# A functional central limit theorem for Polaron path measures

Volker Betz  Volker Betz
FB Mathematik, TU Darmstadt<EMAIL_ADDRESS>and  Steffen
Polzer  Steffen Polzer
FB Mathematik, TU Darmstadt<EMAIL_ADDRESS>
> Abstract. The application of the Feynman-Kac formula to Polaron models of
> quantum theory leads to the path measure of Brownian motion perturbed by a
> pair potential that is translation invariant both in space and time. An
> important problem in this context is the validity of a central limit theorem
> in infinite volume. We show both the existence of the relevant infinite
> volume limits and a functional central limit theorem in a generality that
> includes the Fröhlich polaron for all coupling constants. The proofs are
> based on an extension of a novel method by Mukherjee and Varadhan.
>
> Keywords: Polaron, renewal process, functional central limit theorem, point
> process
>
> 2020 Math. Subj. Class.: 60F17, 60G55, 60K05, 81S40

##  1\. Introduction

The polaron is a model for a quantum particle interacting with a polar
crystal. The interaction affects at least two quantities of physical interest.
On the one hand, the ground state energy of the system is lowered when the
interaction is increased. On the other hand, the particle needs to drag along
the polarization when it moves, and thus appears heavier than it would be
without the interaction: it has an effective mass larger than its bare mass.
For a particular polaron model now known as the Fröhlich polaron, both of
these effects were investigated by Feynman in [Fey55] using his newly invented
path integral method. The central object in this context is the quantity

$z_{t}=\int\exp\Big{(}\alpha\int_{0}^{t}\int_{0}^{t}\frac{\,{\rm
e}^{-|s-r|}\,}{|\mathbf{x}_{s}-\mathbf{x}_{r}|}\,{\rm d}s\,{\rm
d}r\Big{)}{\mathcal{W}}_{0}({\rm d}\mathbf{x}),$ (1.1)

where ${\mathcal{W}}_{0}$ is three-dimensional Brownian motion, and
$\alpha\;\geqslant\;0$ is the coupling constant determining the strength of
the interaction between the particle and the polar crystal. For the ground
state energy, Feynman argues that there exists $E_{\alpha}\;\leqslant\;0$ such
that $z_{t}\sim\,{\rm e}^{-E_{\alpha}t}\,$ as $t\to\infty$, and that
$E_{\alpha}$ is then ground state energy of the polaron. For the effective
mass, he finds that it can be obtained by pinning the Brownian motion at
$\bar{x}$ at the final time $t$ and observing the dependence of $z_{t}$ on
$\bar{x}$ under these circumstances. More precisely, when $z_{t}(\bar{x})$ is
the value of the integral (1.1) with an additional factor of
$\delta(\mathbf{x}_{t}-\bar{x})$ in the integrand, then $z_{t}$ should behave
like $\exp(-E_{\alpha}t-m_{\alpha}\bar{x}^{2}/2t)$ for large $t$, where
$m_{\alpha}$ is the effective mass. Feynman then goes on to derive
quantitative estimates for the ground state energy and the effective mass as
functions of $\alpha$. Among other things, he finds that
$E_{\alpha}\;\leqslant\;-\alpha-1.23(\alpha/10)^{2}$, and that $m_{\alpha}\sim
202(\alpha/10)^{4}$ for large $\alpha$, and remarks that these results compare
well with those obtained earlier by Pekar [Pek49] using the adiabatic
approximation.

The arguments of Feynman, while ingenious, are almost completely non-rigorous.
Since then, there have been many efforts to understand various aspects of the
polaron problem in a mathematically rigorous way. Most fundamental among them
is a firm connection between formula (1.1) and the underlying many-body
quantum system. This connection has emerged over the years in various forms,
and it is difficult to track it back to a single source. We refer to [Møl06]
for a review on functional analytic aspects of the polaron, and to [DS20] for
an outline on the connection to probability theory. For the convenience of the
reader, we also and include a short overview on the topic as an appendix to
the present paper.

The problem of the ground state energy was completely solved by Donsker and
Varadhan in [DV83] at least for the (physically most relevant) limit of large
$\alpha$. Using their large deviation techniques, they find that both
$\psi(\alpha):=\lim_{t\to\infty}\frac{1}{t}\log z_{t}$ and
$g_{0}:=\lim_{\alpha\to\infty}\psi(\alpha)/\alpha^{2}$ exist, and that $g_{0}$
can be calculated using the variational formula of Pekar [Pek49]. Lieb and
Thomas [LT97] give a functional analytic proof of the same result that in
addition yields explicit error estimates.

The problem of the effective mass turned out to be more difficult. Spohn
[Spo87] observed that the mathematically rigorous connection between the
effective mass and the quantity $z_{t}$ should appear through a central limit
theorem: for each $t>0$, one can read (1.1) as the normalization of a
probability measure ${\mathbb{P}}_{t}$ on path space, namely the perturbation
of Brownian motion by the exponentiated double integral; see formula (1.2)
below. The task is then to prove the existence of a limiting probability
measure ${\mathbb{P}}_{\infty}$ as $t\to\infty$, and a central limit theorem
under diffusive rescaling of ${\mathbb{P}}_{\infty}$. The emerging diffusion
constant is the inverse of the effective mass. There are some details that
need consideration, such as the sense in which the limit exists, given that
the interaction is translation invariant. While the treatment in [Spo87] is
still partly non-rigorous, recently Dybalski and Spohn [DS20] put the
connection between a central limit theorem and the effective mass on solid
mathematical ground.

This leaves the problem of actually showing the central limit theorem, and as
a prerequisite the existence of an infinite volume measure. A main obstacle is
the singularity in the integrand of the double integral that prevents e.g.
uniform estimates in the paths. In this paper, we show for the first time that
for all values of the coupling constant $\alpha$, infinite volume measures
related to (1.1) exist, and that a functional central limit theorem holds. We
also give an explicit expression for the diffusion constant that is explicit
enough to read off its strict positivity immediately.

Our results actually hold for more general models than the Fröhlich polaron.
As discussed in the Appendix, various polaron models of quantum theory give
rise to probability measures of the type

$\mathbb{P}_{\alpha,T}(\mathrm{d}\mathbf{x})=\frac{1}{Z_{\alpha,2T}}\exp\bigg{(}\frac{\alpha}{2}\int_{-T}^{T}\int_{-T}^{T}w(|t-s|,\mathbf{x}_{t}-\mathbf{x}_{s})\,\mathrm{d}s\mathrm{d}t\bigg{)}\mathcal{W}(\mathrm{d}\mathbf{x}),$
(1.2)

where $\alpha,T>0$, and where $w$ depends on the specific model,
$Z_{\alpha,2T}$ is the normalizing constant, or partition function, and
${\mathcal{W}}$ is the path measure of Brownian motion, although here we have
to be a bit careful: the translation invariance of the interaction means that
when we e.g. let ${\mathcal{W}}$ be the distribution of Brownian motion
started at time $-T$ in the origin, then the distribution of
$\mathbf{x}\mapsto\mathbf{x}_{0}$ under $\mathbb{P}_{\alpha,T}$ can not be
expected to converge. An elegant solution is to let ${\mathcal{W}}$ be the
measure of two-sided Brownian motion pinned to $0$ at time $0$, and to
consider the family $(\mathbb{P}_{\alpha,T})_{T\;\geqslant\;0}$ on the
$\sigma$-algebra $\mathcal{A}$ generated by the increments
$X_{s,t}(\mathbf{x})=\mathbf{x}_{t}-\mathbf{x}_{s}$. This choice gets rid of
the rather arbitrary pinning at $t=0$, and the measures on the restricted
$\sigma$-algebra will be shown to converge locally in the usual sense.

We call measures of the form (1.2) Polaron path measures. The precise
conditions (A1)-(A3) that we impose on $w$ will be stated at the beginning of
Section 2. The most important among them is the growth condition

$\limsup_{T\to\infty}\frac{Z_{\alpha,2T}}{Z_{\alpha,T}^{2}}<\infty$ (GC)

on the partition function. For suitable $w$, we find a number of equivalent
conditions to (GC), one of them being the existence and strict positivity of
$\lim_{T\to\infty}Z_{\alpha,T}\,{\rm e}^{-\psi(\alpha)T}\,$, where

$\psi(\alpha)=\lim_{T\to\infty}\frac{1}{T}\log Z_{\alpha,T}$

is the free energy associated to (1.2). Under these conditions, we show the
existence of an infinite volume limit ${\mathbb{P}}_{\alpha,\infty}$ of the
family $(\mathbb{P}_{\alpha,T})_{T\;\geqslant\;0}$. Under the additional
condition that $x\mapsto w(t,x)$ is quasiconcave for all $t\;\geqslant\;0$ we
show a functional central limit theorem for ${\mathbb{P}}_{\alpha,\infty}$
under diffusive rescaling.

Central limit theorems for similar models have been proved before. [BS04]
focuses on Polaron path measures that originate from quantum polaron models.
The method rests on not integrating out the field, and instead working
directly with a Markov process on an infinite dimensional state space; the
central limit theorem is then obtained by studying the quantum field as seen
from the moving particle, and by applying a suitable theory of Kipnis and
Varadhan [KV86]. The method needs, among other assumptions, that $w$ is
bounded, and thus does not work for the Fröhlich polaron.

Gubinelli [Gub06] treats general Polaron path measures, and implements an
approach that had already been proposed in [Spo87]: By cutting $[-T,T]$ into
blocks of finite length and considering the space of continuous functions on
each block, he obtains an one-dimensional system of infinite dimensional
spins, where all spins interact with each other through the double integral,
but in case of sufficiently fast decay of $t\mapsto w(t,x)$, that interaction
becomes weak for distant blocks. Dobrushins theory of one-dimensional spin
systems [Dob68, Dob70] then guarantees existence and uniqueness of the
infinite volume limit, and the central limit theorem is a consequence of
sufficiently fast decay of correlations between distant blocks. The conditions
on $w$ are weaker than those from [BS04] in some aspects, but stronger in
others. They also need uniformly bounded potentials and thus exclude the
Fröhlich polaron. Also, the coupling constant $\alpha$ needs to be
sufficiently small for Dobrushins method to work.

The method of Mukherjee [Muk20] also relies on cutting the space of functions
into blocks. He uses it to construct a discrete time Markov chain (with
infinite dimensional state space) that describes the behaviour of the path in
a given block given its behaviour in the previous block, and shows that this
Markov chain fulfills the conditions for a generalized Perron-Frobenius
argument. The central limit theorem is then derived as a consequence of the
CLT for additive functionals of stationary Markov chains. The method works
most cleanly when $t\mapsto w(t,x)$ has compact support, in which case it can
also handle some specific unbounded $w$. It also works when
$\sup_{x}|w(t,x)|\;\leqslant\;Ct^{-2-\delta}$ for constants $C,\delta>0$. Both
conditions exclude the Fröhlich polaron.

Important progress was achieved recently by Mukherjee and Varadhan in the work
[MV19] mentioned already above. The key idea is not technically difficult, but
ingenious: for the case of the Fröhlich polaron, one expands the exponential
in (1.2) into a Taylor series, exchanges the order of integration, and
interprets (1.2) as a mixture of Gaussian measures, where the mixing measure
is a point process on the space of (possibly overlapping) finite subintervals
of ${\mathbb{R}}$. The results of the paper are the existence of
${\mathbb{P}}_{\alpha,\infty}$ and a central limit theorem, valid for either
large enough or small enough $\alpha$. The paper is strictly devoted to the
Fröhlich polaron, and many calculations are specific to the precise form of
$w(t,x)=\,{\rm e}^{-t}\,/|x|$, in particular for writing it as a mixture of
Gaussian measures. Also, the central limit theorem is proved as a ’diagonal
limit’, where the diffusive scaling is applied at the same time as $T$ is sent
to infinity.

We find that the method of [MV19] can be generalized significantly, and
thereby becomes both conceptually simpler and more powerful. The central
object of our theory is a Gibbs measure on marked partitions of the real line,
and it is the infinite volume limit of this measure that needs to be
understood in order to study limits and properties of the measure (1.2). Using
the mixing properties of the limit measure, we prove a full functional central
limit theorem, with the diffusive scaling applied after taking the infinite
volume limit. Its proof relies on mixing properties of the limiting Gibbs
measure on marked partitions and does not need ${\mathbb{P}}_{\alpha,T}$ to be
a mixture of Gaussian measures. Another important result of our paper concerns
the conditions under which the central limit theorem holds. [MV19] give a
sufficient condition (see Condition (G) below) for the validity of their
results, which is however not easy to check in general. We show that condition
(G) is equivalent to several natural conditions, one of them being (GC). In
particular, this allows us to treat the Fröhlich polaron for all coupling
constants with very little additional effort. We also find that the critical
parameter $\lambda$ in (G) is equal to $\psi(\alpha)-\alpha$, which leads to
some intriguing relations between the Gibbs measure on marked partitions, the
ground state energy $E_{\alpha}(0)$ of the quantum polaron, and the overlap
$\langle\Omega,\Psi_{\alpha}\rangle$ between the Fock vacuum $\Omega$ and the
ground state $\Psi_{\alpha}$ of the polaron. We do not explore these relations
in depth in the present paper, but we present a few interesting calculations
in Section 7 below.

The paper is organised as follows: in Section 2, we present our results and
compare them to the results available in the literature. In Section 3 we give
an outline of the proof and discuss where we follow the ideas of [MV19] and
where we go beyond them. Section 4 is the main technical part of the paper:
here we develop a general theory of Gibbs measures on (marked) partitions of
the real line and discuss conditions for existence of infinite volume
measures. Section 5 contains the proof of the central limit theorem. In
Section 6 we prove a sufficient conditions for the validity of (GC), and in
Section 7 we discuss how various quantities for polaron models are connected
to the measures from Section 4.

##  2\. Results

Consider the Polaron path measure (1.2) defined on
$(C({\mathbb{R}},{\mathbb{R}}^{d}),{\mathcal{A}})$, where ${\mathcal{A}}$ is
the $\sigma$-algebra generated by the increment maps

$X_{s,t}:C(\mathbb{R},\mathbb{R}^{d})\to{\mathbb{R}}^{d},\quad\mathbf{x}\mapsto
X_{s,t}(\mathbf{x}):=\mathbf{x}_{s,t}\coloneqq\mathbf{x}_{t}-\mathbf{x}_{s}$

for $s,t\in{\mathbb{R}}$, and where ${\mathcal{W}}$ is the distribution of
Brownian increments (i.e. the restriction of the distribution of a two-sided
Brownian motion to $\mathcal{A}$). For the function $w$ and the parameter
$\alpha$ we make the following assumptions:
(A1): $w$ is measurable, positive (allowing the value $+\infty$), and fulfills
$w(t,-x)=w(t,x)$ for all $t>0$ and $x\in\mathbb{R}^{d}$.
(A2): $Z_{\alpha,T}<\infty$ for all $T\;\geqslant\;0$.
(A3): $\limsup_{T\to\infty}\frac{Z_{\alpha,2T}}{Z_{\alpha,T}^{2}}<\infty$,
i.e. (GC) holds.

######  Remark 2.1.

The positivity condition in (A1) can be relaxed: let
$f:[0,\infty)\to[0,\infty)$ be a measurable function with
$\int_{0}^{\infty}(1+t)f(t)\,{\rm d}t<\infty$. The measures
${\mathbb{P}}_{\alpha,T}$ are invariant under replacing $w$ with $(t,x)\mapsto
w(t,x)+f(t)$ in the double integral, and a a short calculation shows that (GC)
holds after the transformation if and only if it held before the
transformation. Therefore, it is enough to assume that there exists $f$ with
the above integrability condition such that $w+f$ is positive. In other words,
as long as $w$ is bounded below and its negative part decays sufficiently fast
at infinity uniformly in $x$, our results hold. The well-definedness condition
(A2) is trivial for bounded $w$, but becomes an issue for unbounded $w$ such
as in the Fröhlich polaron. While Proposition 2.4 contains some sufficient
conditions under which (A2) holds, we refer to the work of Bley and Thomas
[BT17] for much more general, quantitative estimates on the partition
functions of Feynman-Kac type perturbations of Brownian motion, as well as an
overview over the literature on this topic.

On $(C(\mathbb{R},\mathbb{R}^{d}),\mathcal{A})$ , we say that a family
$(\mathcal{P}_{t})_{t>0}$ of probability measures converges to a probability
measure $\mathcal{P}$ locally in total variation if for all $a<b$ the
restriction of $\mathcal{P}_{t}$ to

$\mathcal{A}_{a}^{b}\coloneqq\sigma\big{(}X_{s,t}:\,s,t\in[a,b]\big{)}$

converges in total variation to the restriction of $\mathcal{P}$ to
$\mathcal{A}_{a}^{b}$ as $t\to\infty$.

######  Theorem 2.2 (Existence of infinite volume measure).

Assume (A1)–(A3). Then there exists a measure $\mathbb{P}_{\alpha,\infty}$ on
$(C(\mathbb{R},\mathbb{R}^{d}),\mathcal{A})$ such that
$\mathbb{P}_{\alpha,T}\to\mathbb{P}_{\alpha,\infty}$ locally in total
variation as $T\to\infty$.

There is an explicit representation for $\mathbb{P}_{\alpha,\infty}$, which
however requires us to introduce and explain several further notions. We refer
to Theorem 3.3 below for details.

For stating the central limit theorem, let

$X_{t}\coloneqq X_{0,t},\qquad\text{and}\qquad
X^{n}_{t}\coloneqq\frac{1}{\sqrt{n}}X_{nt}$

for $n\in{\mathbb{N}}$ and $t\in\mathbb{R}$. Recall that a function
$u:{\mathbb{R}}^{d}\to{\mathbb{R}}\cup\\{\infty\\}$ is called quasiconcave if
its superlevel sets $u^{-1}([a,\infty])$, $a\in\mathbb{R}\cup\\{\infty\\}$ are
convex sets. An example for a quasiconcave function is $x\mapsto w(t,x)$ with
$w$ as in the Fröhlich polaron.

######  Theorem 2.3 (Functional central limit theorem).

Assume (A1)–(A3), and assume that $w(t,\cdot)$ is quasiconcave for all $t>0$.
Then there exists a positive definite matrix $\Sigma\in\mathbb{R}^{d\times d}$
with $\Sigma\;\leqslant\;I_{d}$ (in the sense of quadratic forms) such that
the distribution of $X^{n}$ under $\mathbb{P}_{\alpha,\infty}$ converges
weakly to the distribution of $\sqrt{\Sigma}X$ under $\mathcal{W}$ as
$n\to\infty$.

Again, there is an explicit representation for the covariance matrix $\Sigma$
that we do not have the notation and notions to state yet. We refer to Theorem
3.4 below.

We now discuss sufficient conditions for (GC). Let
$H:D(H)\subseteq\mathscr{H}\to\mathscr{H}$ be a self-adjoint, lower bounded
operator in some Hilbert space $\mathscr{H}$ with inner product
$\langle\cdot,\cdot\rangle$ and let $\Phi\in\mathscr{H}$. We say
$(Z_{\alpha,T})_{T\;\geqslant\;0}$ is of spectral type with Hamiltonian $H$
and state $\Phi$ if

$Z_{\alpha,T}=\langle\Phi,\,{\rm e}^{-TH}\,\Phi\rangle$

for all $T\;\geqslant\;0$. Clearly, this implies the validity of condition
(A2). If the bottom of the spectrum of $H$ is an eigenvalue, any normalized
eigenvector corresponding to that eigenvalue is called a ground state of $H$.
In this case, we say that $H$ has a ground state. We denote by $\nu_{H,\Phi}$
the spectral measure of $H$ with respect to $\Phi$, i.e.
$\nu_{H,\Phi}(A)=\langle\Phi,\mathds{1}_{A}(H)\Phi\rangle$ for all
$A\in\mathcal{B}(\mathbb{R})$.

######  Proposition 2.4.

linebreak
a) Assume that there exists a measurable function $f:[0,\infty)\to[0,\infty)$
satisfying $\int_{0}^{\infty}(1+t)f(t)\,\mathrm{d}t<\infty$ such that
$|w(t,x)|\;\leqslant\;f(t)$ for all $t\;\geqslant\;0$ and
$x\in\mathbb{R}^{d}$. Then assumption (A2) and (A3) are fulfilled.
b) Assume that (A1) holds and that $(Z_{\alpha,T})_{T\;\geqslant\;0}$ is of
spectral type with Hamiltonian $H$ and state $\Phi$ where
$H:D(H)\subseteq\mathscr{H}\to\mathscr{H}$ is a self-adjoint, lower bounded
operator and $\Phi\in\mathscr{H}$. Then

$\psi(\alpha)=-\inf\operatorname{supp}(\nu_{H,\Phi})$

and condition (A3) is satisfied if and only if $-\psi(\alpha)$ is an
eigenvalue of $H$ and $\Phi$ is non-orthogonal to the respective eigenspace.
In particular, if $H$ has a ground state $\Psi$ with
$\langle\Phi,\Psi\rangle\neq 0$ then (A3) is satisfied and $-\psi(\alpha)$ is
the ground state energy.
c) Assume that $w(t,\cdot)=\tilde{w}(t,|\cdot|)$ is rotationally symmetric and
positive and that $\tilde{w}(t,\cdot)$ is decreasing for all
$t\;\geqslant\;0$. Let
$\tilde{w}_{\beta}(r)\coloneqq\sup_{t\;\geqslant\;0}e^{\beta t}\tilde{w}(t,r)$
for $\beta,r\;\geqslant\;0$. Assume there exist $\beta>0$ and $p>1$ such that

$\begin{cases*}\int_{0}^{1}\tilde{w}_{\beta}(r)^{p}\,\mathrm{d}r<\infty\quad&\text{
in case $d=1$}\\\
\int_{0}^{1}r|\log(r)|\cdot\tilde{w}_{\beta}(r)^{p}\,\mathrm{d}r<\infty&\text{
in case $d=2$}\\\
\int_{0}^{1}r\cdot\tilde{w}_{\beta}(r)^{p}\,\mathrm{d}r<\infty&\text{ in case
$d\;\geqslant\;3$}.\end{cases*}$

Then Condition (A2) is fulfilled for all $\alpha\;\geqslant\;0$,
$\psi(\alpha)$ exists and is finite for all $\alpha\;\geqslant\;0$, and there
exists an $\alpha_{0}>0$ such Condition (A3) is fulfilled for all
$\alpha\;\leqslant\;\alpha_{0}$.

###### Proof.

For the proof of a), notice that for all $T>0$

$\int_{-T}^{T}\int_{-T}^{T}|w(t-s,X_{s,t})|\,{\rm d}s\,{\rm
d}t\;\leqslant\;\int_{-T}^{T}\int_{-T}^{T}f(|t-s|)\,{\rm d}s\,{\rm
d}t\;\leqslant\;4T\int_{0}^{\infty}f(\tau)\,{\rm d}\tau<\infty,$

and thus (A2) holds. The estimate

$\int_{-T}^{0}\int_{0}^{T}|w(t-s,X_{s,t})|\,\mathrm{d}t\,\mathrm{d}s\;\leqslant\;\int_{0}^{\infty}\int_{s}^{\infty}f(\tau)\,\mathrm{d}\tau\,\mathrm{d}s=\int_{0}^{\infty}\tau
f(\tau)\,\mathrm{d}\tau<\infty$ (2.1)

leads to

$Z_{\alpha,2T}\;\leqslant\;Z_{\alpha,T}^{2}\exp\bigg{(}\alpha\int_{0}^{\infty}\tau
f(\tau)\,\mathrm{d}\tau\bigg{)}$ (2.2)

for all $\alpha,T>0$. This shows (A3).

For b), let $E\coloneqq\inf\operatorname*{\text{supp}}(\nu_{H,\Phi})$. Since

$Z_{\alpha,T}=e^{-TE}\langle\Phi,e^{-T(H-E)}\Phi\rangle=e^{-TE}\int_{[E,\infty)}\,{\rm
e}^{-T(x-E)}\,\nu_{H,\Phi}({\rm d}x)$

holds for all $T\;\geqslant\;0$, we have

$\psi(\alpha)=\lim_{T\to\infty}\frac{1}{T}\log
Z_{\alpha,T}=-E+\lim_{T\to\infty}\frac{1}{T}\log\int_{[E,\infty)}\,{\rm
e}^{-T(x-E)}\,\nu_{H,\Phi}({\rm d}x)=-E,$

the last equality holding because for each $\varepsilon>0$, we have

$1\;\geqslant\;\int_{[E,\infty)}\,{\rm e}^{-T(x-E)}\,\nu_{\Phi}({\rm
d}x)\;\geqslant\;\,{\rm
e}^{-T\varepsilon}\,\int_{[E,E+\varepsilon)}\nu_{H,\Phi}({\rm d}x),$

with the integral on the right hand side being strictly positive. Thus

$\lim_{T\to\infty}Z_{\alpha,T}e^{-\psi(\alpha)T}=\lim_{T\to\infty}\int_{[E,\infty)}e^{-T(x-E)}\nu_{H,\Phi}({\rm
d}x)=\nu_{H,\Phi}(\\{E\\})$

by dominated convergence. Hence, the limit on the left hand side is positive
if and only if $E$ is an eigenvalue of $H$ and $\Phi$ is non-orthogonal to the
corresponding eigenspace. If the limit is positive,
$\lim_{T\to\infty}Z_{\alpha,2T}/Z_{\alpha,T}^{2}=(\lim_{T\to\infty}Z_{\alpha,T}e^{-\psi(\alpha)T})^{-1}$
and (GC) is satisfied. We will see in Theorem 3.2 that the other direction
holds as well: If (GC) is satisfied then
$\lim_{T\to\infty}Z_{\alpha,T}e^{-\psi(\alpha)T}$ exists and is positive.
The proof of part c) is more involved and is given in Section 6 below. ∎

######  Corollary 2.5.

Theorems 2.2 and 2.3 hold for the Fröhlich polaron, for all values of
$\alpha\;\geqslant\;0$.

###### Proof.

For the Fröhlich polaron, $(Z_{\alpha,T})_{T\;\geqslant\;0}$ is of spectral
type where $H=H_{\alpha}(0)$ is the Fröhlich Hamiltonian at total momentum
zero, and $\Phi=\Omega$ is the Fock vacuum; for instance, set $k=t=0$ in
formulae (2.11) and (2.14) of [DS20] to see this. It is well known that for
all coupling constants $\alpha>0$, $H(0)$ has a spectral gap and an unique (up
to a phase) ground state $\Psi_{\alpha}$, and that
$\langle\Psi_{\alpha},\Omega\rangle\neq 0$; again, we refer to [DS20]. We can
therefore apply Proposition 2.4 b) and obtain the result. ∎

The rather direct connection of Condition (GC) to quantum systems given in 2.4
b) has some interesting implications: it means that independent criteria for
the validity of (GC), such as those from Proposition 2.4 c) or from Theorem
3.2 below, can be used to decide about the existence of ground states in
quantum systems. Used in the other direction, this connection indicates that
the sufficient condition from Proposition 2.4 a) is very close to being also
necessary for the case of bounded $w$: for the massless Nelson model without
infrared cutoff it is known that $w$ is bounded with $w(t,0)\sim|t|^{-2}$, and
that the Hamiltonian has no ground state in Fock space; see Chapter 6.7 of
[LHB11]. Therefore (GC) cannot hold in this case.

Let us end this section by giving a more detailed comparison of our results
with those previously available. We will take Remark 2.1 into account, i.e.
that the assumption of positivity in (A1) can be weakened. Then Proposition
2.4 a) implies that (GC) holds for all cases considered in [BS04] as well as
those included in Assumption 2.1 of [Muk20]. For the alternative Assumption
2.2 of [Muk20], Proposition 2.4 c) guarantees that (GC) holds for small enough
$\alpha$ in all cases except the singular potential in one dimension. Also,
note that Assumption 2.2 of [Muk20] does not carry a restriction on the value
of $\alpha$.

For the central limit theorem, we require quasiconcavity of $w$, which is not
needed in [BS04] and in the cases from [Muk20] where $w$ is bounded.
Quasiconcavity is used for a short proof of both tightness in path space and
finite variance, via Gaussian correlation inequalities. We believe that there
should be other methods of achieving these goals whenever (GC) holds, but at
the moment we do not have them.

Gubinelli [Gub06] also requires $w$ to be bounded, but proves the existence of
an infinite volume limit (but not the central limit theorem) also in some
cases where the criterion of Proposition 2.4 a) does not hold. As discussed
above in the context of the massless Nelson model, we do not expect (GC) to
hold in general in such cases. In this context, the work [OS99] is also of
interest, where in a non-translation invariant version of the model with slow
decay of $t\mapsto w(x,t)$, the existence of at least two different infinite
volume limit measures is proved, depending on boundary conditions. Since we
believe that (A1) - (A3) should already be sufficient for a central limit
theorem, and since such a central limit theorem is not expected to hold in
situations where correlations are so strong that multiple Gibbs measures
exist, we conjecture that in generic cases of bounded potentials violating the
assumptions of Proposition 2.4 a), (GC) will not hold.

The results in [MV19] were obtained using a method that is related in some
aspects to the proof we will give for Proposition 2.4 c), although other
aspects differ significantly. An important common feature is that in both
methods, there is a restriction on the range of $\alpha$ that can be treated:
Proposition 2.4 c) only works for small $\alpha$, while the results in [MV19]
are stated for large enough or small enough $\alpha$. We believe that there is
a gap in the proof of Theorem 4.1 of [MV19]. Namely, it is unclear to us where
the factor of $\frac{\alpha}{\alpha+\lambda}$ comes from in formula (4.18) of
[MV19]; such a factor corresponds to the ’dormant’ period at the beginning of
a cluster, but if (4.18) were true, a dormant period would occur for each
point in the cluster, which is not the case. Without the presence of that
factor, however, the choice (4.19) for $\lambda(\alpha)$ is no longer possible
when $\alpha$ is large, and this creates a gap in the proof of Theorem 4.8 for
the case of large $\alpha$. On the other hand, our Corollary 2.5 shows that
the results of [MV19] are correct as stated and extends them to all
$\alpha\;\geqslant\;0$.

In conclusion, the extended method of Mukherjee and Varadhan in conjunction
with (GC) is a powerful tool to study infinite volume limits and functional
central limit theorems for Polaron path measures including the Fröhlich
polaron for all coupling strengths. One restriction of the method is that $w$
has to be bounded below; this is necessary for positivity properties of the
measure on partitions. Consequently, e.g. the ’anti-polaron’ $w(t,x)=-\,{\rm
e}^{-|t|}\,/|x|$, can not be treated. Another restriction is the requirement
of translation invariance. This excludes interesting cases, e.g. measures
corresponding to particles interacting both with a quantized field and an
external potential; see [Bet03, BHL+02, OS99], for instance.

##  3\. Outline of the proof

Let $\triangle=\\{(s,t)\in\mathbb{R}^{2}:s<t\\}$. We denote by
$(\mathbf{N}(\triangle),\mathcal{N}(\triangle))$ and
$(\mathbf{N}_{f}(\triangle),\mathcal{N}_{f}(\triangle))$ the state spaces of
point processes and finite point processes on $\triangle$ respectively. In the
following, we always assume that Assumptions (A1) and (A2) hold. We start by
modifying the approach taken by Mukherjee and Varadhan for the Polaron [MV19]:
By using the series expansion of the exponential function and exchanging the
order of integration, we write the measure $\mathbb{P}_{\alpha,T}$ as a
mixture of probability measures $\mathbf{P}_{\xi}$,
$\xi\in\mathbf{N}_{f}(\triangle)$ which have a product structure under non-
intersecting “clusters”. Contrary to [MV19] (where the mixing measure is the
distribution of a point process on $\triangle\times(0,\infty)$), our measures
$\mathbf{P}_{\xi}$ are in general not Gaussian (not even for the Fröhlich
polaron). We discuss the connection between both representations in Remark
4.14. It turns out to be useful to absorb a time damping factor into our
Poisson point process. For this, we choose a measurable function
$v:[0,\infty)\times\mathbb{R}^{d}\to(0,\infty]$ and a probability density
$g:[0,\infty)\to(0,\infty)$ with finite first moment such that

$w(t,x)=g(t)v(t,x)$ (3.1)

for all $t\;\geqslant\;0$ and $x\in\mathbb{R}^{d}$ (this representation is
clearly not unique). Let $\Gamma_{\alpha,T}$ be the distribution of a Poisson
point process on $\triangle$ with intensity measure

$\mu_{\alpha,T}(\mathrm{d}s\mathrm{d}t)\coloneqq\alpha\cdot
g(t-s)\mathds{1}_{\\{-T<s<t<T\\}}\mathrm{d}s\mathrm{d}t$

and $c_{\alpha,T}\coloneqq\mu_{\alpha,T}(\triangle)$. If $\tau_{1}$ is a radom
variable with density $g$, then

$c_{\alpha,T}\coloneqq\mu_{\alpha,T}(\triangle)=2\alpha
T-\alpha\int_{0}^{2T}\mathbb{P}(\tau_{1}>s)\,\mathrm{d}s$ (3.2)

(which will be relevant at a later point). We then have for $A\in\mathcal{A}$

$\displaystyle\mathbb{P}_{\alpha,T}(A)$
$\displaystyle=\frac{1}{Z_{\alpha,2T}}\int_{A}\mathcal{W}(\mathrm{d}\mathbf{x})\,\exp\left(\alpha\int_{-T}^{T}\mathrm{d}s\,\int_{s}^{T}\mathrm{d}t\,g(t-s)v\big{(}t-s,\mathbf{x}_{s,t}\big{)}\right)$
$\displaystyle=\frac{1}{Z_{\alpha,2T}}\sum_{n=0}^{\infty}\frac{1}{n!}\int_{\triangle^{n}}\mu_{\alpha,T}^{\otimes
n}(\mathrm{d}s_{1}\mathrm{d}t_{1},\ldots,\mathrm{d}s_{n}\mathrm{d}t_{n})\bigg{[}\int_{A}\mathcal{W}(\mathrm{d}\mathbf{x})\prod_{i=1}^{n}v\big{(}t_{i}-s_{i},\mathbf{x}_{s_{i},t_{i}}\big{)}\bigg{]}$
$\displaystyle=\frac{e^{c_{\alpha,T}}}{Z_{\alpha,2T}}\int_{\mathbf{N}_{f}(\triangle)}\Gamma_{\alpha,T}(\mathrm{d}\xi)\int_{A}\mathcal{W}(\mathrm{d}\mathbf{x})\prod_{(s,t)\in\operatorname{supp}(\xi)}v\big{(}t-s,\mathbf{x}_{s,t}\big{)}.$
(3.3)

Note that already here, we used the symmetry and nonngativity conditions from
Condition (A1). For
$\xi=\sum_{i=1}^{n}\delta_{(s_{i},t_{i})}\in\mathbf{N}_{f}(\triangle)$, we
define

$F(\xi)\coloneqq\mathbb{E}_{\mathcal{W}}\bigg{[}\prod_{i=1}^{n}v(t_{i}-s_{i},X_{s_{i},t_{i}})\bigg{]}$

and the measure $\mathbf{P}_{\xi}$ on
$(C(\mathbb{R},\mathbb{R}^{d}),\mathcal{A})$ by

$\mathbf{P}_{\xi}(\mathrm{d}\mathbf{x})\coloneqq\frac{1}{F(\xi)}\prod_{i=1}^{n}v(t_{i}-s_{i},\mathbf{x}_{s_{i},t_{i}})\,\mathcal{W}(\mathrm{d}\mathbf{x})$

(where $1/\infty\coloneqq 0$). Additionally, we define the measure
$\widehat{\Gamma}_{\alpha,T}$ on
$(\mathbf{N}_{f}(\triangle),\mathcal{N}_{f}(\triangle))$ by

$\widehat{\Gamma}_{\alpha,T}(\mathrm{d}\xi)\coloneqq\frac{e^{c_{\alpha,T}}}{Z_{\alpha,2T}}F(\xi)\,\Gamma_{\alpha,T}(\mathrm{d}\xi).$

We then have $F(\xi)<\infty$ for $\Gamma_{\alpha,T}$ almost all
$\xi\in\mathbf{N}_{f}(\triangle)$ (choose $A=C(\mathbb{R},\mathbb{R}^{d})$ in
(3)) and

$\mathbb{P}_{\alpha,T}(\cdot)=\int_{\mathbf{N}_{f}(\triangle)}\widehat{\Gamma}_{\alpha,T}(\mathrm{d}\xi)\,\mathbf{P}_{\xi}(\cdot).$
(3.4)

In particular $\mathbf{P}_{\xi}$ is a probability measure for
$\widehat{\Gamma}_{\alpha,T}$ almost all $\xi\in\mathbf{N}_{f}(\triangle)$ and
$\widehat{\Gamma}_{\alpha,T}$ is a probability measure. We have now
interchanged the task of proving convergence of the path measures
$\mathbb{P}_{\alpha,T}$ with proving convergence of the distributions of
suitable point processes. Notice that

$\mathbf{Z}_{\alpha,2T}\coloneqq\int_{\mathbf{N}_{f}(\triangle)}\Gamma_{\alpha,T}(\mathrm{d}\xi)\,F(\xi)=Z_{\alpha,2T}e^{-c_{\alpha,T}}$
(3.5)

acts as a normalization constant in the definition of the reweighted measure
$\widehat{\Gamma}_{\alpha,T}$. One can view
$\xi=\sum_{i=1}^{n}\delta_{(s_{i},t_{i})}\in\mathbf{N}_{f}(\triangle)$ as a
collection of intervals $[s_{i},t_{i}]$, $1\;\leqslant\;i\;\leqslant\;n$. We
call $\xi$ a cluster, if these intervals are overlapping in the sense that
$\bigcup_{i=1}^{n}[s_{i},t_{i}]$ is an interval. For an interval
$I\subseteq\mathbb{R}$ we say that $\xi$ falls into $I$ if
$[s_{i},t_{i}]\subseteq I$ for all $1\;\leqslant\;i\;\leqslant\;n$. Due to
independence of increments under $\mathcal{W}$, the measures
$\mathbf{P}_{\xi}$ have a product structure along non-intersecting clusters:
Let $I_{1},\ldots,I_{n}$ be intervals such that for $i\neq j$ the intersection
$I_{i}\cap I_{j}$ contains at most one point. Let
$\xi_{1},\ldots,\xi_{n}\in\mathbf{N}_{f}(\triangle)$ be such that $\xi_{k}$
falls into $I_{k}$ for all $1\;\leqslant\;k\;\leqslant\;n$ and let
$\xi=\sum_{k=1}^{n}\xi_{k}$. Then

$F(\xi)=\prod_{k=1}^{n}F(\xi_{k}).$

If we additionally assume that $F(\xi_{k})<\infty$ for all
$k\in\\{1,\ldots,n\\}$ then the processes of increments $(X_{s,t})_{s,t\in
I_{1}},\ldots,(X_{s,t})_{s,t\in I_{n}}$ are independent under
$\mathbf{P}_{\xi}$ and for all $k\in\\{1,\ldots,n\\}$

$\mathbf{P}_{\xi}\circ\Big{(}(X_{s,t})_{s,t\in
I_{k}}\Big{)}^{-1}=\mathbf{P}_{\xi_{k}}\circ\Big{(}(X_{s,t})_{s,t\in
I_{k}}\Big{)}^{-1}.$

In particular, for $a<b$, $\xi\in\mathbf{N}_{f}(\triangle)$ and
$A\in\mathcal{A}_{a}^{b}$, the probability $\mathbf{P}_{\xi}(A)$ only depends
on all clusters of $\xi$ that intersect $[a,b]$. We denote by $R_{a}^{b}$ the
restriction to all clusters that intersect $[a,b]$, that is for
$\xi=\sum_{i\in I}\delta_{(s_{i},t_{i})}\in\mathbf{N}(\triangle)$

$R_{a}^{b}\xi\coloneqq\sum_{i\in J_{a}^{b}(\xi)}\delta_{(s_{i},t_{i})}\text{
with }J_{a}^{b}(\xi)\coloneqq\max\big{\\{}J\subseteq I:\,\bigcup_{i\in
J}[s_{i},t_{i}]\cup[a,b]\text{ is an interval}\big{\\}}.$

Then for all $A\in\mathcal{A}_{a}^{b}$

$\mathbb{P}_{\alpha,T}(A)=\int_{\mathbf{N}_{f}(\triangle)}\widehat{\Gamma}_{\alpha,T}(\mathrm{d}\xi)\,\mathbf{P}_{R_{a}^{b}\xi}(A).$
(3.6)

In Section 4, we prove Theorem 2.2 i.e. the existence of an infinite volume
measure by showing that for all $a<b$ the measures
$\widehat{\Gamma}_{\alpha,T}\circ(R_{a}^{b})^{-1}$ converge in total variation
as $T\to\infty$ provided that (GC) holds. We show:

######  Proposition 3.1.

Assume that Conditions (A1)-(A3) hold. Then there exists a stationary (with
respect to translations along the diagonal) measure
$\widehat{\Gamma}_{\alpha,\operatorname{st}}$ on
$(\mathbf{N}(\triangle),\mathcal{N}(\triangle))$ such that for all $a<b$ we
have
$\widehat{\Gamma}_{\alpha,T}\circ(R_{a}^{b})^{-1}\to\widehat{\Gamma}_{\alpha,\operatorname{st}}\circ(R_{a}^{b})^{-1}$
in total variation as $T\to\infty$.

We give a short summary of the argument used. Consider a Poisson point process
$\eta=\sum_{i=1}^{\infty}\delta_{(s_{i},t_{i})}$ with $s_{1}<s_{2}<\ldots$ on
$\triangle$ with intensity measure $\alpha
g(t-s)\mathds{1}_{\\{-T<s<t\\}}\,\mathrm{d}s\mathrm{d}t$. Here $s_{i}$ and
$t_{i}$ can be interpreted at the arrival and departure of the $i$-th customer
of a $M/G/\infty$-queue (Poisson arrivals, general service time distribution,
infinitely many servers) started empty at time $-T$ with arrival intensity
$\alpha$ and service time distribution with density $g$. Conditionally on the
event that no customer is present at $T$, the process of all customers
arriving between $-T$ and $T$ has distribution $\Gamma_{\alpha,T}$. We can
decompose the $M/G/\infty$-queue into successive busy cycles (a dormant
period, in which no customers are present, followed by an active period in
which at least one customer is present). We denote by $\xi_{1}$ the first
cluster of the queue, i.e. the process of all customers arriving during the
first busy cycle, shifted in time such that the first arrival is at time zero.
We denote by $T_{1}$ the length of the first busy cycle, i.e. the sum of the
first dormant and the first active period. As identified by Mukherjee and
Varadhan in [MV19] for the Polaron measure, the condition

$\text{There exists a }\lambda\in\mathbb{R}\text{ such that
}\mathbb{E}_{\alpha}\big{[}e^{-\lambda T_{1}}F(\xi_{1})\big{]}=1\text{ and
}\mathbb{E}_{\alpha}\big{[}T_{1}e^{-\lambda T_{1}}F(\xi_{1})\big{]}<\infty$
(G)

is sufficient to conclude Proposition 3.1. In Section 4 we show this in much
greater generality, not needing any specific properties of the distribution
$\Xi_{\alpha}$. The argument is as follows: The condition
$\mathbb{E}_{\alpha}\big{[}e^{-\lambda T_{1}}F(\xi_{1})\big{]}=1$ guarantees
that
$\widehat{\Xi}_{\alpha}(\mathrm{d}\xi)\coloneqq\tfrac{\alpha}{\alpha+\lambda}e^{-\lambda
a(\xi)}F(\xi)\Xi_{\alpha}(\mathrm{d}\xi)$ defines a probability measure (where
$a(\xi)$ denotes the length of the active period of $\xi$). As the sum of all
active and dormant periods intersecting $[-T,T]$ is $2T$ (up to a correction
term that is irrelevant due to memorylessness) we can multiply $F$ with
factors $e^{-\lambda a_{i}}$ and $e^{-\lambda d_{j}}$ for each active period
$a_{i}$ and each dormant period $d_{j}$ intersecting $[-T,T]$ without changing
the Gibbs measure $\widehat{\Gamma}_{\alpha,T}$. As $F$ is multiplicative in
the clusters, the measure $\widehat{\Gamma}_{\alpha,T}$ can hence be obtained
by simply reweighting the distribution of exponentially distributed dormant
periods and $\Xi_{\alpha}$ distributed clusters: Starting in $-T$ we alternate
independently drawn $\operatorname{Exp}(\alpha+\lambda)$ distributed dormant
periods with $\widehat{\Xi}_{\alpha}$ distributed clusters. Conditionally on
the event that no customer is present at $T$, the process of customers
arriving between $-T$ and $T$ has distribution $\widehat{\Gamma}_{\alpha,T}$.
The second condition $\mathbb{E}_{\alpha}\big{[}T_{1}e^{-\lambda
T_{1}}F(\xi_{1})\big{]}<\infty$ yields the finiteness of the expected value of
the cluster length for the reweighted cluster distribution. This allows us to
apply renewal theory to deduce the existence of a stationary version
$\widehat{\Gamma}_{\alpha,\text{st}}$ of the process obtained by alternating
the reweighted distributions of dormant periods and active clusters. Due to
the memorylessness property of the exponential distribution, the measure
$\widehat{\Gamma}_{\alpha,T}$ can alternatively be seen as the process of all
customers under $\widehat{\Gamma}_{\alpha,\text{st}}$ that arrive between $-T$
and $T$ conditionally on the boundary condition that the system is dormant at
$-T$ and $T$. Unsurprisingly, locally the effect of the boundary condition
vanishes as $T\to\infty$. Condition (G) is rather difficult to verify
directly, but we find that it is equivalent to several natural conditions,
including (GC). Remember that we denote by
$\psi(\alpha)\coloneqq\lim_{T\to\infty}\log(Z_{\alpha,T})/T\in\mathbb{R}\cup\\{\infty\\}$
the free energy (the limit exists by the version of Feketes lemma for
measurable superadditive functions). We set $e^{-\infty}\coloneqq 0$. In
Section 4 we show:

######  Theorem 3.2.

Assume (A1) and (A2) hold. Then the following conditions are equivalent:

  1. (1)

There exists a $\lambda\in\mathbb{R}$ such that
$\mathbb{E}_{\alpha}\big{[}e^{-\lambda T_{1}}F(\xi_{1})\big{]}=1$ and
$\mathbb{E}_{\alpha}\big{[}T_{1}e^{-\lambda T_{1}}F(\xi_{1})\big{]}<\infty$

  2. (2)

$\limsup_{T\to\infty}Z_{\alpha,2T}/Z_{\alpha,T}^{2}<\infty$

  3. (3)

$\liminf_{T\to\infty}\widehat{\Gamma}_{\alpha,T}(\text{the system is dormant
at }0)>0$

  4. (4)

$\liminf_{T\to\infty}Z_{\alpha,T}e^{-\psi(\alpha)T}>0$.

If (1)-(4) hold then $\lambda=\psi(\alpha)-\alpha$ is the unique real number
satisfying (G) and

$\lim_{T\to\infty}Z_{\alpha,T}e^{-\psi(\alpha)T}=\frac{1}{\psi(\alpha)\mathbb{E}_{\alpha}[T_{1}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})]}.$

We prove Theorem 3.2 through another application of renewal theory. More
specifically, we use that the function $f:[0,\infty)\to\mathbb{R}$ defined by

$f(T)\coloneqq
e^{-(\psi(\alpha)-\alpha)T}\mathbf{Z}_{\alpha,T}\cdot\mathbb{P}_{\alpha}(\text{the
}M/G/\infty\text{-queue is dormant at $T$})$

satisfies the renewal equation

$f(T)=\mathbb{E}\big{[}\mathds{1}_{\\{T_{1}\;\leqslant\;t\\}}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})f(T-T_{1})\big{]}+e^{-\psi(\alpha)T}\quad\text{
for all }T\;\geqslant\;0.$

Combining the previous considerations, we obtain

######  Theorem 3.3 (Existence of infinite volume measure).

Assume that (A1)-(A3) hold. Then there exists a measure
$\mathbb{P}_{\alpha,\infty}$ on $(C(\mathbb{R},\mathbb{R}^{d}),\mathcal{A})$
such $\mathbb{P}_{\alpha,T}\to\mathbb{P}_{\alpha,\infty}$ locally in total
variation as $T\to\infty$. For any $a<b$ the restriction of
$\mathbb{P}_{\alpha,\infty}$ to $\mathcal{A}_{a}^{b}$ is given by

$\mathbb{P}_{\alpha,\infty}(A)=\int_{\mathbf{N}(\triangle)}\widehat{\Gamma}_{\alpha,\operatorname{st}}(\mathrm{d}\xi)\,\mathbf{P}_{R_{a}^{b}\xi}(A)$

for all $A\in\mathcal{A}_{a}^{b}$ where
$\widehat{\Gamma}_{\alpha,\operatorname{st}}$ is the stationary distribution
of the process obtained by alternating $\operatorname{Exp}(\psi(\alpha))$
distributed dormant periods and the distribution

$\widehat{\Xi}_{\alpha}(\mathrm{d}\xi)=\frac{\alpha}{\psi(\alpha)}e^{-(\psi(\alpha)-\alpha)a(\xi)}F(\xi)\,\Xi_{\alpha}(\mathrm{d}\xi)$

of active clusters.

For the Fröhlich Polaron path measure, Mukherjee and Varadhan [MV19] show (for
$\alpha$ such that (G) holds) an ordinary central limit theorem in the
“diagonal limit”, using that their counterparts $\mathbf{P}_{\xi,u}$ of our
measures $\mathbf{P}_{\xi}$ are Gaussian. We use a different approach that
allows us to show Theorem 2.3, i.e. a full functional central limit theorem,
for more general potentials $w$. Roughly speaking, we use the following rather
natural arguments (assuming that (GC) is satisfied):

  * •

Let $\hat{\eta}_{s}\sim\widehat{\Gamma}_{\alpha,\operatorname{st}}$ be a
stationary version of the process obtained by alternating the reweighted
dormant periods and clusters. Let $s_{1}<t_{1}<\ldots<s_{k}<t_{k}$. For large
$n$ the processes
$R_{ns_{1}}^{nt_{1}}\hat{\eta}_{s},\ldots,R_{ns_{k}}^{nt_{k}}\hat{\eta}_{s}$
are approximately independent due to the renewal structure. By the product
structure of the measures $\mathbf{P}_{\xi}$ along non-intersecting clusters,
the increments $X_{ns_{1},nt_{1}},\ldots,X_{ns_{k},nt_{k}}$ are approximately
independent under $\mathbb{P}_{\alpha,\infty}$ for large $n$.

  * •

For $a<b$ and large $n$, approximately
$n(b-a)/\hat{\mathbb{E}}_{\alpha}[\hat{T}_{1}]$ renewal points fall into
$[na,nb]$. By the product structure, the increment
$\frac{1}{\sqrt{n}}X_{an,bn}$ is under $\mathbb{P}_{\alpha,\infty}$
approximately the sum of $n(b-a)/\hat{\mathbb{E}}_{\alpha}[\hat{T}_{1}]$
independent increments.

  * •

To prove an ordinary central limit theorem, we still need that the variance of
an increment along a single renewal period is finite. For the functional
central limit theorem, we additionally need tightness of the family of
probability measures
$\\{\mathbb{P}_{\alpha,\infty}\circ(X^{n})^{-1}:\,n\in{\mathbb{N}}\\}$. We
assume that $w(t,\cdot)$ is quasiconcave for all $t>0$ and apply the Gaussian
correlation inequality in order to show these two prerequisites.

While the assumption of quasiconcavity of $w(t,\cdot)$ is satisfied for the
Fröhlich polaron and several other cases, it is conceivable that at least the
finite variance along a single renewal period (and thus the ordinary central
limit theorem) can be obtained under weaker assumptions on $w$. We do not
pursue this any further here and settle for:

######  Theorem 3.4 (Functional Central limit theorem).

Assume that (A1)-(A3) hold, and that $w(t,\cdot)$ is quasiconcave for all
$t>0$. Then the distribution of $X^{n}$ under $\mathbb{P}_{\alpha,\infty}$
converges weakly to the distribution of $\sqrt{\Sigma}X$ under $\mathcal{W}$
where

$\Sigma\coloneqq\frac{\hat{\mathbb{E}}_{\alpha}[\Sigma(\hat{d}_{1},\hat{\xi}_{1})]}{\hat{\mathbb{E}}_{\alpha}[\widehat{T}_{1}]}\quad\text{with
}\Sigma(\hat{d}_{1},\hat{\xi}_{1})\coloneqq\hat{d}_{1}I_{d}+\mathbf{E}_{\hat{\xi}_{1}}\big{[}X_{0,\hat{a}_{1}}\cdot
X_{0,\hat{a}_{1}}^{T}\big{]},$ (3.7)

and
$(\hat{d}_{1},\hat{\xi}_{1})\sim\operatorname{Exp}(\psi(\alpha))\otimes\widehat{\Xi}_{\alpha}$,
$\hat{a}_{1}$ is the length of the cluster $\hat{\xi}_{1}$ and
$\widehat{T}_{1}=\hat{a}_{1}+\hat{d}_{1}$. Additionally,
$\Sigma\;\leqslant\;I_{d}$ in the sense of quadratic forms.

Here $\hat{d}_{1}I_{d}$ is the contribution of a dormant period on which the
process is a Brownian motion. Equation (3.7) directly entails that $\Sigma$ is
positive definite. In order to show that $\Sigma\;\leqslant\;I_{d}$, we use
quasiconcavity of $w(t,\cdot)$ and the Gaussian correlation inequality.

If we replace $w$ with the potential $(t,x)\mapsto w(t,x)+g(t)$ our partition
function becomes $Z_{\alpha,2T}e^{c_{\alpha,T}}$. By Equation (3.2),

$e^{c_{\alpha,T}}\sim e^{2\alpha T-\alpha\mathbb{E}[\tau_{1}]}$ (3.8)

as $T\to\infty$. Thus, adding $g$ to $w$ does not change whether (GC) holds or
not. After the transformation of the potential, we have $v>1$ and thus
$\mathbb{E}_{\alpha}[F(\xi_{1})]>1$, which turns out to be beneficial for
showing (G) for sufficiently small $\alpha$: Then, by the intermediate value
theorem, the condition $\mathbb{E}_{\alpha}[F(\xi_{1})]<\infty$ is sufficient
such that (G) is satisfied. We apply this in Section 6 and prove Proposition
2.4 c). Besides giving a similar, slightly stronger, upper estimate on $F$
(providing necessarily a different proof which does not rely on the specific
choice of $w$) our approach is different from that used in [MV19], replacing
the Markov chain argument with an application of the optional stopping
theorem, and thus using an argument that does not need $g$ to be the density
of an exponential distribution. At the end of Section 6 (see Section 7) we
will use the fact that $\psi(\alpha)-\alpha$ is the unique solution to (G)
(assuming that (GC) holds) and perform a few heuristic calculations that
connect the free energy to properties of the reweighted point process.

##  4\. Gibbs measures on partitions of $[-T,T]$

It will be notationally convenient to identify a point process on $\triangle$
that alternates between dormant periods and active clusters with a random
partition of $\mathbb{R}$ into dormant and active intervals, each active
interval beeing equipped with the corresponding cluster as a “state”. We will
consider general systems which alternate between a dormant state and (possible
multiple) active states. Starting dormant at time zero, the system switches
into an active state after an exponentially distributed waiting time. After an
independently drawn waiting time (whose distribution might depend on the
state) the system turns dormant again. Iterating this procedure generates a
partition of $[0,\infty)$, where each interval of the partition is in a
certain state. For a visualization see Figure 1. By an application of renewal
theory, there exists a stationary version of the process. We will use its
distribution as a reference measure for a certain class of Gibbs measures
under the dormant boundary condition. We will assume that dormant intervals do
not contribute energy and that the Hamiltonian is additive in the active
states. Given a growth condition on the partition function, we will see that
the Gibbs measures can be obtained by simply reweighting the distributions of
the independent components mentioned above. This yields the existence of an
infinite volume measure.

Before continuing, we give a short summary of some standard results from
renewal theory. A renewal process $(S_{n})_{n\;\geqslant\;0}$ is a sequence of
random variables such that the so called interarrival times
$(T_{n})_{n\;\geqslant\;1}\coloneqq(S_{n}-S_{n-1})_{n\;\geqslant\;1}$ form an
iid sequence of a.s. positive random variables and such that $S_{0}=0$ almost
surely. For $t\;\geqslant\;0$ let
$N(t)\coloneqq\\#\\{n:\,S_{n}\;\leqslant\;t\\}$ be the number of renewal
points in the interval $[0,t]$. The renewal function
$U:[0,\infty)\to[0,\infty)$, $U(t)\coloneqq\mathbb{E}[N(t)]$ is the expected
number of renewal points (including the origin) in the interval $[0,t]$. Then
it holds

######  Theorem 4.1.

[Asm03, p. 140] Irrespective whether $\mathbb{E}[T_{1}]$ is finite or infinite
(setting $1/\infty=0$) we have as $t\to\infty$

$N(t)/t\to 1/\mathbb{E}[T_{1}]\text{ almost surely and }U(t)/t\to
1/\mathbb{E}[T_{1}].$

Let $\mu$ be a Radon measure on $[0,\infty)$ with $\mu(\\{0\\})=0$. Let
$z:[0,\infty)\to\mathbb{R}$ be a locally bounded measurable function. Then the
renewal equation

$Z(t)=\int_{[0,t]}Z(t-s)\,\mu(\mathrm{d}s)+z(t)\quad\text{for all
}t\;\geqslant\;0$

has an unique locally bounded measurable solution
$Z_{z}:[0,\infty)\to\mathbb{R}$ given by the convolution of $z$ with the
renewal measure $\sum_{n=0}^{\infty}\mu^{*n}$. We will apply the renewal
theorem in the following form:

######  Theorem 4.2 (Renewal theorem).

[ANT78] If $\mu$ is a probability measure that is absolutely continuous with
respect to the Lebesgue measure and $g:[0,\infty)\to[0,\infty)$ is bounded and
Lebesgue integrable with $g(t)\to 0$ as $t\to\infty$ then

$\lim_{t\to\infty}\sup_{|z|\;\leqslant\;g}\Big{|}Z_{z}(t)-\frac{1}{\int_{0}^{\infty}s\,\mu(\mathrm{d}s)}\int_{0}^{\infty}z(s)\,\mathrm{d}s\Big{|}=0.$

For sub-probability measures, the following holds:

######  Proposition 4.3.

[Asm03, p. 163] Assume $\mu([0,\infty))<1$. If $z$ is bounded and
$\lim_{t\to\infty}z(t)$ exists then

$\lim_{t\to\infty}Z_{z}(t)=\frac{\lim_{t\to\infty}z(t)}{1-\mu([0,\infty))}.$

The forward recurrence time $A_{t}$ of the renewal process $(S_{n})_{n}$ at
time $t$ is the timespan from $t$ up to the next renewal after $t$, i.e.

$A_{t}\coloneqq\inf\\{S_{n}-t:\,n\in{\mathbb{N}}\text{ s.t. }S_{n}>t\\}.$

If $T_{1}$ has a density and $\mathbb{E}[T_{1}]<\infty$, one can apply the
renewal theorem in order to show that the distribution of $A_{t}$ converges in
total variation as $t\to\infty$.

Let $\mathcal{X}$ be a Polish space which will serve as our state space. Let
$\alpha>0$ and $\Xi$ be a probability distribution on
$(0,\infty)\times\mathcal{X}$. Let $(d_{n})_{n}$ be an iid sequence of
$\operatorname{Exp}(\alpha)$ distributed random variables and
$((a_{n},\chi_{n}))_{n}$ be an iid sequence of $\Xi$ distributed random
variables, independent of $(d_{n})_{n}$. We will always assume that $\Xi$ is
such that $\mathbb{E}[a_{1}]<\infty$. Here $d_{n}$ should be interpreted as
the $n$-th dormant period and $a_{n}$ as the $n$-th active period with state
$\chi_{n}$. If, for example, we look at the process that represents the
$M/G/\infty$-queue (started empty at time $0$) we choose
$\mathcal{X}=\mathbf{N}_{f}(\triangle)$ and $\chi_{n}=\xi_{n}$ to be the
process of customers arriving during the $n$-th busy period (shifted in time
such that the first customer arrives at time 0). For $n\in{\mathbb{N}}_{0}$,
we define

$X_{2n}\coloneqq\sum_{k=1}^{n}d_{k}+a_{k},\quad\quad X_{2n+1}\coloneqq
d_{n+1}+\sum_{k=1}^{n}d_{k}+a_{k},$

consider the point process

$\eta\coloneqq\sum_{n=1}^{\infty}\delta_{(X_{2n-1},X_{2n},\chi_{n})}$

on $\triangle\times\mathcal{X}$, and denote its distribution by
$\Gamma^{\Xi}_{\alpha}$. The point process shall be interpreted as “on the
interval $[X_{2n-1},X_{2n})$ the system is in state $\chi_{n}$ and on the
interval $[X_{2n},X_{2n+1})$ the system is dormant”. We call the renewal
process $(S_{n})_{n}\coloneqq(X_{2n})_{n}$ the embedded renewal process of
$\eta$, see Figure 1 for a visualization. For $n\in{\mathbb{N}}$ we denote by
$T_{n}\coloneqq d_{n}+a_{n}$ the $n$-th interarrival time of embedded renewal
process.

Figure 1. Example of a partition with statespace
$\mathcal{X}=\\{{\color[rgb]{1,0,0}\text{red}},{\color[rgb]{0,0,1}\text{blue}},{\color[rgb]{0,1,0}\text{green}}\\}$.
Dormant periods are in black.

For $t>0$ let

$\tau_{t}\coloneqq\inf\\{n:\,S_{n}\;\geqslant\;t\\}$

be the hitting time of $[t,\infty)$ of the embedded renewal process of $\eta$.
Notice that $\tau_{t}$ coincides with the number of renewal points in $[0,t)$
(counting the origin as a renewal point). The distribution of $T_{1}$ is
absolutely continuous with respect to the Lebesgue measure as $a_{1}$ and
$d_{1}\sim\operatorname{Exp}(\alpha)$ are independent. Since

$\mathbb{P}(\eta\text{ is dormant at
}t)=\mathbb{E}\big{[}\mathds{1}_{\\{T_{1}\;\leqslant\;t\\}}\mathbb{P}(\eta\text{
is dormant at }t-T_{1})\big{]}+\mathbb{P}(d_{1}>t)$

for all $t\;\geqslant\;0$ the renewal theorem implies

$\lim_{t\to\infty}\mathbb{P}(\eta\text{ is dormant at
}t)=\mathbb{E}[d_{1}]/\mathbb{E}[T_{1}].$

For $a\;\leqslant\;b$ let
$M_{a,b}\coloneqq\\{(s,t,x)\in\triangle\times\mathcal{X}:\,[s,t]\cap[a,b]\neq\emptyset\\}$
and let

$R_{a}^{b}:\mathbf{N}(\triangle\times\mathcal{X})\to\mathbf{N}(\triangle\times\mathcal{X}),\quad\mu\mapsto\mu(\,\cdot\,\cap
M_{a,b})$

be the restriction to all marked intervals that intersect the interval
$[a,b]$. Let us consider again the example of the process $\eta$ representing
the $M/G/\infty$-queue. Then $R_{a}^{b}\eta$ contains not only all individuals
that arrive or depart during $[a,b]$ but all clusters that intersect $[a,b]$.
In a similar manner, we define $R_{0}^{\infty}$ to be the restriction to all
marked intervals intersecting $[0,\infty)$. For $t>0$ we define

$\theta_{t}:\mathbf{N}(\triangle\times\mathcal{X})\to\mathbf{N}(\triangle\times\mathcal{X}),\quad\sum_{i\in
I}\delta_{(s_{i},t_{i},x_{i})}\mapsto\sum_{i\in
I}\delta_{(s_{i}-t,t_{i}-t,x_{i})}$

to be the translation by $-t$. We call a probability measure $\mathcal{P}$ on
$(\mathbf{N}(\triangle\times\mathcal{X}),\mathcal{N}(\triangle\times\mathcal{X}))$
stationary, if $\mathcal{P}=\mathcal{P}\circ\theta_{t}^{-1}$ for all
$t\in\mathbb{R}$. We call a point process on $\triangle\times\mathcal{X}$
stationary if its distribution is stationary. By an application of renewal
theory, we obtain the existence of a stationary version
$\Gamma^{\Xi}_{\alpha,\operatorname{st}}$ of $\Gamma^{\Xi}_{\alpha}$.

######  Proposition 4.4.

There exists a stationary point process $\eta^{s}$ on
$\triangle\times\mathcal{X}$ such that the distribution of
$R_{0}^{\infty}\theta_{t}\eta$ converges in total variation to the
distribution of $R_{0}^{\infty}\eta^{s}$ as $t\to\infty$. Additionally, we
have

$R_{0}^{\infty}\eta^{s}\overset{d}{=}\gamma+\theta_{-T_{0}}\eta$

where $(T_{0},\gamma)$ is a
$(0,\infty)\times\mathbf{N}_{f}(\triangle\times\mathcal{X})$ valued random
variable independent of $\eta$ with

$\mathbb{P}\big{(}(T_{0},\gamma)\in
C\big{)}=\frac{1}{\mathbb{E}[T_{1}]}\int_{0}^{\infty}\mathbb{P}\big{(}(T_{1}-t,\theta_{t}R_{t}^{T_{1}}\eta)\in
C\big{)}\,\mathrm{d}t$ (4.1)

for all
$C\in\mathcal{B}((0,\infty))\otimes\mathcal{N}_{f}(\triangle\times\mathcal{X})$.

###### Proof.

First, we show convergence of the distribution of
$R_{0}^{\infty}\theta_{t}\eta$ as $t\to\infty$. The proof follows the general
idea: “Condition with respect to a suitable chosen initial condition and show
convergence of the distribution of the initial condition”, see e.g. [DVJ08,
p.224 ff.]
If $A_{t}$ is the forward recurrence time of the embedded renewal process at
$t$ and $\gamma_{t}\coloneqq\theta_{t}R_{t}^{t+A_{t}}\eta$ is the re-shifted
marked interval ending in $t+A_{t}$ then we have for all
$B\in\mathcal{N}(\triangle\times\mathcal{X})$

$\mathbb{P}(R_{0}^{\infty}\theta_{t}\eta\in
B)=\int_{(0,\infty)\times\mathbf{N}_{f}(\triangle\times\mathcal{X})}\mathbb{P}(\zeta+\theta_{-s}\eta\in
B)\,\big{(}\mathbb{P}\circ(A_{t},\gamma_{t})^{-1}\big{)}(\mathrm{d}s\,\mathrm{d}\zeta).$

Now for all
$C\in\mathcal{B}((0,\infty))\otimes\mathcal{N}_{f}(\triangle\times\mathcal{X})$
and $t\;\geqslant\;0$

$\mathbb{P}((A_{t},\gamma_{t})\in
C)=\mathbb{E}\big{[}\mathds{1}_{\\{T_{1}\;\leqslant\;t\\}}\mathbb{P}\big{(}(A_{t-T_{1}},\gamma_{t-T_{1}})\in
C\big{)}\big{]}+\mathbb{P}\big{(}(T_{1}-t,\theta_{t}R_{t}^{T_{1}}\eta)\in
C\big{)}.$

By the renewal theorem (with the majorant $g(t)\coloneqq\mathbb{P}(T_{1}>t)$
for $t\;\geqslant\;0$), the distribution of $(A_{t},\gamma_{t})$ converges in
total variation to the distribution defined by the right hand side of (4.1) as
$t\to\infty$. This implies the convergence of the distribution of
$R_{0}^{\infty}\theta_{t}\eta$ in total variation as $t\to\infty$. By
construction, the limit $\mathcal{P}$ satisifes
$\mathcal{P}\circ(R_{0}^{\infty}\theta_{t})^{-1}=\mathcal{P}$ for all $t>0$
and can be extended to a stationary distribution on
$\mathbf{N}(\triangle\times\mathcal{X})$. ∎

We denote by $\Gamma_{\alpha,\operatorname{st}}^{\Xi}$ the distribution of
$\eta^{s}$. Below we will use $\Gamma_{\alpha,\operatorname{st}}^{\Xi}$ as a
reference measure for certain Gibbs measures. For $t>0$ we denote by
$\Gamma^{\Xi}_{\alpha,t}$ the distribution of $R_{-t}^{t}\theta_{t}\eta$
conditionally on $\\{\eta\text{ is dormant at $2t$}\\}$. Notice that, by the
memorylessness property of the exponential distribution,
$\Gamma^{\Xi}_{\alpha,t}$ is also the distribution of the process
$R_{-t}^{t}\eta^{s}$ conditionally on $\\{\eta^{s}\text{ is dormant at $-t$
and $t$}\\}$. Unsurprisingly, locally the effect of the boundary condition
vanishes as $t\to\infty$. Given probability measures $(\mathcal{P}_{t})_{t}$,
$\mathcal{P}$ on
$\big{(}\mathbf{N}(\triangle\times\mathcal{X}),\mathcal{N}(\triangle\times\mathcal{X})\big{)}$
we say $\mathcal{P}_{t}\to\mathcal{P}$ locally in total variation if, for any
$a<b$, we have
$\mathcal{P}_{t}\circ(R_{a}^{b})^{-1}\to\mathcal{P}\circ(R_{a}^{b})^{-1}$ in
total variation as $t\to\infty$.

######  Proposition 4.5.

We have $\Gamma^{\Xi}_{\alpha,t}\to\Gamma_{\alpha,\operatorname{st}}^{\Xi}$
locally in total variation as $t\to\infty$.

###### Proof.

Let $a<b$. To shorten notation, we denote for $t>0$ by
$\sigma_{t}\coloneqq\tau_{t+b}$ the hitting time of $[t+b,\infty)$ of the
embedded renewal process of $\eta$ and set

$N_{t}(\eta)\coloneqq\begin{cases}0\quad&\text{if $\eta$ is dormant at $t$}\\\
1\quad&\text{if $\eta$ is active at $t$}.\end{cases}$

As a consequence of the renewal theorem, we have

$\mathbb{P}(N_{t}(\eta)=0)\to\mathbb{E}[d_{1}]/\mathbb{E}[T_{1}]\eqqcolon c$
(4.2)

as $t\to\infty$. Let $\varepsilon>0$ and let $t_{1}>0$ be such that for all
$t\;\geqslant\;t_{1}$

$\mathbb{P}(N_{2t}(\eta)=0)\;\geqslant\;c/2,\quad|\mathbb{P}(N_{2t}(\eta)=0)-c|\;\leqslant\;\varepsilon.$
(4.3)

For $t\;\geqslant\;0$ let $s_{t}\coloneqq b+t+\sqrt{t}$. Let
$t_{2}\;\geqslant\;0$ be such that for all $t\;\geqslant\;t_{2}$ we have
$t-b-\sqrt{t}\;\geqslant\;t_{1}$. For $t\;\geqslant\;t_{2}$ we then have on
$\\{S_{\sigma_{t}}\;\leqslant\;s_{t}\\}$ that
$2t-S_{\sigma_{t}}\;\geqslant\;t_{1}$ holds. By conditioning with respect to
the process up the first renewal after $b+t$, we get for all
$A\in\mathcal{N}(\triangle\times\mathcal{X})$ and $t\;\geqslant\;t_{2}$

$\displaystyle\big{|}\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A,S_{\sigma_{t}}\;\leqslant\;s_{t},N_{2t}(\eta)=0)-\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A,S_{\sigma_{t}}\;\leqslant\;s_{t})\mathbb{P}(N_{2t}(\eta)=0)\big{|}$
$\displaystyle\;\leqslant\;\int_{\Omega}\mathbb{P}(\mathrm{d}\omega)\,\Big{[}\mathds{1}_{\\{S_{\sigma_{t}}\;\leqslant\;s_{t}\\}}(\omega)\big{|}\mathbb{P}(N_{2t-S_{\sigma_{t}}(\omega)}(\eta)=0)-\mathbb{P}(N_{2t}(\eta)=0)\big{|}\,\Big{]}$
$\displaystyle\;\leqslant\;2\varepsilon.$

On $\\{S_{\sigma_{t}}>s_{t}\\}$ the forward recurrence time at $t+b$ is at
least $\sqrt{t}$. By convergence of the distribution of the forward recurrence
time there exists a $t_{3}\;\geqslant\;t_{2}$ such that for all
$t\;\geqslant\;t_{3}$

$\mathbb{P}(S_{\sigma_{t}}>s_{t})\;\leqslant\;\varepsilon.$

We then have for all $t\;\geqslant\;t_{3}$

$\displaystyle\big{|}\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A,N_{2t}(\eta)=0)-\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A)\mathbb{P}(N_{2t}(\eta)=0)\big{|}$
$\displaystyle\;\leqslant\;\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A,S_{\sigma_{t}}>s_{t},N_{2t}(\eta)=0)$
$\displaystyle+|\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A,S_{\sigma_{t}}\;\leqslant\;s_{t},N_{2t}(\eta)=0)-\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A,S_{\sigma_{t}}\;\leqslant\;s_{t})\cdot\mathbb{P}(N_{2t}(\eta)=0)|$
$\displaystyle+\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A,S_{\sigma_{t}}>s_{t})\cdot\mathbb{P}(N_{2t}(\eta)=0)$
$\displaystyle\;\leqslant\;4\varepsilon$

and hence

$|\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A|N_{2t}(\eta)=0)-\mathbb{P}(R_{a}^{b}\theta_{t}\eta\in
A)|\;\leqslant\;\frac{4\varepsilon}{\mathbb{P}(N_{2t}(\eta)=0)}\;\leqslant\;\frac{4\varepsilon}{c/2}.$

Combining this with Proposition 4.4 yields the claim. ∎

We will now consider Gibbs measures with respect to the stationary
distribution $\Gamma_{\alpha,\text{st}}^{\Xi}$ under dormant boundary
conditions. We will assume that dormant intervals have no energy contribution
and that the Hamiltonian is additive in the active states. Let
$E:\mathcal{X}\to\mathbb{R}\cup\\{-\infty\\}$ be measurable. We define
$\mathcal{H}:\mathbf{N}_{f}(\triangle\times\mathcal{X})\to\mathbb{R}\cup\\{-\infty\\}$
by

$\mathcal{H}\Big{(}\sum_{k=1}^{n}\delta_{(s_{k},t_{k},x_{k})}\Big{)}\coloneqq\sum_{k=1}^{n}E(x_{k})\quad\text{for
}\sum_{k=1}^{n}\delta_{(s_{k},t_{k},x_{k})}\in\mathbf{N}_{f}(\triangle\times\mathcal{X}).$

In the following, we set $e^{\infty}\coloneqq\infty\eqqcolon\log(\infty)$.

######  Proposition 4.6.

Assume that there exists a $\mu\in\mathbb{R}$ such that $\mathbb{E}[e^{-\mu
T_{1}-E(\chi_{1})}]<\infty$. Then

$t\mapsto\mathbf{Z}_{2t}\coloneqq\int_{\mathbf{N}_{f}(\triangle\times\mathcal{X})}e^{-\mathcal{H}(\zeta)}\,\Gamma^{\Xi}_{\alpha,t}(\mathrm{d}\zeta)$

is locally bounded, more precisely there exists a $\lambda\in\mathbb{R}$ and a
$C>0$ such that $\mathbf{Z}_{t}\;\leqslant\;Ce^{\lambda t}$ for all
$t\;\geqslant\;0$.

###### Proof.

By the dominated convergence theorem, there exists a $\lambda\in\mathbb{R}$
such that

$c\coloneqq\mathbb{E}[e^{-\lambda T_{1}-E(\chi_{1})}]\;\leqslant\;1.$

If $A_{t}$ denotes for $t\;\geqslant\;0$ the forward recurrence time of the
embedded renewal process of $\eta$ at time $t$ we have
$T_{1}+\ldots+T_{\tau_{t}}=t+A_{t}$. By the memorylessness property of the
exponential distribution, $(A_{t},\chi_{\tau_{t}})$ is independent of
$R_{0}^{t}\eta$ conditionally on $\\{\eta\text{ is dormant at }t\\}$ and we
have for all $t\;\geqslant\;0$

$\displaystyle\mathbf{Z}_{t}e^{-\lambda t}$
$\displaystyle=\frac{1}{c\cdot\mathbb{P}(\eta\text{ is dormant at
}t)}\mathbb{E}\Big{[}\prod_{i=1}^{\tau_{t}}e^{-\lambda
T_{i}-E(\chi_{i})}\mathds{1}_{\\{\eta\text{ is dormant at }t\\}}\Big{]}$
$\displaystyle\;\leqslant\;\frac{1}{c\cdot\mathbb{P}(\eta\text{ is dormant at
}t)}\mathbb{E}\Big{[}\prod_{i=1}^{\tau_{t}}e^{-\lambda
T_{i}-E(\chi_{i})}\Big{]}.$

As $(\prod_{i=1}^{n}e^{-\lambda T_{i}-E(\chi_{i})})_{n\;\geqslant\;1}$ is a
supermartingale with respect to the filtration generated by
$((d_{n},a_{n},\chi_{n}))_{n\;\geqslant\;1}$ we get by an application of the
optional stopping theorem and Fatou’s lemma

$\mathbb{E}\Big{[}\prod_{i=1}^{\tau_{t}}e^{-\lambda
T_{i}-E(\chi_{i})}\Big{]}\;\leqslant\;\liminf_{n\to\infty}\mathbb{E}\Big{[}\prod_{i=1}^{\tau_{t}\wedge
n}e^{-\lambda T_{i}-E(\chi_{i})}\Big{]}\;\leqslant\;1.$

The claim follows as $\lim_{t\to\infty}\mathbb{P}(\eta\text{ is dormant at
}t)>0$. ∎

From now on, we will always assume that $t\mapsto\mathbf{Z}_{t}$ is locally
bounded. For $t>0$ we define the probability measure
$\widehat{\Gamma}^{\Xi}_{\alpha,t}$ by

$\widehat{\Gamma}^{\Xi}_{\alpha,t}(\mathrm{d}\zeta)=\frac{1}{\mathbf{Z}_{2t}}e^{-\mathcal{H}(\zeta)}\,\Gamma^{\Xi}_{\alpha,t}(\mathrm{d}\zeta).$

To make contact with the mixing measure in the representation introduced in
Section 3, consider $\mathcal{X}=\mathbf{N}_{f}(\triangle)$ and $(d_{n})_{n}$,
$(a_{n})_{n}$ to be the sequences of dormant and active periods of the
$M/G/\infty$ queue and $(\chi_{n})_{n}=(\xi_{n})_{n}$ to be the sequence of
clusters (i.e. the reshifted processes of customers arriving during the
respective active periods) and $E=-\log(F)$ with

$F\Big{(}\sum_{i=1}^{n}\delta_{(s_{i},t_{i})}\Big{)}\coloneqq\mathbb{E}_{\mathcal{W}}\Big{[}\prod_{i=1}^{n}v(t_{i}-s_{i},X_{s_{i},t_{i}})\Big{]}.$

We give a condition under which the reweighted measure
$\widehat{\Gamma}^{\Xi}_{\alpha,t}$ can be obtained by simply reweighting the
distributions of the dormant periods $(d_{k})_{k}$ and active periods and
states $(a_{k},\chi_{k})_{k}$. If the reweighted active periods have finite
expectation, this will in particular imply local convergence in total
variation of the measures $\widehat{\Gamma}^{\Xi}_{\alpha,t}$ as $t\to\infty$.

######  Proposition 4.7.

Assume $\lambda\in\mathbb{R}$ satisfies

$\mathbb{E}\big{[}e^{-\lambda T_{1}-E(\chi_{1})}\big{]}=1.$

Then

$\widehat{\Gamma}^{\Xi}_{\alpha,t}=\Gamma^{\widehat{\Xi}}_{\alpha+\lambda,t}\quad\text{
where
}\quad\widehat{\Xi}(\mathrm{d}a\,\mathrm{d}\chi)\coloneqq\frac{\alpha}{\alpha+\lambda}e^{-\lambda
a-E(\chi)}\,\Xi(\mathrm{d}a\,\mathrm{d}\chi)$

for all $t>0$. Additionally,

$\mathbf{Z}_{t}=e^{\lambda
t}\cdot\frac{\Gamma_{\alpha}^{\widehat{\Xi}}(\text{the system is dormant at
$t$})}{\Gamma_{\alpha}^{\Xi}(\text{the system is dormant at $t$})}.$

###### Proof.

First, notice that $\widehat{\Xi}$ indeed defines a probability measure as

$1=\mathbb{E}\big{[}e^{-\lambda
T_{1}-E(\chi_{1})}\big{]}=\mathbb{E}\big{[}e^{-\lambda
d_{1}}\big{]}\mathbb{E}\big{[}e^{-\lambda
a_{1}-E(\chi_{1})}\big{]}=\frac{\alpha}{\alpha+\lambda}\mathbb{E}\big{[}e^{-\lambda
a_{1}-E(\chi_{1})}\big{]}.$

Let $(\hat{d}_{n})_{n}$ be an iid sequence of
$\operatorname{Exp}(\alpha+\lambda)$ distributed random variables. Let
$((\hat{a}_{n},\hat{\chi}_{n}))_{n}$ be an iid sequence of $\widehat{\Xi}$
distributed random variables, independent of $(\hat{d}_{n})_{n}$, and
$\hat{\eta}$ be the process constructed in the same manner as in the beginning
of the section. As usual, we denote for $t\in\mathbb{R}$ by $\hat{\tau}_{t}$
and $\tau_{t}$ the hitting times of $[t,\infty)$ of the embedded renewal
processes of $\hat{\eta}$ and $\eta$ respectively. For $n\in{\mathbb{N}}_{0}$,
define $\delta_{n}\coloneqq 2t-S_{n}$ and $\hat{\delta}_{n}\coloneqq
2t-\hat{S}_{n}$. Then

$\\{\tau_{2t}=n+1,\,\eta\text{ is dormant at
$2t$}\\}=\\{S_{n}\;\leqslant\;2t,d_{n+1}>\delta_{n}\\}.$

The distribution of
$\hat{Y}=(\hat{d}_{1},\ldots,\hat{d}_{n+1},\hat{a}_{1},\hat{\chi}_{1},\ldots,\hat{a}_{n},\hat{\chi}_{n})$
has a density with respect to the distribution of
$Y=(d_{1},\ldots,d_{n+1},a_{1},\chi_{1},\ldots,a_{n},\chi_{n})$ given by

$(\tilde{d}_{1},\ldots,\tilde{d}_{n+1},\tilde{a}_{1},\tilde{\chi}_{1},\ldots,\tilde{a}_{n},\tilde{\chi}_{n})\mapsto\frac{\alpha+\lambda}{\alpha}e^{-\lambda\tilde{d}_{n+1}}\prod_{i=1}^{n}e^{-\lambda(\tilde{d}_{i}+\tilde{a}_{i})}e^{-E(\tilde{\chi}_{i})}.$

For measurable $f:\mathbf{N}_{f}(\triangle\times\mathcal{X})\to[0,\infty)$ we
have that
$f(R_{-t}^{t}\theta_{t}\hat{\eta})\mathds{1}_{\\{\hat{S}_{n}\;\leqslant\;2t,\hat{d}_{n+1}>\delta_{n}\\}}=h(\hat{Y})$
for a suitable measurable function $h$. By writing the following expected
value as an integral with respect to the image measure of $\hat{Y}$ and using
that

$e^{-\lambda(T_{1}+\ldots+T_{n}+d_{n+1})}=e^{-2\lambda
t-\lambda(d_{n+1}-\delta_{n})}$

one obtains

$\displaystyle\mathbb{E}\big{[}f(R_{-t}^{t}\theta_{t}\hat{\eta})\mathds{1}_{\\{\hat{S}_{n}\;\leqslant\;2t,\,\hat{d}_{n+1}>\hat{\delta}_{n}\\}}\big{]}$
$\displaystyle=\tfrac{\alpha+\lambda}{\alpha}e^{-2\lambda
t}\mathbb{E}\big{[}f(R_{-t}^{t}\theta_{t}\eta)\mathds{1}_{\\{S_{n}\;\leqslant\;2t,d_{n+1}>\delta_{n}\\}}e^{-\sum_{i=1}^{n}E(\chi_{i})}e^{-\lambda(d_{n+1}-\delta_{n})}\big{]}.$

By the memorylessness property of the exponential distribution, we have for
all $t\;\geqslant\;0$

$\mathbb{E}\big{[}e^{-\lambda(d_{n+1}-t)}\mathds{1}_{\\{d_{n+1}>t\\}}\big{]}=\frac{\alpha}{\alpha+\lambda}\mathbb{E}\big{[}\mathds{1}_{\\{d_{n+1}>t\\}}\big{]}.$

By conditioning with respect to
$(d_{1},\ldots,d_{n},a_{1},\chi_{1},\ldots,a_{n},\chi_{n})$ we hence obtain

$\mathbb{E}\big{[}f(R_{-t}^{t}\theta_{t}\hat{\eta})\mathds{1}_{\\{\hat{S}_{n}\;\leqslant\;2t,\,\hat{d}_{n+1}>\hat{\delta}_{n}\\}}\big{]}=e^{-2\lambda
t}\mathbb{E}\big{[}f(R_{-t}^{t}\theta_{t}\eta)\mathds{1}_{\\{S_{n}\;\leqslant\;2t,d_{n+1}>\delta_{n}\\}}e^{-\mathcal{H}(R_{0}^{2t}\eta)}\big{]}.$

By summing over all $n\in{\mathbb{N}}_{0}$ we get

$\mathbb{E}\big{[}f(R_{-t}^{t}\theta_{t}\hat{\eta})\mathds{1}_{\\{\hat{\eta}\text{
is dormant at }2t\\}}\big{]}=e^{-2\lambda
t}\mathbb{E}\big{[}f(R_{-t}^{t}\theta_{t}\eta)\mathds{1}_{\\{\eta\text{ is
dormant at }2t\\}}e^{-\mathcal{H}(R_{0}^{2t}\eta)}\big{]}$

and hence

$\int_{\mathbf{N}_{f}(\triangle\times\mathcal{X})}f(\zeta)\,\Gamma_{\alpha+\lambda,t}^{\widehat{\Xi}}(\mathrm{d}\zeta)=e^{-2\lambda
t}\frac{\mathbb{P}(\eta\text{ is dormant at }2t)}{\mathbb{P}(\hat{\eta}\text{
is dormant at
}2t)}\int_{\mathbf{N}_{f}(\triangle\times\mathcal{X})}f(\zeta)e^{-\mathcal{H}(\zeta)}\,\Gamma^{\Xi}_{\alpha,t}(\mathrm{d}\zeta)$

which yields the claims. ∎

As a corollary of Proposition 4.7 and Proposition 4.5 we obtain our first main
result, namely a sufficient condition for the existence of an infinite volume
measure for the Gibbs measures $\widehat{\Gamma}^{\Xi}_{\alpha,T}$. This
implies the existence of the infinite volume measure for our path measures
$\mathbb{P}_{\alpha,T}$.

######  Corollary 4.8.

Assume $\lambda\in\mathbb{R}$ satisfies

$\mathbb{E}\big{[}e^{-\lambda T_{1}-E(\chi_{1})}\big{]}=1\quad\text{ and
}\quad\mathbb{E}\big{[}T_{1}e^{-\lambda T_{1}-E(\chi_{1})}\big{]}<\infty.$
(G’)

Then
$\widehat{\Gamma}^{\Xi}_{\alpha,t}\to\Gamma_{\alpha+\lambda,\operatorname{st}}^{\widehat{\Xi}}\eqqcolon\widehat{\Gamma}_{\alpha,\operatorname{st}}^{\Xi}$
locally in total variation.

Note that Condition (G’) specializes to (G) in the context of Section 3. We
will now show that the existence of a real number $\lambda$ satisfying (G’) is
equivalent to several natural conditions, including exponential growth of the
partition function with $t$.

######  Proposition 4.9.

The limit $\varphi\coloneqq\lim_{t\to\infty}\log(\mathbf{Z}_{t})/t$ exists in
$\mathbb{R}\cup\\{\infty\\}$.

###### Proof.

For $t>0$ let

$f(t)\coloneqq\mathbb{E}[e^{-\mathcal{H}(R_{0}^{t}\eta)}\mathds{1}_{\\{\eta\text{
is dormant at }t\\}}]=\mathbf{Z}_{t}\mathbb{P}(\eta\text{ is dormant at }t).$

By the memorylessness property of the exponential distribution, conditionally
on the event $\\{\eta\text{ is dormant at }t\\}$ the process $R_{0}^{t}\eta$
is independent of $R_{t}^{t+s}\eta$ for all $s,t>0$ and

$\mathbb{P}(\theta_{t}R_{t}^{t+s}\eta\in\,\cdot\,|\eta\text{ is dormant at
}t)=\mathbb{P}(R_{0}^{s}\eta\in\,\cdot\,).$

This yields for all $s,t>0$

$f(t+s)=f(t)f(s)+\mathbb{E}\big{[}e^{-\mathcal{H}(R_{0}^{t+s}\eta)}\mathds{1}_{\\{\eta\text{
is dormant at }t+s\\}}\mathds{1}_{\\{\eta\text{ is active at }t\\}}\big{]}$

which implies, by the version of Feketes lemma for measurable superadditive
functions, $\log(f(t))/t\to\sup_{s>0}\log(f(s))/s$ as $t\to\infty$. Since
$\lim_{t\to\infty}\mathbb{P}(\eta\text{ is dormant at }t)>0$, the claim
follows. ∎

######  Lemma 4.10.

Let $f:(0,\infty)\to(0,\infty)$ be a function such that
$a\coloneqq\lim_{t\to\infty}\log(f(t))/t$ exists in $\mathbb{R}$ and such that
$\limsup_{t\to\infty}f(t)e^{-at}<\infty$. Then we have

$\liminf_{t\to\infty}f(t)e^{-at}>0\iff\limsup_{t\to\infty}\frac{f(2t)}{f(t)^{2}}<\infty.$

###### Proof.

Define $g(t)\coloneqq f(t)e^{-at}$ for $t>0$. First, assume that
$\liminf_{t\to\infty}g(t)>0$. Then

$\limsup_{t\to\infty}\frac{f(2t)}{f(t)^{2}}=\limsup_{t\to\infty}\frac{g(2t)}{g(t)^{2}}<\infty.$

Now, assume that $\limsup_{t\to\infty}f(2t)/f(t)^{2}<\infty$. Then there
exists a $c>1$ and a $T>0$ such that $g(2t)\;\leqslant\;cg(t)^{2}$ for all
$t\;\geqslant\;T$. Assume that $\liminf_{t\to\infty}g(t)=0$ would hold. Then,
there would exists a $t_{0}\;\geqslant\;T$ such that $cg(t_{0})<1$. Set
$t_{k}\coloneqq 2^{k}t_{0}$ for $k\in{\mathbb{N}}$. Then we have with
$b\coloneqq(cg(t_{0}))^{1/t_{0}}$

$g(t_{k})\;\leqslant\;c^{2^{k}-1}g(t_{0})^{2^{k}}=\frac{1}{c}(cg(t_{0}))^{t_{k}/t_{0}}<b^{t_{k}}$

for all $k\in{\mathbb{N}}$. However, as $\log(g(t))/t\to 0$ as $t\to\infty$
and $\log(b)<0$ there exists a $k\in{\mathbb{N}}$ such that
$g(t_{k})>e^{t_{k}\log(b)}=b^{t_{k}}$. ∎

######  Theorem 4.11.

Recall that
$\varphi\coloneqq\lim_{t\to\infty}\log(\mathbf{Z}_{t})/t\in\mathbb{R}\cup\\{\infty\\}$.
The following conditions are equivalent:

  1. (1)

Condition (G’) holds, i.e. there exists a $\lambda\in\mathbb{R}$ such that

$\mathbb{E}\big{[}e^{-\lambda
T_{1}-E(\chi_{1})}\big{]}=1\qquad\text{and}\qquad\mathbb{E}\big{[}T_{1}e^{-\lambda
T_{1}-E(\chi_{1})}\big{]}<\infty.$

  2. (2)

$\limsup_{t\to\infty}\mathbf{Z}_{2t}/\mathbf{Z}_{t}^{2}<\infty$

  3. (3)

$\liminf_{t\to\infty}\widehat{\Gamma}_{\alpha,t}^{\Xi}(\text{the system is
dormant at }0)>0$

  4. (4)

$\liminf_{t\to\infty}\mathbf{Z}_{t}e^{-\varphi t}>0$.

If (1)-(4) hold, then $\lambda=\varphi<\infty$ is the unique real number
satisfying Condition 1 and

$\lim_{t\to\infty}\mathbf{Z}_{t}e^{-\varphi
t}=\frac{\alpha}{\alpha+\lambda}\frac{\mathbb{E}[T_{1}]}{\hat{\mathbb{E}}[\widehat{T}_{1}]}.$

###### Proof.

We will use the same notation as in the proof of Proposition 4.5. First, we
will point out that Conditions (2) and (3) are equivalent. Using the same
argument as in the proof of Proposition 4.9 we obtain

$\displaystyle\widehat{\Gamma}_{\alpha,t}^{\Xi}(N_{0}=0)$
$\displaystyle=\frac{1}{\mathbf{Z}_{2t}\mathbb{P}(N_{2t}(\eta)=0)}\mathbb{E}[e^{-\mathcal{H}(R_{0}^{t}\eta)}\mathds{1}_{\\{N_{t}(\eta)=0\\}}e^{-\mathcal{H}(R_{t}^{2t}\eta)}\mathds{1}_{\\{N_{2t}(\eta)=0\\}}]$
$\displaystyle=\frac{1}{\mathbf{Z}_{2t}\mathbb{P}(N_{2t}(\eta)=0)}\mathbb{E}[e^{-\mathcal{H}(R_{0}^{t}\eta)}\mathds{1}_{\\{N_{t}(\eta)=0\\}}]^{2}$
$\displaystyle=\frac{\mathbb{P}(N_{t}(\eta)=0)^{2}}{\mathbb{P}(N_{2t}(\eta)=0)}\frac{\mathbf{Z}_{t}^{2}}{\mathbf{Z}_{2t}}.$

The equivalence of both conditions follows from

$\lim_{t\to\infty}\frac{\mathbb{P}(N_{t}(\eta)=0)^{2}}{\mathbb{P}(N_{2t}(\eta)=0)}=\frac{\mathbb{E}[d_{1}]}{\mathbb{E}[T_{1}]}.$

Similar to the proof of Proposition 4.10, one can see that Condition (2)
implies $\varphi<\infty$. The equivalence of Conditions (2) and (4) follows
from Lemma 4.10 since

$\log(\mathbf{Z}_{t}\mathbb{P}(N_{t}(\eta)=0))/t\;\leqslant\;\varphi$ (4.4)

holds by superadditivity (as seen in in the proof of Proposition 4.9).
Finally, we show that Conditions (1) and (4) are equivalent. Assume that there
exists a $\lambda\in\mathbb{R}$ as in Condition (1). By Proposition 4.7, we
then have

$\mathbf{Z}_{t}e^{-\lambda
t}=\frac{\hat{\mathbb{P}}(N_{t}(\hat{\eta})=0)}{\mathbb{P}(N_{t}(\eta)=0)}\to\frac{\hat{\mathbb{E}}_{\alpha}[\hat{d}_{1}]/\hat{\mathbb{E}}_{\alpha}[\hat{T}_{1}]}{\mathbb{E}_{\alpha}[d_{1}]/\mathbb{E}_{\alpha}[T_{1}]}=\frac{\alpha}{\alpha+\lambda}\frac{\mathbb{E}[T_{1}]}{\hat{\mathbb{E}}[\hat{T}_{1}]}$

as $t\to\infty$. As the limit is finite and positive, this additionally
implies that $\log(\mathbf{Z}_{t})-t\lambda$ converges to a real number as
$t\to\infty$ and hence, $\lambda=\varphi<\infty$. For the other direction,
assume that

$\liminf_{t\to\infty}\mathbf{Z}_{t}e^{-\varphi t}>0.$

Then in particular $\varphi<\infty$. We define $f:[0,\infty)\to(0,\infty)$ by

$\displaystyle f(t)\coloneqq e^{-\varphi
t}\mathbb{E}[e^{-\mathcal{H}(R_{0}^{t}\eta)}\mathds{1}_{\\{N_{t}(\eta)=0\\}}]=\mathbf{Z}_{t}e^{-\varphi
t}\mathbb{P}(N_{t}(\eta)=0).$

By the usual renewal argument

$\displaystyle f(t)$
$\displaystyle=\mathbb{E}[\mathds{1}_{\\{T_{1}\;\leqslant\;t\\}}e^{-\varphi
T_{1}-E(\chi_{1})}f(t-T_{1})]+e^{-\varphi t}\mathbb{P}(d_{1}>t)$ (4.5)
$\displaystyle=\mathbb{E}[\mathds{1}_{\\{T_{1}\;\leqslant\;t\\}}e^{-\varphi
T_{1}-E(\chi_{1})}f(t-T_{1})]+e^{-(\varphi+\alpha)t}.$

That is, $f$ satisfies a renewal equation with respect to the image measure of
$T_{1}$ under the (possibly non probability-) measure $e^{-\varphi
T_{1}-E(\chi_{1})}\mathrm{d}\mathbb{P}$. Notice that Equation (4.4) implies
$f(t)\;\leqslant\;1$ for all $t\;\geqslant\;0$ and thus $\varphi>-\alpha$.
Equation (4.5) in combination with Fatou’s Lemma implies

$\mathbb{E}\Big{[}\liminf_{t\to\infty}\mathds{1}_{\\{T_{1}\;\leqslant\;t\\}}e^{-\varphi
T_{1}-E(\chi_{1})}f(t-T_{1})\Big{]}\;\leqslant\;\liminf_{t\to\infty}f(t).$
(4.6)

Since

$\liminf_{t\to\infty}f(t)=\liminf_{t\to\infty}\mathbf{Z}_{t}e^{-\varphi
t}\mathbb{P}(N_{t}(\eta)=0)=\tfrac{\mathbb{E}[d_{1}]}{\mathbb{E}[T_{1}]}\liminf_{t\to\infty}\mathbf{Z}_{t}e^{-\varphi
t}>0$

we can divide both sides of Inequality (4.6) by $\liminf_{t\to\infty}f(t)$ and
obtain

$\mathbb{E}[e^{-\varphi T_{1}-E(\chi_{1})}]\;\leqslant\;1.$

If $\mathbb{E}[e^{-\varphi T_{1}-E(\chi_{1})}]<1$ or
$\mathbb{E}[T_{1}e^{-\varphi T_{1}-E(\chi_{1})}]=\infty$ would hold, then $f$
(as the solution to the renewal equation (4.5)) would converge to zero by
renewal theory (compare Proposition 4.3, Theorem 4.2). We thus conclude that
Condition (1) holds. ∎

We finish our treatment of the abstract alternating processes by first
considering the case that (G’) is not satisfied and then giving two sufficient
criteria for (G’).

######  Proposition 4.12.

Assume $\varphi<\infty$. Irrespective whether (G’) is satisfied, we have

$\varphi=\min\Big{\\{}\lambda\in\mathbb{R}:\,\mathbb{E}\big{[}e^{-\lambda
T_{1}-E(\chi_{1})}\big{]}\;\leqslant\;1\Big{\\}}.$

If (G’) does not hold then $\lim_{t\to\infty}\mathbf{Z}_{t}e^{-\varphi t}=0$.

###### Proof.

Assume that $\mathbb{E}[e^{-\varphi T_{1}-E(\chi_{1})}]>1$. Then there would
exist a $N\in{\mathbb{N}}$ such that

$\mathbb{E}\big{[}e^{-\varphi
T_{1}-E(\chi_{1})}\mathds{1}_{\\{T_{1}\;\leqslant\;N,\,E(\chi_{1})\;\geqslant\;-N\\}}\big{]}>1.$

By the intermediate value theorem, there would exists a $\lambda>0$ such that

$\mathbb{E}\big{[}e^{-(\lambda+\varphi)T_{1}-E(\chi_{1})}\mathds{1}_{\\{T_{1}\;\leqslant\;N,\,E(\chi_{1})\;\geqslant\;-N\\}}\big{]}=1.$
(4.7)

For $t\;\geqslant\;0$ set

$f_{N}(t)\coloneqq
e^{-(\varphi+\lambda)t}\mathbb{E}\Big{[}\prod_{i=1}^{\tau_{t}-1}e^{-E(\chi_{i})}\mathds{1}_{\\{T_{i}\;\leqslant\;N,\,E(\chi_{i})\;\geqslant\;-N\\}}\mathds{1}_{\\{N_{t}(\eta)=0\\}}\Big{]}.$

As in the proof of Theorem 4.11, one can derive a renewal equation for
$f_{N}$. Equation (4.7) in combination with the renewal theorem would imply
convergence of $f_{N}(t)$ to a positive real number as $t\to\infty$. This,
however, contradicts

$f_{N}(t)\;\leqslant\;e^{-(\varphi+\lambda)t}\mathbb{E}\Big{[}\prod_{i=1}^{\tau_{t}-1}e^{-E(\chi_{i})}\mathds{1}_{\\{N_{t}(\eta)=0\\}}\Big{]}=e^{-(\varphi+\lambda)t}\mathbf{Z}_{t}\mathbb{P}(\eta\text{
is dormant at }t)$

for all $t>0$. Thus $\mathbb{E}[e^{-\varphi
T_{1}-E(\chi_{1})}]\;\leqslant\;1$. If $\mathbb{E}\big{[}e^{-\varphi
T_{1}-E(\chi_{1})}\big{]}<1$ or $\mathbb{E}\big{[}T_{1}e^{-\varphi
T_{1}-E(\chi_{1})}\big{]}=\infty$ holds then $f$ as defined in the proof of
Theorem 4.11 converges to zero and thus

$\lim_{t\to\infty}\mathbf{Z}_{t}e^{-\varphi t}=0.$

Similiarily it can be shown, that $\varphi$ is indeed the minimum: Assume
there would exists a $\varepsilon\in(0,\varphi+\alpha)$ such that

$\mathbb{E}[e^{-(\varphi-\varepsilon)T_{1}-E(\chi_{1})}]<1.$

Then, by Proposition 4.3, we would have $e^{\varepsilon t}f(t)\to 0$ as
$t\to\infty$ i.e. exponential decay of $t\mapsto\mathbf{Z}_{t}e^{-\varphi t}$.
∎

######  Proposition 4.13.

The following two conditions are sufficient for (G’) to hold:

  1. (1)

There exists a $\mu\in\mathbb{R}$ such that

$1<\mathbb{E}\big{[}e^{-\mu T_{1}-E(\chi_{1})}\big{]}<\infty.$

  2. (2)

We have $\varphi<\infty$ and there exists an $\varepsilon>0$ such that

$\mathbb{E}\big{[}e^{-(\varphi-\varepsilon)T_{1}-E(\chi_{1})}\big{]}<\infty.$

###### Proof.

Assume there exists a $\mu\in\mathbb{R}$ such that $1<\mathbb{E}\big{[}e^{-\mu
T_{1}-E(\chi_{1})}\big{]}<\infty$. Then, by the intermediate value theorem,
there exists a $\lambda>\mu$ such that $\mathbb{E}\big{[}e^{-\lambda
T_{1}-E(\chi_{1})}\big{]}=1$. Then $\mathbb{E}\big{[}T_{1}e^{-\lambda
T_{1}-E(\chi_{1})}\big{]}<\infty$ holds as well, as
$t\in\mathcal{O}(e^{(\lambda-\mu)t})$ for $t\to\infty$. The second claim
follows from the first one and Proposition 4.12. ∎

We are now in the position to prove Theorem 3.2 and Theorem 3.3 by applying
the previous results to the measures introduced in Section 3. We consider
$\mathcal{X}=\mathbf{N}_{f}(\triangle)$ and $(d_{n})_{n}$, $(a_{n})_{n}$ to be
the sequences of dormant and active periods of the $M/G/\infty$-queue and
$(\chi_{n})_{n}=(\xi_{n})_{n}$ to be the sequence of clusters of the queue.
Additionally, we choose $E=-\log(F)$. Since the first active period
$a_{1}\eqqcolon a(\xi_{1})$ is a function of the first cluster $\xi_{1}$, we
reduce notation and denote by $\Xi_{\alpha}$ (as in Section 3) the
distribution of $\xi_{1}$ (and not the joint distribution of
$(a_{1},\xi_{1})$).

###### Proof of Theorem 3.2 and Theorem 3.3.

The number of customers present at time $T>0$ in the $M/G/\infty$-queue
(started empty at time zero) is $\operatorname{Poi}(\beta(\alpha,T))$
distributed, where

$\beta(\alpha,T)=\alpha\int_{0}^{T}\int_{T}^{\infty}g(t-s)\,\mathrm{d}t\mathrm{d}s=\alpha\int_{0}^{T}\mathbb{P}(\tau_{1}>r)\,\mathrm{d}r.$

Hence,

$\mathbb{E}_{\alpha}[T_{1}]=\frac{\mathbb{E}_{\alpha}[d_{1}]}{\lim_{T\to\infty}\mathbb{P}_{\alpha}(\eta\text{
is dormant at
}T)}=\frac{1/\alpha}{\lim_{T\to\infty}e^{-\beta(\alpha,T)}}=\frac{e^{\alpha\mathbb{E}[\tau_{1}]}}{\alpha}.$

As mentioned before

$e^{c_{\alpha,T}}\sim e^{2\alpha
T-\alpha\mathbb{E}[\tau_{1}]}=(\alpha\mathbb{E}[T_{1}])^{-1}e^{2\alpha T}$

as $T\to\infty$. In particular, since

$\mathbf{Z}_{\alpha,2T}\coloneqq\int_{\mathbf{N}_{f}(\triangle)}\mathrm{d}\Gamma_{\alpha,T}(\xi)\,F(\xi)=Z_{\alpha,2T}e^{-c_{\alpha,T}}$

we have

$\varphi(\alpha)\coloneqq\lim_{T\to\infty}\log(\mathbf{Z}_{\alpha,T})/T=\psi(\alpha)-\alpha.$

In combination with Theorem 4.11 the above yields Theorem 3.2. We have

$\mathbb{E}_{\alpha}\Big{[}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})\Big{]}=1,\quad\mathbb{E}_{\alpha}\Big{[}T_{1}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})\Big{]}<\infty$

and thus Corollary 4.8 and Equation (3.6) yield for all $a<b$

$\mathbb{P}_{\alpha,T}(A)\to\int_{\mathbf{N}(\triangle)}\widehat{\Gamma}_{\alpha,\operatorname{st}}(\mathrm{d}\xi)\mathbf{P}_{R_{a}^{b}\xi}(A)\eqqcolon\mathbb{P}_{\alpha,\infty}^{a,b}(A)$

uniformly in $A\in\mathcal{A}_{a}^{b}$ as $T\to\infty$. Consistency of the
family of measures $(\mathbb{P}_{\alpha,\infty}^{a,b})_{a<b}$ implies the
existence of the measure $\mathbb{P}_{\alpha,\infty}$. ∎

######  Remark 4.14.

Let us assume that $w(t,\cdot)=\tilde{w}(t,|\cdot|)$ is rotationally symmetric
and positive and that $r\mapsto\tilde{w}(t,\sqrt{r})$ is completely monotone
on $(0,\infty)$ with $\tilde{w}(t,0)=\lim_{r\to 0}\tilde{w}(t,r)$ for all
$t>0$. We will represent the measure $\mathbb{P}_{\alpha,T}$ as a mixture of
Gaussian measures. For the Polaron measure, this will make contact between our
representation and the representation introduced in [MV19]. Additionally, this
representation can be used in order to show that (under these stronger
assumptions on $w$) the convergence of the finite dimensional distributions in
the proof of the central limit theorem below is even in total variation. By
Bernsteins Theorem, there exists for all $t>0$ a Radon measure
$\tilde{\mu}_{t}$ on $[0,\infty)$ such that for all $r\;\geqslant\;0$

$\tilde{v}(t,\sqrt{2r})=\int_{[0,\infty)}\tilde{\mu}_{t}(\mathrm{d}u)\,e^{-ur}$

where $v(t,\cdot)\eqqcolon\tilde{v}(t,|\cdot|)$. With
$\mu_{t}\coloneqq\tilde{\mu}_{t}\circ\sqrt{\,\cdot\,}^{\,-1}$ we have for all
$t>0$ and $x\in\mathbb{R}^{d}$

$v(t,x)=\int_{[0,\infty)}\mu_{t}(\mathrm{d}u)\,e^{-u^{2}|x|^{2}/2}.$

Hence, we can further rewrite for $A\in\mathcal{A}$

$\displaystyle\mathbf{P}_{\xi}(A)=$
$\displaystyle\frac{1}{F(\xi)}\int_{A}\mathcal{W}(\mathrm{d}\mathbf{x})\,\prod_{i=1}^{n}v(t_{i}-s_{i},\mathbf{x}_{s_{i},t_{i}})$
$\displaystyle=\frac{1}{F(\xi)}\int_{[0,\infty)^{n}}\bigotimes_{i=1}^{n}\mu_{t_{i}-s_{i}}(\mathrm{d}u_{i})\int_{A}\mathcal{W}(\mathrm{d}\mathbf{x})\,e^{-\frac{1}{2}\sum_{i=1}^{n}u_{i}^{2}|\mathbf{x}_{s_{i},t_{i}}|^{2}}.$
(4.8)

We normalize the inner expression by marking a point process with distribution
$\widehat{\Gamma}_{\alpha,T}$ accordingly. Given a locally compact Polish
space $E$ we can identify a probability measure on
$(\mathbf{N}_{f}(E),\mathcal{N}_{f}(E))$ with a symmetric probability measure
on $E^{\cup}\coloneqq\bigcup_{n=0}^{\infty}E^{n}$. Let $E_{1}$ and $E_{2}$ be
two locally compact Polish spaces and
$\kappa:E_{1}^{\cup}\times\mathcal{B}(E_{2}^{\cup})\to[0,1]$ be a probability
kernel satisfying

  1. (1)

$\kappa(x,E_{2}^{n})=1$ for all $x\in E_{1}^{n},n\in{\mathbb{N}}_{0}$

  2. (2)

$\kappa(\sigma x,\sigma A)=\kappa(x,A)$ for all $x\in
E_{1}^{n},A\in\mathcal{B}(E_{2}^{n})$, $\sigma\in S_{n}$ and
$n\in{\mathbb{N}}$

where $\sigma x\coloneqq(x_{\sigma(1)},\ldots,x_{\sigma(n)})$ for $x\in
E_{i}^{n}$, a permutation $\sigma\in S_{n}$ and $i\in\\{1,2\\}$. Given a
probability measure $\mathcal{P}$ on $\mathbf{N}_{f}(E_{1})$ we can define the
marked distribution $\mathcal{P}\otimes\kappa$ in the following way: Draw a
sample $\sum_{i=1}^{n}\delta_{x_{i}}$ according to $\mathcal{P}$, draw marks
$(y_{1},\ldots,y_{n})$ according to $\kappa((x_{1},\ldots,x_{n}),\cdot)$ and
obtain $\sum_{i=1}^{n}\delta_{(x_{i},y_{i})}$. That is, under the
identification mentioned above,

$\big{(}\mathcal{P}\otimes\kappa\big{)}(\mathrm{d}x\,\mathrm{d}y)=\mathcal{P}(\mathrm{d}x)\kappa(x,\mathrm{d}y).$

We apply this to our case and define for
$\zeta=\sum_{i=1}^{n}\delta_{(s_{i},t_{i},u_{i})}\in\mathbf{N}_{f}(\triangle\times[0,\infty))$
the centered Gaussian measure $\mathbf{Q}_{\zeta}$ on
$(C(\mathbb{R},\mathbb{R}^{d}),\mathcal{A})$ by

$\mathbf{Q}_{\zeta}(\mathrm{d}\mathbf{x})=\frac{1}{\phi(\zeta)}\exp\Big{(}-\frac{1}{2}\sum_{i=1}^{n}u_{i}^{2}|\mathbf{x}_{s_{i},t_{i}}|^{2}\Big{)}\,\mathcal{W}(\mathrm{d}\mathbf{x})$

where $\phi(\zeta)$ is a normalization constant and
$\kappa:\triangle^{\cup}\times\mathcal{B}([0,\infty)^{\cup})\to[0,1]$ by

$\kappa(\xi,\mathrm{d}u)\coloneqq\frac{\phi(\xi,u)}{F(\xi)}\,\bigotimes_{i=1}^{n}\mu_{t_{i}-s_{i}}(\mathrm{d}u_{i})$

with $\phi(\xi,u)\coloneqq 0$ for $\xi\in\triangle^{n}$, $u\in[0,\infty)^{m}$
with $n\neq m$. By Equation (4.14) we then obtain

$\mathbb{P}_{\alpha,T}(\cdot)=\int_{\mathbf{N}_{f}(\triangle\times[0,\infty))}\big{(}\widehat{\Gamma}_{\alpha,T}\otimes\kappa\big{)}(\mathrm{d}\zeta)\,\mathbf{Q}_{\zeta}(\cdot).$

Notice that local convergence of the measures $\widehat{\Gamma}_{\alpha,T}$
immediately implies local convergence of the measures
$\widehat{\Gamma}_{\alpha,T}\otimes\kappa$. Provided that (GC) is satisfied,
one can obtain $\widehat{\Gamma}_{\alpha,T}\otimes\kappa$ by marking the
tilted clusters independently of each other. That is, starting in $-T$ we
alternate independently drawn $\operatorname{Exp}(\psi(\alpha))$ distributed
dormant periods with $\widehat{\Xi}_{\alpha}\otimes\kappa$ distributed marked
clusters. Conditionally on the event that the system is dormant at $T$, the
process of marked customers arriving between $-T$ and $T$ has distribution
$\widehat{\Gamma}_{\alpha,T}\otimes\kappa$. In particular, the explicit form
of the infinite volume measure is the same if we replace the measures
$\mathbf{P}_{\xi}$ with the measures $\mathbf{Q}_{\zeta}$ and
$\widehat{\Xi}_{\alpha}$ with $\widehat{\Xi}_{\alpha}\otimes\kappa$.
For the special case of the Polaron, i.e. $w(t,x)\coloneqq e^{-t}/|x|$ and
$w(t,0)\coloneqq\infty$ for $t\;\geqslant\;0$ and
$x\in\mathbb{R}^{3}\setminus\\{0\\}$, one chooses $g(t)=e^{-t}$ for
$t\;\geqslant\;0$. Since

$\frac{1}{|x|}=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\mathrm{d}u\,e^{-u^{2}|x|^{2}/2}$

for all $x\in\mathbb{R}^{3}\setminus\\{0\\}$, one obtains the representation
derived in [MV19] of the Fröhlich polaron as a mixture of Gaussian measures.

##  5\. Proof of the functional central limit theorem

We equip $C(\mathbb{R},\mathbb{R}^{d})$ with the topology of locally uniform
convergence. A function $f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{\infty\\}$ is
called quasiconcave (quasiconvex) if for all $a\in\mathbb{R}\cup\\{\infty\\}$
the superlevel set $f^{-1}([a,\infty])$ is convex (the sublevel set
$f^{-1}((-\infty,a])$ is convex). Provided that $w(t,\cdot)$ is quasiconcave
for all $t>0$ and that Conditions (A1)-(A3) are met, we are going to show that
the distribution of $X^{n}$ under $\mathbb{P}_{\alpha,\infty}$ converges
weakly to the distribution of a centered Gaussian process with stationary and
independent increments. In case that $w(t,\cdot)$ is rotationally symmetric
for all $t>0$, this Gaussian process is a rescaled Brownian motion.

We start by showing tightness of
$\\{\mathbb{P}_{\alpha,\infty}\circ(X^{n})^{-1}:\,n\in{\mathbb{N}}\\}$. For
$n\in{\mathbb{N}}$ we define the modulus of continuity by

$\omega_{n}:C(\mathbb{R},\mathbb{R}^{d})\times(0,\infty)\to[0,\infty),\quad\omega_{n}(\mathbf{x},\delta)\coloneqq\sup_{s,t\in[-n,n],|s-t|<\delta}|\mathbf{x}_{s,t}|.$

The Gaussian correlation inequality [Roy14] states that for all convex sets
$A_{1},A_{2}\subseteq\mathbb{R}^{n}$ that are symmetric about the origin and
any centered Gaussian measure $\mu$ on $\mathbb{R}^{n}$

$\mu(A_{1}\cap A_{2})\;\geqslant\;\mu(A_{1})\mu(A_{2}).$

It is well known that this implies
$\mathbb{E}_{\mu}[fg]\;\geqslant\;\mathbb{E}_{\mu}[f]\mathbb{E}_{\mu}[g]$ for
all non-negative, symmetric, quasiconcave functions $f,g$ and any centered
Gaussian measure $\mu$ on $\mathbb{R}^{n}$. We generalize this and obtain the
following proposition:

######  Proposition 5.1.

Let $X$ be a $n$-dimensional centered Gaussian vector.

  1. (1)

If $f_{1},\ldots,f_{k}:\mathbb{R}^{n}\to[0,\infty]$ are symmetric (with
respect to point reflections in the origin) and quasiconcave and
$f_{k+1}:\mathbb{R}^{n}\to[0,\infty]$ is symmetric and quasiconvex then

$\mathbb{E}\Big{[}\prod_{i=1}^{k+1}f_{i}(X)\Big{]}\;\leqslant\;\mathbb{E}\Big{[}\prod_{i=1}^{k}f_{i}(X)\Big{]}\cdot\mathbb{E}[f_{k+1}(X)].$

  2. (2)

If $f_{1},\ldots,f_{m}:\mathbb{R}^{n}\to[0,\infty]$ are symmetric and
quasiconcave then

$\mathbb{E}\Big{[}\prod_{i=1}^{m}f_{i}(X)\Big{]}\;\geqslant\;\prod_{i=1}^{k}\mathbb{E}\Big{[}\prod_{j\in
J_{i}}f_{j}(X)\Big{]}.$

for any partition $\\{1,\ldots,m\\}=J_{1}\dot{\cup}\ldots\dot{\cup}J_{k}$.

###### Proof.

We only show the first statement, the proof of the second statement can be
conducted similarly. As a direct consequence of the Gaussian correlation
inequality, one obtains for $A_{1}\subseteq\mathbb{R}^{n}$ symmetric and
convex and $A_{2}\subseteq\mathbb{R}^{n}$ symmetric such that $A_{2}^{c}$ is
convex

$\mathbb{P}(X\in A_{1},X\in A_{2})\;\leqslant\;\mathbb{P}(X\in
A_{1})\mathbb{P}(X\in A_{2}).$

We write

$f_{i}(X)=\int_{0}^{\infty}\mathds{1}_{[0,f_{i}(X)]}(s)\,\mathrm{d}s,\quad\text{
for }1\;\leqslant\;i\;\leqslant\;k,\quad
f_{k+1}(X)=\int_{0}^{\infty}\mathds{1}_{[0,f_{k+1}(X))}(s)\,\mathrm{d}s.$

Now, for all $s_{1},\ldots,s_{k+1}>0$ the sets
$\bigcap_{i=1}^{k}f_{i}^{-1}([s_{i},\infty])$ and
$(f_{k+1}^{-1}((s_{k+1},\infty]))^{c}$ are symmetric and convex. We hence get
with the Gaussian correlation inequality

$\displaystyle\mathbb{E}\Big{[}\prod_{i=1}^{k+1}f_{i}(X)\Big{]}$
$\displaystyle=\int_{0}^{\infty}\ldots\int_{0}^{\infty}\mathbb{P}(f_{1}(X)\;\geqslant\;s_{1},\ldots,f_{k+1}(X)>s_{k+1})\,\mathrm{d}s_{1}\ldots\mathrm{d}s_{k+1}$
$\displaystyle\;\leqslant\;\int_{0}^{\infty}\ldots\int_{0}^{\infty}\mathbb{P}(f_{1}(X)\;\geqslant\;s_{1},\ldots,f_{k}(X)\;\geqslant\;s_{k})\mathbb{P}(f_{k+1}(X)>s_{k+1})\,\mathrm{d}s_{1}\ldots\mathrm{d}s_{k+1}$
$\displaystyle=\mathbb{E}\Big{[}\prod_{i=1}^{k}f_{i}(X)\Big{]}\cdot\mathbb{E}[f_{k+1}(X)].\qed$

######  Corollary 5.2.

Assume that (A1) holds and that $w(t,\cdot)$ is quasiconcave for all $t>0$.
Then for all $\xi\in\mathbf{N}_{f}(\triangle)$ with $F(\xi)<\infty$ and all
$a<b$

$\mathbf{E}_{\xi}\big{[}|X_{a,b}|^{2}\big{]}\;\leqslant\;\mathbb{E}_{\mathcal{W}}\big{[}|X_{a,b}|^{2}\big{]}=d(b-a).$

###### Proof.

For $\xi=\sum_{i=1}^{k}\delta_{(s_{i},t_{i})}\in\mathbf{N}_{f}(\triangle)$
with $F(\xi)<\infty$ we apply Proposition 5.1 to the quasiconvex function
$f_{k+1}:\mathbb{R}^{dk+d}\to[0,\infty)$,
$f_{k+1}(x_{1},\ldots,x_{k+1})\coloneqq|x_{k+1}|^{2}$, the quasiconcave
functions $f_{1},\ldots,f_{k}:\mathbb{R}^{dk+d}\to(0,\infty]$ defined by

$f_{i}(x_{1},\ldots,x_{k+1})\coloneqq v(t_{i}-s_{i},x_{i}),$

for $1\;\leqslant\;i\;\leqslant\;k$ and the Gaussian vector
$(X_{s_{1},t_{1}},\ldots,X_{s_{k},t_{k}},X_{a,b})$. ∎

######  Remark 5.3.

Assume that (A1) and (A2) hold. If $w(t,\cdot)$ is quasiconcave for all $t>0$
then Proposition 5.1 implies that for
$\xi_{1},\xi_{2}\in\mathbf{N}_{f}(\triangle)$

$F(\xi_{1}+\xi_{2})\;\geqslant\;F(\xi_{1})F(\xi_{2})$

i.e. in that sense $\log(F)$ is superadditive. In particular, if (GC) holds,
then the interaction energy between left and right half axis satisfies

$\mathbb{E}_{\mathcal{W}}\bigg{[}\exp\bigg{(}\alpha\int_{-\infty}^{0}\int_{0}^{\infty}w(t-s,X_{s,t})\,\mathrm{d}t\mathrm{d}s\bigg{)}\bigg{]}<\infty.$
(5.1)

In order to see this, we write the Poisson point process
$\eta\sim\Gamma_{\alpha,T}$ as the sum $\eta=\eta_{1}+\eta_{2}+\eta_{3}$ where
$\eta_{1},\eta_{2},\eta_{3}$ are independent Poisson point processes with
intensity measures

$\displaystyle\mu^{1}_{\alpha,T}(\mathrm{d}s\mathrm{d}t)$
$\displaystyle\coloneqq\alpha\cdot
g(t-s)\mathds{1}_{\\{-T<s<t<0\\}}\mathrm{d}s\mathrm{d}t$
$\displaystyle\mu^{2}_{\alpha,T}(\mathrm{d}s\mathrm{d}t)$
$\displaystyle\coloneqq\alpha\cdot
g(t-s)\mathds{1}_{\\{0<s<t<T\\}}\mathrm{d}s\mathrm{d}t$
$\displaystyle\mu^{3}_{\alpha,T}(\mathrm{d}s\mathrm{d}t)$
$\displaystyle\coloneqq\alpha\cdot
g(t-s)\mathds{1}_{\\{-T<s<0<t<T\\}}\mathrm{d}s\mathrm{d}t$

respectively. Then

$\displaystyle\mathbf{Z}_{\alpha,2T}$
$\displaystyle=\mathbb{E}[F(\eta_{1}+\eta_{2}+\eta_{3})]$
$\displaystyle\;\geqslant\;\mathbb{E}[F(\eta_{1})]\mathbb{E}[F(\eta_{2})]\mathbb{E}[F(\eta_{3})]=\mathbf{Z}_{\alpha,T}^{2}\mathbb{E}[F(\eta_{3})].$

By the same calculations as in Section 3 we have

$\mathbb{E}[F(\eta_{3})]=e^{-b_{\alpha,T}}\mathbb{E}_{\mathcal{W}}\bigg{[}\exp\bigg{(}\alpha\int_{-T}^{0}\int_{0}^{T}w(t-s,X_{s,t})\,\mathrm{d}t\mathrm{d}s\bigg{)}\bigg{]}$

with $b_{\alpha,T}=\mu_{\alpha,T}^{3}(\triangle)$ and the statement follows by
the monotone convergence theorem and boundedness of $T\mapsto b_{\alpha,T}$.
Notice that

$\frac{Z_{\alpha,T}^{2}}{Z_{\alpha,2T}}=\mathbb{E}_{\alpha,T}\bigg{[}\exp\bigg{(}-\alpha\int_{-T}^{0}\int_{0}^{T}w(t-s,X_{s,t})\,\mathrm{d}t\mathrm{d}s\bigg{)}\bigg{]}$

Therefore, heuristically we would expect (GC) to hold if and only if (assuming
an infinite volume measure $\mathbb{P}_{\alpha,\infty}$ exists)

$\mathbb{E}_{\alpha,\infty}\bigg{[}\exp\bigg{(}-\alpha\int_{-\infty}^{0}\int_{0}^{\infty}w(t-s,X_{s,t})\,\mathrm{d}t\mathrm{d}s\bigg{)}\bigg{]}>0$

i.e.
$\mathbb{P}_{\alpha,\infty}\big{(}\int_{-\infty}^{0}\int_{0}^{\infty}w(t-s,X_{s,t})\,\mathrm{d}t\mathrm{d}s<\infty\big{)}>0$;
it would be interesting to have a rigorous proof (or counterexample) for this
connection, as well as some understanding how it relates to Equation (5.1).

######  Lemma 5.4.

Assume that (A1) holds and that $w(t,\cdot)$ is quasiconcave for all $t>0$.
Then for all $n\in{\mathbb{N}}$, $\varepsilon,\delta>0$ and
$\xi\in\mathbf{N}_{f}(\triangle)$ with $F(\xi)<\infty$

$\mathbf{P}_{\xi}\big{(}\\{\mathbf{x}\in
C(\mathbb{R},\mathbb{R}^{d}):\,\omega_{n}(\mathbf{x},\delta)>\varepsilon\\}\big{)}\;\leqslant\;\mathcal{W}\big{(}\\{\mathbf{x}\in
C(\mathbb{R},\mathbb{R}^{d}):\,\omega_{n}(\mathbf{x},\delta)>\varepsilon\\}\big{)}.$

###### Proof.

We enumerate

$[-n,n]\cap\mathbb{Q}=\\{q_{1},q_{2},q_{3},\ldots\\}.$

Let $\xi=\sum_{i=1}^{m}\delta_{(s_{i},t_{i})}\in\mathbf{N}_{f}(\triangle)$
with $F(\xi)<\infty$ and $k\in{\mathbb{N}}$. We define
$f_{1},\ldots,f_{m}:\mathbb{R}^{d(k^{2}+m)}\to(0,\infty]$ by

$f_{i}(x_{11},x_{12},\ldots,x_{kk},y_{1},\ldots,y_{m})\coloneqq
v(t_{i}-s_{i},y_{i})$

for all $1\;\leqslant\;i\;\leqslant\;m$ and
$f_{m+1}:\mathbb{R}^{d(k^{2}+m)}\to[0,\infty)$ by

$f_{m+1}(x_{11},x_{12},\ldots,x_{kk},y_{1},\ldots,y_{m})\coloneqq\begin{cases}1\text{
if }&\exists i,j\in\\{1,\ldots,k\\}:\,|x_{ij}|>\varepsilon\\\ &\text{ and
}|q_{i}-q_{j}|<\delta\\\ 0\text{ else}\end{cases}.$

Then $f_{m+1}$ is symmetric and quasiconvex and $f_{1},\ldots,f_{m}$ are
symmetric and quasiconcave. If we define

$Y\coloneqq(X_{q_{1},q_{1}},\ldots,X_{q_{1},q_{k}},X_{q_{2},q_{1}},\ldots,X_{q_{k},q_{k}},X_{s_{1},t_{1}},\ldots,X_{s_{m},t_{m}})$

we hence get with the Gaussian correlation inequality

$\displaystyle\mathbf{P}_{\xi}\big{(}\big{\\{}\mathbf{x}\in
C(\mathbb{R},\mathbb{R}^{d}):\,(\exists
i,j\in\\{1,\ldots,k\\}:|q_{i}-q_{j}|<\delta\text{ and
}|\mathbf{x}_{q_{i},q_{j}}|>\varepsilon)\big{\\}}\big{)}$
$\displaystyle=\frac{1}{F(\xi)}\mathbb{E}_{\mathcal{W}}\Big{[}\prod_{i=1}^{m+1}f_{i}(Y)\Big{]}$
$\displaystyle\;\leqslant\;\mathbb{E}_{\mathcal{W}}[f_{m+1}(Y)]$
$\displaystyle=\mathcal{W}\big{(}\\{\mathbf{x}\in
C(\mathbb{R},\mathbb{R}^{d}):\,(\exists
i,j\in\\{1,\ldots,k\\}:|q_{i}-q_{j}|<\delta\text{ and
}|\mathbf{x}_{q_{i},q_{j}}|>\varepsilon)\big{\\}}\big{)}.$

Writing

$\displaystyle\\{\mathbf{x}\in
C(\mathbb{R},\mathbb{R}^{d}):\,\omega_{n}(\mathbf{x},\delta)>\varepsilon\\}$
$\displaystyle=\bigcup_{k=1}^{\infty}\big{\\{}\mathbf{x}\in
C(\mathbb{R},\mathbb{R}^{d}):\,(\exists
i,j\in\\{1,\ldots,k\\}:|q_{i}-q_{j}|<\delta\text{ and
}|\mathbf{x}_{q_{i},q_{j}}|>\varepsilon)\big{\\}}$

and using continuity of measures from below yields the claim. ∎

######  Lemma 5.5.

Assume that $w(t,\cdot)$ is quasiconcave for all $t>0$ and that Conditions
(A1)-(A3) are met. Then the family of measures
$\big{\\{}\mathbb{P}_{\alpha,\infty}\circ(X^{n})^{-1}:\,n\in{\mathbb{N}}\big{\\}}$
is tight.

###### Proof.

As $\mathbb{P}_{\alpha,\infty}(X^{n}_{0}=0)=1$ for all $n\in{\mathbb{N}}$ the
tightness is equivalent to

$\forall
m\in{\mathbb{N}}\,\forall\eta>0\,\forall\varepsilon>0\,\exists\delta>0\,\forall
n\in{\mathbb{N}}:\,\mathbb{P}_{\alpha,\infty}(\omega_{m}(X^{n},\delta)>\varepsilon)<\eta.$

By Lemma 5.4, we have for $n,m\in{\mathbb{N}}$ and $\varepsilon,\delta>0$

$\displaystyle\mathbb{P}_{\alpha,\infty}(\omega_{m}(X^{n},\delta)>\varepsilon)$
$\displaystyle=\mathbb{P}_{\alpha,\infty}(\omega_{mn}(X,\delta
n)>\sqrt{n}\varepsilon)$
$\displaystyle=\mathbb{E}_{\alpha}\big{[}\mathbf{P}_{R_{0}^{mn}\hat{\eta}_{s}}\big{(}\omega_{mn}(X,\delta
n)>\sqrt{n}\varepsilon\big{)}\big{]}$
$\displaystyle\;\leqslant\;\mathbb{E}_{\alpha}\big{[}\mathcal{W}\big{(}\omega_{mn}(X,\delta
n)>\sqrt{n}\varepsilon\big{)}\big{]}$
$\displaystyle=\mathcal{W}\big{(}\omega_{mn}(X,\delta
n)>\sqrt{n}\varepsilon\big{)}$
$\displaystyle=\mathcal{W}(\omega_{m}(X,\delta)>\varepsilon)$

where we used Brownian scaling in the last step. The claim follows by the fact
that a single probability measure on a Polish space is tight. ∎

In order to prove the functional central limit theorem, it is left to show
convergence of the finite dimensional distributions. We will use the following
lemma.

######  Lemma 5.6.

Assume Conditions (A1)-(A3). Let $s_{1}<t_{1}<s_{2}<t_{2}<\ldots<s_{k}<t_{k}$.
Then, as $n\to\infty$,

$\sup_{A_{1},\ldots,A_{k}\in\mathcal{B}(\mathbb{R}^{d})}\Big{|}\mathbb{P}_{\alpha,\infty}(X^{n}_{s_{1},t_{1}}\in
A_{1},\ldots,X^{n}_{s_{k},t_{k}}\in
A_{k})-\prod_{i=1}^{k}\mathbb{P}_{\alpha,\infty}(X^{n}_{s_{i},t_{i}}\in
A_{i})\Big{|}\to 0.$

###### Proof.

Let
$((\hat{T}_{0},\hat{\gamma}),(\hat{d}_{1},\hat{\xi}_{1}),(\hat{d}_{2},\hat{\xi}_{2}),\ldots)$
be a sequence of independent $(0,\infty)\times\mathbf{N}_{f}(\triangle)$
valued random variables such that

  * •

For all $k\;\geqslant\;1$ the random variable $(\hat{d}_{k},\hat{\xi}_{k})$ is
$\operatorname{Exp}(\psi(\alpha))\otimes\widehat{\Xi}_{\alpha}$ distributed

  * •

$(\hat{T}_{0},\hat{\gamma})$ is as in Proposition 4.4 such that
$R_{0}^{\infty}\hat{\eta}_{s}\overset{d}{=}\hat{\gamma}+\theta_{-\hat{T}_{0}}\hat{\eta}$
(where $\hat{\eta}$ denotes the process obtained by alternating dormant
periods $\hat{d}_{k}$ and active clusters $\hat{\xi}_{k}$ (starting dormant)
and $\hat{\eta}_{s}$ denotes a stationary version of $\hat{\eta}$).

By stationarity, we may assume w.l.o.g. that $s_{1}=0$. Let
$A_{1},\ldots,A_{k}\in\mathcal{B}(\mathbb{R}^{d})$. For $n\in{\mathbb{N}}$, we
define $f_{n},g_{n}:\mathbf{N}_{f}(\triangle)\to[0,1]$ by

$\displaystyle f_{n}(\xi)$
$\displaystyle\coloneqq\mathbf{P}_{\xi}(X^{n}_{0,t_{1}}\in
A_{1},\ldots,X_{s_{k-1},t_{k-1}}^{n}\in A_{k-1})$ $\displaystyle g_{n}(\xi)$
$\displaystyle\coloneqq\mathbf{P}_{\xi}(X_{0,t_{k}-s_{k}}^{n}\in A_{k}).$

For $j\;\geqslant\;1$ let (as usual) $\hat{T}_{j}$ denote the sum of the
$j$-th dormant and active period. Let
$(\hat{S}_{j}^{s})_{j\;\geqslant\;0}\coloneqq(\hat{T}_{0},\hat{T}_{0}+\hat{T}_{1},\ldots)$
denote the embedded delayed renewal process of $R_{0}^{\infty}\hat{\eta}_{s}$
and, for $n\in{\mathbb{N}}$,

$Y_{n}\coloneqq\inf\\{\hat{S}^{s}_{j}-nt_{k-1}:\,j\;\geqslant\;0\text{ such
that }\hat{S}^{s}_{j}>nt_{k-1}\\}$

denote the forward recurrence time at time $nt_{k-1}$. Let $\varepsilon>0$. By
Proposition 4.4 and the convergence of the distribution of $Y_{n}$ as
$n\to\infty$ (it does not matter that the renewal process is delayed by
$\hat{T}_{0}$, see e.g. [Asm03, p. 155]), there exists a $N\in{\mathbb{N}}$
and a $T>0$ such that for all $n\;\geqslant\;N$ and $t\;\geqslant\;T$

$\displaystyle\hat{\mathbb{P}}_{\alpha}\big{(}Y_{n}>T\big{)}<\varepsilon,\quad
n(s_{k}-t_{k-1})>2T,\,$
$\displaystyle\big{\|}\hat{\mathbb{P}}_{\alpha}\circ(R_{0}^{\infty}\hat{\eta}_{s})^{-1}-\hat{\mathbb{P}}_{\alpha}\circ(R_{0}^{\infty}\theta_{t}\hat{\eta})^{-1}\big{\|}<\varepsilon.$

As $n(s_{k}-t_{k-1})-Y_{n}(\omega)>T$ for $\omega\in\\{Y_{n}\;\leqslant\;T\\}$
and $n\;\geqslant\;N$ and since

$\mathbb{P}_{\alpha,\infty}(X^{n}_{s_{k},t_{k}}\in
A_{k})=\hat{\mathbb{E}}_{\alpha}\big{[}g_{n}(R_{0}^{n(t_{k}-s_{k})}\hat{\eta}_{s})\big{]}$

conditioning with respect to the process up to the first renewal after
$nt_{k-1}$ yields

$\displaystyle\Big{|}\mathbb{P}_{\alpha,\infty}(X^{n}_{0,t_{1}}\in
A_{1},\ldots,X^{n}_{s_{k},t_{k}}\in A_{k})$
$\displaystyle-\mathbb{P}_{\alpha,\infty}(X^{n}_{0,t_{1}}\in
A_{1},\ldots,X^{n}_{s_{k-1},t_{k-1}}\in
A_{k-1})\mathbb{P}_{\alpha,\infty}(X^{n}_{s_{k},t_{k}}\in A_{k})\Big{|}$
$\displaystyle\;\leqslant\;2\varepsilon+\int_{\hat{\Omega}}\hat{\mathbb{P}}_{\alpha}(\mathrm{d}\omega)\,\mathds{1}_{\\{Y_{n}\;\leqslant\;T\\}}(\omega)f_{n}(R_{0}^{nt_{k-1}}\hat{\eta}_{s}(\omega))$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\cdot\Big{|}\hat{\mathbb{E}}_{\alpha}\big{[}g_{n}\big{(}R_{0}^{n(t_{k}-s_{k})}\theta_{n(s_{k}-t_{k-1})-Y_{n}(\omega)}\,\hat{\eta}\big{)}\big{]}-\mathbb{P}_{\alpha,\infty}(X^{n}_{s_{k},t_{k}}\in
A_{k})\Big{|}$ $\displaystyle\;\leqslant\;4\varepsilon$

for all $n\;\geqslant\;N$. The claim follows by inductively applying this
argument. ∎

For a proof of the following theorem by Rényi [Rén63] (for $d=1$) that can
directly be generalized to higher dimensions we refer the reader to [Gut13, p.
346 f.].

######  Theorem 5.7 (Anscombe-Rényi).

Let $(X_{n})_{n}$ be an iid sequence of centered random vectors with
$\mathbb{E}[|X_{1}|^{2}]<\infty$ and $(N_{t})_{t}$ be a family of
$\mathbb{N}$-valued random variables such that $N_{t}/t$ converges in
probability to a constant $\theta>0$ as $t\to\infty$. Then

$\frac{1}{\sqrt{t}}\sum_{k=1}^{N_{t}}X_{k}\overset{d}{\rightarrow}\mathcal{N}(0,\theta\Sigma)$

as $t\to\infty$ where $\Sigma\coloneqq\mathbb{E}[X_{1}\cdot X_{1}^{T}]$.

######  Lemma 5.8.

Assume that $w(t,\cdot)$ is quasiconcave for all $t>0$ and that (A1)-(A3)
hold. Let $s_{1}<t_{1}<\ldots<s_{k}<t_{k}$. Then

$\mathbb{P}_{\alpha,\infty}\circ(X^{n}_{s_{1},t_{1}},\ldots,X^{n}_{s_{k},t_{k}})^{-1}\Rightarrow\bigotimes_{i=1}^{k}\mathcal{N}\big{(}0,(t_{i}-s_{i})\Sigma\big{)}$

as $n\to\infty$, where

$\Sigma\coloneqq\frac{\hat{\mathbb{E}}_{\alpha}[\Sigma(\hat{d}_{1},\hat{\xi}_{1})]}{\hat{\mathbb{E}}_{\alpha}[\widehat{T}_{1}]}\quad\text{with
}\Sigma(\hat{d}_{1},\hat{\xi}_{1})\coloneqq\hat{d}_{1}I_{d}+\mathbf{E}_{\hat{\xi}_{1}}\big{[}X_{0,\hat{a}_{1}}\cdot
X_{0,\hat{a}_{1}}^{T}\big{]},$

and
$(\hat{d}_{1},\hat{\xi}_{1})\sim\operatorname{Exp}(\psi(\alpha))\otimes\widehat{\Xi}_{\alpha}$,
$\hat{a}_{1}$ is the length of the cluster $\hat{\xi}_{1}$ and
$\widehat{T}_{1}=\hat{a}_{1}+\hat{d}_{1}$.

###### Proof.

By Lemma 5.6, it is sufficient to show that for all $a<b$

$\mathbb{P}_{\alpha,\infty}\circ(X^{n}_{a,b})^{-1}\Rightarrow\mathcal{N}\big{(}0,(b-a)\Sigma\big{)}$

as $n\to\infty$. By stationarity, we may assume w.l.o.g. that $a=0$. In order
to reduce notation we will additionally assume that $b=1$.
Let
$\big{(}(\hat{T}_{0},\hat{\gamma},\hat{Y}^{0}),(\hat{d}_{1},\hat{\xi}_{1},\hat{Y}^{1}),(\hat{d}_{2},\hat{\xi}_{2},\hat{Y}^{2}),\ldots\big{)}$
be a sequence of independent $(0,\infty)\times\mathbf{N}_{f}(\triangle)\times
C(\mathbb{R},\mathbb{R}^{d})$ valued random variables such that

  * •

For all $k\;\geqslant\;1$ the random variable $(\hat{d}_{k},\hat{\xi}_{k})$ is
$\operatorname{Exp}(\psi(\alpha))\otimes\widehat{\Xi}_{\alpha}$ distributed
and $\hat{Y}^{k}$ has conditionally on $(\hat{d}_{k},\hat{\xi}_{k})$
distribution $\mathbf{P}_{\theta_{-\hat{d}_{k}}\hat{\xi}_{k}}$

  * •

$(\hat{T}_{0},\hat{\gamma})$ is as in Proposition 4.4 such that
$R_{0}^{\infty}\hat{\eta}_{s}\overset{d}{=}\hat{\gamma}+\theta_{-\hat{T}_{0}}\hat{\eta}$
(where $\hat{\eta}$ denotes as usual the process obtained by alternating
dormant periods $\hat{d}_{k}$ and active clusters $\hat{\xi}_{k}$ (starting
dormant) and $\hat{\eta}_{s}$ denotes a stationary version of $\hat{\eta}$)

  * •

$\hat{Y}^{0}$ has conditionally on $(\hat{T}_{0},\hat{\gamma})$ distribution
$\mathbf{P}_{\hat{\gamma}}$.

In particular

$\hat{\mathbb{E}}_{\alpha}\Big{[}\hat{Y}_{0,\hat{T}_{1}}^{1}\cdot(\hat{Y}_{0,\hat{T}_{1}}^{1})^{T}\Big{]}=\hat{\mathbb{E}}_{\alpha}[\Sigma(\hat{d}_{1},\hat{\xi}_{1})]$

where $\hat{T}_{k}$ denotes for $k\in{\mathbb{N}}$ as usual the sum of the
$k$-th active and dormant period. Notice that the existence and finiteness of
the second moments follows from

$\hat{\mathbb{E}}_{\alpha}\Big{[}\big{|}\hat{Y}^{1}_{0,\hat{T}_{1}}\big{|}^{2}\Big{]}\;\leqslant\;d\cdot\hat{\mathbb{E}}_{\alpha}[\hat{T}_{1}]<\infty$

(remember Corollary 5.2). Let
$(\hat{S}_{n})_{n\;\geqslant\;0}=(0,\hat{T}_{1},\hat{T}_{1}+\hat{T}_{2},\ldots)$
denote the embedded renewal process of $\hat{\eta}$. For $t\;\geqslant\;0$ let

$\hat{X}_{t}\coloneqq\sum_{k=1}^{n}\hat{Y}^{k}_{0,\hat{T}_{k}}+\hat{Y}^{n+1}_{0,t-\hat{S}_{n}}\quad\text{
where $n\in{\mathbb{N}}_{0}$ is such that }t\in[\hat{S}_{n},\hat{S}_{n+1}).$

For $t\;\geqslant\;0$ let $\hat{A}_{t}$ denote the forward recurrence time of
the embedded renewal process of $\hat{\eta}$. By conditioning with respect to
$(\hat{T}_{0},\hat{\gamma},\hat{Y}^{0})$ we get with $V_{n}\coloneqq
n-\hat{T}_{0}$ for $f:\mathbb{R}^{d}\to\mathbb{R}$ continuous and bounded

$\displaystyle\mathbb{E}_{\alpha,\infty}\big{(}f(X^{n}_{1})\big{)}=\hat{\mathbb{E}}_{\alpha}\Big{[}\mathbf{E}_{R_{0}^{n}\hat{\eta}_{s}}\big{[}f(X^{n}_{1})\big{]}\Big{]}$
$\displaystyle=\int_{\hat{\Omega}}\hat{\mathbb{P}}_{\alpha}(\mathrm{d}\omega_{1})\int_{\hat{\Omega}}\hat{\mathbb{P}}_{\alpha}(\mathrm{d}\omega_{2})\,f\bigg{(}\tfrac{1}{\sqrt{n}}\hat{Y}^{0}_{0,\hat{T}_{0}(\omega_{1})\wedge
n}(\omega_{1})$
$\displaystyle\quad+\mathds{1}_{\\{\hat{T}_{0}(\omega_{1})<n\\}}\bigg{[}\tfrac{1}{\sqrt{n}}\sum_{k=1}^{\hat{\tau}_{V_{n}(\omega_{1})}(\omega_{2})}\hat{Y}^{k}_{0,\hat{T}_{k}(\omega_{2})}(\omega_{2})-\tfrac{1}{\sqrt{n}}\hat{X}_{V_{n}(\omega_{1}),V_{n}(\omega_{1})+\hat{A}_{V_{n}(\omega_{1})}(\omega_{2})}(\omega_{2})\bigg{]}\bigg{)}.$
(5.2)

Now, fix $\omega_{1}\in\hat{\Omega}$ and set
$r\coloneqq\hat{T}_{0}(\omega_{1})$. We have by Theorem 4.1

$\frac{\hat{\tau}_{n-r}}{n}\to\frac{1}{\hat{\mathbb{E}}[\widehat{T}_{1}]}$

almost surely as $n\to\infty$. By Theorem 5.7

$\frac{1}{\sqrt{n}}\sum_{k=1}^{\hat{\tau}_{n-r}}\hat{Y}_{0,\hat{T}_{k}}^{k}\overset{d}{\rightarrow}\mathcal{N}(0,\Sigma)$

as $n\to\infty$. For $\varepsilon,\delta>0$ let $N\in{\mathbb{N}}$ be such
that for all $n\;\geqslant\;N$

$\|\hat{\mathbb{P}}_{\alpha}\circ(\hat{X}_{n-r,n-r+\hat{A}_{n-r}})^{-1}-\hat{\mathbb{P}}_{\alpha}\circ(\hat{Y}^{0}_{0,\hat{T}_{0}})^{-1}\|<\delta$

and
$\hat{\mathbb{P}}_{\alpha}(|\hat{Y}^{0}_{0,\hat{T}_{0}}|>\sqrt{n}\varepsilon)<\delta$.
Then, for all $n\;\geqslant\;N$

$\hat{\mathbb{P}}_{\alpha}\Big{(}\tfrac{1}{\sqrt{n}}\big{|}\hat{X}_{n-r,n-r+\hat{A}_{n-r}}\big{|}>\varepsilon\Big{)}<2\delta.$

Hence, by Slutsky’s theorem, the inner integral in Equation (5) converges for
all $\omega_{1}\in\hat{\Omega}$ to
$\int_{\mathbb{R}^{d}}\mathcal{N}(0,\Sigma)(\mathrm{d}x)f(x)$. Thus, by
dominated convergence,

$\lim_{n\to\infty}\mathbb{E}_{\alpha,\infty}\big{(}f(X^{n}_{1})\big{)}=\int_{\mathbb{R}^{d}}\mathcal{N}(0,\Sigma)(\mathrm{d}x)f(x).\qed$

###### Proof of Theorem 3.4.

We briefly give the argument why we could exclude the case $t_{i}=s_{i+1}$ for
some $1\;\leqslant\;i\;\leqslant\;k-1$ in Lemma 5.8. Let $(n_{k})_{k}$ be a
strictly increasing sequence of natural numbers. By tightness, there exists a
subsequence $(n_{k_{j}})_{j}$ and a measure $\mathcal{P}$ on
$C(\mathbb{R},\mathbb{R}^{d})$ such that

$\mathbb{P}_{\alpha,\infty}\circ(X^{n_{k_{j}}})^{-1}\Rightarrow\mathcal{P}$

as $j\to\infty$. By the continuous mapping theorem and Lemma 5.8, for any
$s_{1}<t_{1}<\ldots<s_{k}<t_{k}$

$\mathcal{P}\circ(X_{s_{1},t_{1}},\ldots
X_{s_{k},t_{k}})^{-1}=\tilde{\mathcal{P}}\circ(X_{s_{1},t_{1}},\ldots
X_{s_{k},t_{k}})^{-1}$

where $\tilde{\mathcal{P}}$ denotes the distribution of $\sqrt{\Sigma}X$ under
$\mathcal{W}$. By approximation, we obtain for all
$s_{1}<t_{1}\;\leqslant\;\ldots\;\leqslant\;s_{k}<t_{k}$

$\mathcal{P}\circ(X_{s_{1},t_{1}},\ldots
X_{s_{k},t_{k}})^{-1}=\tilde{\mathcal{P}}\circ(X_{s_{1},t_{1}},\ldots
X_{s_{k},t_{k}})^{-1}$

i.e. $\mathcal{P}=\tilde{\mathcal{P}}$. Hence, each subsequence of
$(\mathbb{P}_{\alpha,\infty}\circ(X^{n})^{-1})_{n}$ has a subsequence that
converges weakly to $\tilde{\mathcal{P}}$. This implies that
$(\mathbb{P}_{\alpha,\infty}\circ(X^{n})^{-1})_{n}$ converges weakly to
$\tilde{\mathcal{P}}$. It is left to show $\Sigma\;\leqslant\;I_{d}$. By the
Gaussian correlation inequality in the form of Proposition 5.1, we have for
all $x\in\mathbb{R}^{d}$

$\mathbf{E}_{\hat{\xi}_{1}}\big{[}\langle
x,X_{0,\hat{a}_{1}}\rangle^{2}\big{]}\;\leqslant\;\mathbb{E}_{\mathcal{W}}[\langle
x,X_{0,\hat{a}_{1}}\rangle^{2}]=\hat{a}_{1}|x|^{2}$

and hence

$\displaystyle\big{\langle}x,\hat{\mathbb{E}}_{\alpha}[\Sigma(\hat{d}_{1},\hat{\xi}_{1})]x\big{\rangle}$
$\displaystyle=\hat{\mathbb{E}}_{\alpha}[\hat{d}_{1}]|x|^{2}+\hat{\mathbb{E}}_{\alpha}\bigg{[}\sum_{i,j=1}^{d}x_{i}x_{j}\mathbf{E}_{\hat{\xi}_{1}}\Big{[}X_{0,\hat{a}_{1}}^{i}X_{0,\hat{a}_{1}}^{j}\Big{]}\bigg{]}$
$\displaystyle=\hat{\mathbb{E}}_{\alpha}[\hat{d}_{1}]|x|^{2}+\hat{\mathbb{E}}_{\alpha}\Big{[}\mathbf{E}_{\hat{\xi}_{1}}\big{[}\langle
x,X_{0,\hat{a}_{1}}\rangle^{2}\big{]}\Big{]}$
$\displaystyle\;\leqslant\;\mathbb{E}[\hat{T}_{1}]|x|^{2}.\qed$

######  Remark 5.9.

If $w(t,\cdot)$ is additionally rotationally symmetric for all $t>0$ then
$\Sigma$ is a multiple of the unit matrix, i.e. the limiting distribution is a
rescaled Wiener measure.

##  6\. Proof of Proposition 2.4 c)

In this section we assume that $w(t,\cdot)$ is rotationally symmetric and
positive (allowing the value $+\infty$) for all $t\;\geqslant\;0$, and that
the function $\tilde{w}$ defined by $w(\cdot,\cdot)=\tilde{w}(\cdot,|\cdot|)$
is decreasing in the second variable. Remember that we defined
$\tilde{w}_{\beta}=\sup_{t\;\geqslant\;0}\,{\rm e}^{\beta
t}\,\tilde{w}(t,\cdot)$ for $\beta>0$. Let $\beta>0$ and $p>1$ be such that
the integrability condition in Proposition 2.4 c) holds. We choose
$g(t)=\beta\,{\rm e}^{-\beta t}\,$ in the decomposition (3.1) and obtain

$v(t,x)=\frac{1}{\beta}\,{\rm e}^{\beta
t}\,w(t,x)\;\leqslant\;\frac{1}{\beta}\tilde{w}_{\beta}(|x|)$

for all $t\;\geqslant\;0$ and $x\in\mathbb{R}^{d}$. Notice that
$\tilde{w}_{\beta}$ is decreasing and (as $\tilde{w}_{\beta}$ additionally
satisfies the integrability condition) that $\tilde{w}_{\beta}(r)<\infty$ for
all $r\in(0,\infty)$. Without loss of generality we can assume that $v(t,x)>1$
for all $t\;\geqslant\;0$ and $x\in\mathbb{R}^{d}$. The reason is that the
function $\bar{w}$ defined by $\bar{w}(t,x)=w(t,x)+\beta\,{\rm e}^{-\beta
t}\,$ satisfies the assumptions of Proposition 2.4 c) if $w$ does, and leads
to $v>1$. As mentioned at the end of Section 3, (GC) holds for $\bar{w}$ if
and only if it holds for $w$.

Let $(\sigma_{n})_{n\;\geqslant\;0}$ be the iid sequence of
$\operatorname{Exp}(\alpha)$ distributed interarrival times and let
$(\tau_{n})_{n\;\geqslant\;1}$ be the iid sequence of
$\operatorname{Exp}(\beta)$ distributed service times (which is independent of
$(\sigma_{n})_{n}$) of our queue. For $n\in{\mathbb{N}}$ let

$s_{n}\coloneqq\sum_{k=0}^{n-1}\sigma_{k},\quad t_{n}\coloneqq
s_{n}+\tau_{n}.$

That is, $s_{n}$ is the arrival and $t_{n}$ the departure of the $n$-th
customer and $\sigma_{n}$ is the time that passes between the arrival of the
$n$-th and the $n+1$-th customer. Then the first cluster $\xi_{1}$ is given by

$\xi_{1}=\sum_{i=1}^{N}\delta_{(s_{i}-s_{1},t_{i}-s_{1})}$

where

$N=\inf\big{\\{}n\in{\mathbb{N}}:\tau_{k}<s_{n+1}-s_{k}\,\text{ for all
}1\;\leqslant\;k\;\leqslant\;n\big{\\}}$

is the number of customers in the first cluster. Notice that $N$ is a stopping
time with respect to the filtration generated by
$((\sigma_{n},\tau_{n}))_{n\;\geqslant\;1}$. We define the function

$h:(0,\infty)\times(0,\infty)\to(0,\infty],\quad
h(t_{1},t_{2})\coloneqq\mathbb{E}_{\mathcal{W}}[v(t_{1},X_{t_{2}})].$

######  Lemma 6.1.

We have

$\prod_{i=1}^{N}h(\tau_{i},\tau_{i})\;\leqslant\;F(\xi_{1})\;\leqslant\;\prod_{i=1}^{N}h(\tau_{i},\sigma_{i}\wedge\tau_{i}).$

###### Proof.

The lower estimate follows immediately from the Gaussian correlation
inequality in the form of Proposition 5.1. For the upper bound we distinguish
between two cases: If $t_{1}\;\leqslant\;s_{2}$ we immediately get by
independence of increments

$F(\xi_{1})=\mathbb{E}_{\mathcal{W}}\Big{[}\prod_{i=1}^{N}v(\tau_{i},X_{s_{i},t_{i}})\Big{]}=h(\tau_{1},\tau_{1})\mathbb{E}_{\mathcal{W}}\Big{[}\prod_{i=2}^{N}v(\tau_{i},X_{s_{i},t_{i}})\Big{]}.$

In case that $t_{1}>s_{2}$ we have by independence of $X_{s_{1},s_{2}}$ and
$(X_{s,t})_{s,t\;\geqslant\;s_{2}}$

$F(\xi_{1})=\int_{C(\mathbb{R},\mathbb{R}^{d})}\mathcal{W}(\mathrm{d}\mathbf{x})\,\mathbb{E}_{\mathcal{W}}\big{[}v\big{(}\tau_{1},X_{s_{1},s_{2}}+\mathbf{x}_{s_{2},t_{1}}\big{)}\big{]}\prod_{i=2}^{N}v\big{(}\tau_{i},\mathbf{x}_{s_{i},t_{i}}\big{)}.$

The assumptions on $w$ imply
$\mathbb{E}_{\mathcal{W}}\big{[}v\big{(}\tau_{1},X_{s_{1},s_{2}}+x\big{)}\big{]}\;\leqslant\;\mathbb{E}_{\mathcal{W}}\big{[}v\big{(}\tau_{1},X_{s_{1},s_{2}}\big{)}\big{]}$
for all $x\in\mathbb{R}^{d}$ and thus

$F(\xi_{1})\;\leqslant\;h(\tau_{1},\sigma_{1})\mathbb{E}_{\mathcal{W}}\Big{[}\prod_{i=2}^{N}v(\tau_{i},X_{s_{i},t_{i}})\Big{]}.$

Iterating this procedure completes the proof. ∎

For the Fröhlich polaron path measure we choose $\beta=1$ and obtain by an
integration in spherical coordinates

$h(t_{1},t_{2})=\sqrt{\frac{2}{\pi}}\frac{1}{\sqrt{t_{2}}}$

for all $t_{1},t_{2}>0$. Hence, our lower estimate for $F$ becomes the lower
estimate given in [MV19]. If one sorts the customers in the first cluster by
the time of their departure, starting with the customer that departs last, and
modifies our proof accordingly, one obtains an upper estimate for $F$ that is
sharper then the estimate given in [MV19].

######  Lemma 6.2.

Assume that $w$, $\beta$ and $p$ satisfy the assumptions of Proposition 2.4
c). Then for all $\alpha>0$,

${\mathbb{E}}_{\alpha}\big{[}h(\tau_{1},\sigma_{1}\wedge\tau_{1})^{p}\big{]}\;\leqslant\;(\alpha+\beta)^{(d+2)/4}c(w,\beta,p),$

where the constant $c(w,\beta,p)$ is independent of $\alpha$.

###### Proof.

By Jensens inequality,

$\mathbb{E}_{\alpha}[h(\tau_{1},\sigma_{1}\wedge\tau_{1})^{p}]\;\leqslant\;\mathbb{E}_{\alpha}\Big{[}\mathbb{E}_{\mathcal{W}}\big{[}v(\tau_{1},X_{\sigma_{1}\wedge\tau_{1}})^{p}\big{]}\Big{]}\;\leqslant\;\beta^{-p}\cdot\mathbb{E}_{\alpha}\Big{[}\mathbb{E}_{\mathcal{W}}\big{[}\tilde{w}_{\beta}(|X_{\sigma_{1}\wedge\tau_{1}}|)^{p}\big{]}\Big{]}.$

For $a,b>0$ we have

$\displaystyle\int_{0}^{\infty}\mathrm{d}t\,t^{-d/2}\exp\big{(}-a/t-b\cdot
t\big{)}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}t\,t^{d-3}\exp\big{(}-at^{2}-b/t^{2}\big{)}$
$\displaystyle=\left(b/a\right)^{(d-2)/4}K_{(d-2)/2}(2\sqrt{ab})$

(the second equality can be obtained by substituting $t=(b/a)^{1/4}e^{u}$)
where, for $\gamma\in\mathbb{R}$ and $x>0$

$K_{\gamma}(x)=\int_{0}^{\infty}e^{-x\cosh(u)}\cosh(\gamma u)\,\mathrm{d}u$

denotes the modified Bessel function of second kind. With appropriately chosen
constants, $c_{d},\tilde{c}_{d}>0$ we obtain

$\displaystyle\mathbb{E}_{\alpha}\Big{[}\mathbb{E}_{\mathcal{W}}\Big{[}\tilde{w}_{\beta}(|X_{\sigma_{1}\wedge\tau_{1}}|)^{p}\Big{]}\Big{]}$
$\displaystyle=c_{d}(\alpha+\beta)\int_{0}^{\infty}\mathrm{d}t\,e^{-(\alpha+\beta)t}\int_{0}^{\infty}\mathrm{d}r\,r^{d-1}\tilde{w}_{\beta}(r)^{p}e^{-r^{2}/2t}t^{-d/2}$
$\displaystyle=\tilde{c}_{d}(\alpha+\beta)^{(d+2)/4}\int_{0}^{\infty}\mathrm{d}r\,r^{d-1}\tilde{w}_{\beta}(r)^{p}r^{-(d-2)/2}K_{(d-2)/2}(\sqrt{2(\alpha+\beta)}\cdot
r)$
$\displaystyle\;\leqslant\;\tilde{c}_{d}(\alpha+\beta)^{(d+2)/4}\int_{0}^{\infty}\mathrm{d}r\,\tilde{w}_{\beta}(r)^{p}\cdot
r^{d/2}K_{(d-2)/2}(\sqrt{2\beta}\cdot r).$ (6.1)

Now, as $r\to\infty$, we have [AS72, p. 378]

$\displaystyle K_{(d-2)/2}(\sqrt{2\beta}\cdot
r)\in\mathcal{O}\big{(}r^{-1/2}e^{-\sqrt{2\beta}\cdot r}\big{)},$

and since $\tilde{w}_{\beta}$ is decreasing and thus bounded on $[1,\infty)$,
the integral (6.1) is finite on $[1,\infty)$. For $r\to 0$ we have [AS72, p.
375]

$\displaystyle K_{(d-2)/2}(\sqrt{2\beta}\cdot
r)\in\begin{cases}\mathcal{O}\big{(}-\ln(r)\big{)}\quad&\text{ for }d=2\\\ \\\
\mathcal{O}\big{(}r^{-|d-2|/2}\big{)}\quad&\text{ for }d\neq 2.\end{cases}$

The integrability assumptions in 2.4 c) then guarantee that the integral (6.1)
is finite on $[0,1]$, too, proving the claim. ∎

######  Lemma 6.3.

Let $(X_{n})_{n}$ be an iid sequence of positive random variables adapted to
some filtration $(\mathcal{F}_{n})_{n}$. Let $X_{n+1}$ be independent of
$\mathcal{F}_{n}$ for all $n\in{\mathbb{N}}$. Let $\tau$ be an a.s. finite
stopping time with respect to $(\mathcal{F}_{n})_{n}$. Assume that $p>1$ is
such that $\mathbb{E}[X^{p}]<\infty$. Then

$\mathbb{E}\Big{[}\prod_{i=1}^{\tau}X_{i}\Big{]}\;\leqslant\;\mathbb{E}\bigg{[}\Big{(}\mathbb{E}\big{[}X_{1}^{p}\big{]}^{q/p}\Big{)}^{\tau}\bigg{]}^{1/q}$

where $q$ denotes the conjugated Hölder index of $p$.

###### Proof.

For $n\in{\mathbb{N}}$ define

$Y_{n}\coloneqq\prod_{i=1}^{n}\frac{X_{i}^{p}}{\mathbb{E}[X_{1}^{p}]}.$

Then $(Y_{n})_{n}$ is a martingale with respect to $(\mathcal{F}_{n})_{n}$. By
the optional stopping theorem, we have $\mathbb{E}[Y_{\tau\wedge n}]=1$ for
all $n\in{\mathbb{N}}$. By Fatou’s Lemma

$\mathbb{E}[Y_{\tau}]\;\leqslant\;\liminf_{n\to\infty}\mathbb{E}[Y_{\tau\wedge
n}]=1.$

If we denote by $q$ the conjugated Hölder index to $p$ we get with Hölders
inequality

$\displaystyle\mathbb{E}\Big{[}\prod_{i=1}^{\tau}X_{i}\Big{]}$
$\displaystyle=\mathbb{E}\bigg{[}\mathbb{E}[X_{1}^{p}]^{\tau/p}\prod_{i=1}^{\tau}\frac{X_{i}}{\mathbb{E}[X_{1}^{p}]^{1/p}}\bigg{]}$
$\displaystyle\;\leqslant\;\mathbb{E}\Big{[}\mathbb{E}[X_{1}^{p}]^{\tau
q/p}\Big{]}^{1/q}\mathbb{E}[Y_{\tau}]^{1/p}$
$\displaystyle\;\leqslant\;\mathbb{E}\Big{[}\mathbb{E}[X_{1}^{p}]^{\tau
q/p}\Big{]}^{1/q}.\qed$

###### Proof of Proposition 2.4 c).

We first show that $Z_{\alpha,T}<\infty$ for all $\alpha,T>0$ and that
$\psi(\alpha)<\infty$ for all $\alpha>0$. Notice that, by the series expansion
of the exponential function and by a change of the order of integration (in
the same manner as in the beginning of Section 3), one obtains Equation (3.5)
even without assuming that $\mathbb{P}_{\alpha,T}$ defines a probability
measure (i.e. without assuming that $Z_{\alpha,T}<\infty$). Since
$T_{1}\;\geqslant\;\sigma_{0}+\ldots+\sigma_{N-1}+\tau_{N}$, Lemma 6.1 implies

$\mathbb{E}_{\alpha}[e^{-\mu
T_{1}}F(\xi_{1})]\;\leqslant\;\mathbb{E}_{\alpha}\Big{[}\prod_{i=1}^{N}e^{-\mu(\sigma_{i}\wedge\tau_{i})}h(\tau_{i},\sigma_{i}\wedge\tau_{i})\Big{]}.$
(6.2)

for all $\mu\;\geqslant\;0$. By Lemma 6.2, we have
$\mathbb{E}_{\alpha}[h(\tau_{1},\sigma_{1}\wedge\tau_{1})]<\infty$. Hence,
there exists a $\mu\;\geqslant\;0$ such that
$\mathbb{E}_{\alpha}[e^{-\mu(\sigma_{1}\wedge\tau_{1})}h(\tau_{1},\sigma_{1}\wedge\tau_{1})]\;\leqslant\;1$.

As in the proof of Lemma 6.3, an application of the optional stopping theorem
to the supermartingale
$\big{(}\prod_{i=1}^{n}e^{-\mu(\sigma_{i}\wedge\tau_{i})}h(\tau_{i},\sigma_{i}\wedge\tau_{i})\big{)}_{n\in{\mathbb{N}}}$
gives $\mathbb{E}_{\alpha}[e^{-\mu T_{1}}F(\xi_{1})]\;\leqslant\;1$ for this
$\mu$. By Proposition 4.6, there exist $C,\lambda>0$ such that
$\mathbf{Z}_{\alpha,T}\;\leqslant\;C\,{\rm e}^{\lambda T}\,$ for all
$T\;\geqslant\;0$. Since $e^{c_{\alpha,T}}\sim e^{2\alpha
T-\alpha\mathbb{E}[\tau_{1}]}$ this yield the existence of
$\widetilde{C},\tilde{\lambda}>0$ such that

$Z_{\alpha,T}\;\leqslant\;\widetilde{C}e^{\tilde{\lambda}T}$

for all $T\;\geqslant\;0$. By superadditivity

$\psi(\alpha)=\lim_{T\to\infty}\frac{\log(Z_{\alpha,T})}{T}=\sup_{T>0}\frac{\log(Z_{\alpha,T})}{T}\;\leqslant\;\tilde{\lambda}.$

For showing the validity of (GC), we show that (G) holds, and as we already
know that (A1) and (A2) hold, we may then apply Theorem 3.2. Since we assumed
$v>1$, we have $\mathbb{E}_{\alpha}[F(\xi_{1})]>1$ for all $\alpha>0$, and by
Proposition 4.13 it is thus sufficient to show that
$\mathbb{E}_{\alpha}[F(\xi_{1})]<\infty$ for sufficiently small $\alpha$. By
Lemma 6.1 it is sufficient to show that

$\mathbb{E}_{\alpha}\Big{[}\prod_{i=1}^{N}h(\tau_{i},\sigma_{i}\wedge\tau_{i})\Big{]}<\infty$

for sufficiently small $\alpha$. Let $r_{\alpha}$ be the radius of convergence
of the probability generating function of $N$. By Lemma 6.3, it is sufficient
to show that

$\mathbb{E}_{\alpha}\big{[}h(\tau_{1},\sigma_{1}\wedge\tau_{1})^{p}\big{]}<r_{\alpha}^{p/q}$
(6.3)

for sufficiently small $\alpha$. One can convince oneself (e.g. by looking at
the known formula of the probability generating function of $N$ for this
particular choice of $g$, see [GS95]) that $\lim_{\alpha\to
0}r_{\alpha}=\infty$. On the other hand, Lemma 6.2 shows that the left hand
side of (6.3) remains bounded as $\alpha\to 0$. This shows the claim. ∎

Figure 2. Expected value $\mathbb{E}_{\alpha}\big{[}e^{-\lambda
T_{1}}F(\xi_{1})\big{]}$ in dependency of $\alpha$ and $\lambda$ under the
assumptions of Proposition 2.4 c) and assuming that $v>1$. In the shaded
region we do not know whether the expected value is finite or infinite

######  Remark 6.4.

Let us consider the situation from Proposition 2.4 c). We choose $\beta$ and
$g$ as in the proof of Proposition 2.4 c) and assume, as in the proof, that
$v>1$. By Proposition 4.12 we have for all $\alpha>0$

$\psi(\alpha)-\alpha=\min\big{\\{}\lambda\in\mathbb{R}:\,\mathbb{E}_{\alpha}\big{[}e^{-\lambda
T_{1}}F(\xi_{1})\big{]}\;\leqslant\;1\big{\\}}.$

Using the known formula for the Laplace transform of an active period for this
choice of $g$, see [GS95], one obtains that the behavior of
$\mathbb{E}_{\alpha}\big{[}e^{-\lambda T_{1}}F(\xi_{1})\big{]}$ as a function
of $\alpha$ and $\lambda$ is as depicted in Figure 2. We call $\alpha$ “good”
for $w$ if (GC) is satisfied. Notice that $\alpha$ is good for $w$ if and only
if $1$ is good for $\alpha w$. A small calculation shows (where we denote the
dependency on $w$ by another subscript)

$\displaystyle\partial_{\alpha}\widehat{\Gamma}_{1,T,\alpha w}(\text{the
system is dormant at $0$})\;\leqslant\;0$
$\displaystyle\iff\mathbb{E}_{\widehat{\Gamma}_{1,T,\alpha
w}}[N_{\triangle}]\;\geqslant\;\mathbb{E}_{\widehat{\Gamma}_{1,T,\alpha
w}}[N_{\triangle}|\text{the system is dormant at $0$}]$

where $N_{\triangle}:\mathbf{N}_{f}(\triangle)\to{\mathbb{N}}_{0}$,
$\mu\mapsto\mu(\triangle)$ denotes the number of points in $\triangle$. Hence,
it seems plausible that $\widehat{\Gamma}_{1,T,\alpha w}(\text{the system is
dormant at $0$})$ is decreasing in $\alpha$ for all $T>0$, which would imply
(by Theorem 3.2) that the set of $\alpha$ that are good for $w$ is of the form
$(0,b)$ or $(0,b]$ for some $b\in(0,\infty]$. Then, if $b<\infty$, the
function $\alpha\mapsto\lim_{T\to\infty}Z_{\alpha,T}e^{-\psi(\alpha)T}$ would
be non-analytic (the limit exists irrespective whether (GC) is satisfied or
not by Theorem 4.11 and Proposition 4.12).

##  7\. Relations between $\psi(\alpha)$ and
$\widehat{\Gamma}_{\alpha,\operatorname{st}}$

In this short section we present a few formal calculations that relate the
free energy $\psi$ to certain expectations with respect to the tilted
stationary measure $\widehat{\Gamma}_{\alpha,\operatorname{st}}$. Although we
exchange limits and integrals in an uncontrolled way in several places, we
expect the resulting formulae to be correct. They show that there is an
intricate relationship between $\psi$ and the expected value of several
natural random variables with respect to
$\widehat{\Gamma}_{\alpha,\operatorname{st}}$ which seems well worth exploring
further in the future.

We assume that (A1)–(A3) are satisfied for all $\alpha$ and hence

$\mathbb{E}_{\alpha}\big{[}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})\big{]}=1$

for all $\alpha>0$. Multiplying the potential by $e^{c}$ for some
$c\in\mathbb{R}$ yields with the number $N$ of customers in the first cluster

$\mathbb{E}_{\alpha}\big{[}e^{-(\psi(e^{c}\alpha)-\alpha)T_{1}}e^{cN}F(\xi_{1})\big{]}=1.$

Assuming that $\psi$ is differentiable and that we may differentiate under the
integral we obtain by differentiation with respect to $c$

$\alpha\psi^{\prime}(\alpha)\mathbb{E}_{\alpha}\big{[}T_{1}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})\big{]}=\mathbb{E}_{\alpha}\big{[}Ne^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})]$

or short
$\alpha\psi^{\prime}(\alpha)\hat{\mathbb{E}}_{\alpha}[\widehat{T}_{1}]=\hat{\mathbb{E}}_{\alpha}[\widehat{N}]$.
In other words, $\psi(\alpha)=1/\hat{\mathbb{E}}_{\alpha}[\hat{d}_{1}]$
determines the length of dormant periods in the reweighted process and
$\alpha\psi^{\prime}(\alpha)$ determines the number of points per unit of time
in the reweighted process. Notice that

$\psi^{\prime}(\alpha)=\frac{\hat{\mathbb{E}}_{\alpha}[\widehat{N}]/\hat{\mathbb{E}}_{\alpha}[\widehat{T}_{1}]}{\mathbb{E}_{\alpha}[N]/\mathbb{E}_{\alpha}[T_{1}]}$
(7.1)