=n_{1},\eta_{x_{2}}=n_{2}]$
$\displaystyle=\frac{w_{L}(n_{1})Z_{L-1,N-n_{1}}}{Z_{L,N}}\frac{1}{Z_{L-1,N-n_{1}}}w_{L}(n_{2})\sum_{\xi\in\Omega_{L-2,N-n_{1}-n_{2}}}\left(\prod_{z}w_{L}(\xi_{z})\right)$
$\displaystyle=\pi_{L,N}[\eta_{x_{1}}=n_{1}]\,\pi_{L-1,N-n_{1}}[\eta_{x_{2}}=n_{2}]\,.$

This identity immediately generalises to

$\displaystyle\pi_{L,N}[\eta_{x_{1}}=n_{1},\ldots,\eta_{x_{m}}=n_{m}]=\prod_{j=1}^{m}\pi_{L-j+1,N-\sum_{k=1}^{j-1}n_{j}}[\eta_{x_{j}}=m_{j}]\,,$

which, by taking the thermodynamic limit on the right hand side, completes the
proof.

It is possible to drop the assumption that the variance of the limiting
weights is positive in the case $w(n)=\mathds{1}\\{n=0\\}$. If we fix
$\rho>0$, we can use the same argument as in the super-critical case above and
put all particles on a single site. In this case the right hand side of (56)
vanishes since

$\displaystyle-\frac{1}{L}\log\bar{\nu}_{\varphi_{L},L}^{\otimes
L}\left[\sum_{x=1}^{L}\eta_{x}=0\right]=-\log\nu_{\varphi_{L},L}\left[\eta_{x}=0\right]\to\log
w(0)=0\,,$

as $L\to\infty$. In this case we do not use the local central limit theorem.
Otherwise, the proof remains unchanged. ∎

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