# Proof of Zamolodchikov conjecture for semi-classical conformal blocks on
torus

Harini Desiraju  ,  Promit Ghosal  and  Andrei Prokhorov  Sydney Mathematical
Research Institute (SMRI), School of Mathematics and Statistics, University of
Sydney, Camperdown, NSW 2006, Australia<EMAIL_ADDRESS>University of Chicago, Department of Statistics, 5747 S Ellis Ave, Chicago, IL
60637, USA<EMAIL_ADDRESS>University of Michigan, 500 S State St, Ann
Arbor, MI 48109, USA, and St. Petersburg University, 7/9 Universitetskaya nab.
199034 St. Petersburg, Russia<EMAIL_ADDRESS>
###### Abstract.

In 1986, Zamolodchikov conjectured an exponential structure for the semi-
classical limit of conformal blocks on a sphere. This paper provides a
rigorous proof of the analog of Zamolodchikov conjecture for Liouville
conformal blocks on a one-punctured torus, using their probabilistic
construction and show the existence of a positive radius of convergence of the
semi-classical limit. As a consequence, we obtain a closed form expression for
the solution of the Lamé equation, and show a relation between its accessory
parameter and the classical action of the non-autonomous elliptic Calogero-
Moser model evaluated at specific values of the solution.

###### Contents

  1. 1 Introduction
  2. 2 Probabilistic conformal blocks
  3. 3 Derivative gaussian multiplicative chaos measure
  4. 4 Semi-classical limits for deformed conformal blocks
  5. 5 Semi-classical limit of undeformed conformal blocks
  6. 6 Asymptotics of semi-classical conformal block as $\tau\to\mathbf{i}\infty$
  7. A Useful notation and elliptic functions
  8. B Non-autonomous elliptic Calogero-Moser model

##  1\. Introduction

Conformal field theory (CFT), a subclass of quantum field theory provides a
framework for constructing random functions on Riemannian manifolds, which
transform covariantly under conformal (i.e., angle-preserving) mappings. Since
the groundbreaking work by Belavin, Polyakov, and Zamolodchikov (BPZ) in 1984
[BPZ84], two-dimensional (2D) CFT has evolved into a major area of theoretical
physics. It has significant applications in 2D statistical physics, string
theory and has profound implications for mathematics. See [Rib14] for a review
and references therein.

An important subclass of conformal field theories (CFTs) is known as Liouville
CFTs (LCFTs). These CFTs are characterized by a parameter called the central
charge, denoted as $c$, which for LCFTs lies in the range $c\in[25,\infty)$.
LCFT originated from Polyakov’s work on two-dimensional quantum gravity and
bosonic string theory in 1981 [Pol81]. It was rigorously constructed using the
path integral formalism of quantum field theory, initially on the sphere in
[Dav+16] and subsequently on other Riemann surfaces in [DRV16, HRV18, GRV19].
We refer the reader to [RV] for a review.

The paper by Belavin, Polyakov, and Zamolodchikov in 1984 introduced a
schematic approach known as the conformal bootstrap. This program aims to
exactly solve the correlation functions of a given two-dimensional CFT by
expressing them in terms of its three-point sphere correlation functions and
specific power series called conformal blocks. These conformal blocks are
entirely determined by the Virasoro algebra, which encodes the infinitesimal
local conformal symmetries. The conformal blocks depend on the particular CFT
solely through the central charge.

Recently a construction of Liouville conformal blocks using rigorous path
integral formulation based on probabilistic ideas was introduced by [Gho+24].
The construction is via the Gaussian multiplicative chaos (GMC) measure, a
random measure defined by exponentiating the Gaussian free field (see e.g.
[DRV16, Ber17, Aru20]). This paper explores the semi-classical limit of the
probabilistic representation of Liouville conformal blocks, focusing on the
case of torus with a single marked point. For a given in the upper half-plane,
we consider the flat torus $\mathbb{T}_{\tau}$ characterized by the modular
parameter $\tau$, and the nome $q:=e^{\mathbf{i}\pi\tau}$. The $1$-point toric
correlation function of LCFT, rigorously constructed in [DRV16], has the form
$\langle e^{\alpha\Phi(0)}\rangle_{\tau}$, where $\langle\ldots\rangle_{\tau}$
is the average over the random field $\phi$ for LCFT on $\mathbb{T}_{\tau}$
and $\alpha$ is called the vertex insertion weight. Conformal bootstrap
formula for torus which is recently shown in [Gui+21, Theorem 9] expresses
$\langle e^{\alpha\Phi(0)}\rangle_{\tau}$ via one point toric conformal block
$\mathcal{G}^{\alpha}_{\gamma,P}(q)$,

$\displaystyle\langle
e^{\alpha\Phi(0)}\rangle_{\tau}=\int_{\mathbb{R}}C_{\gamma}(\alpha,Q-\mathbf{i}P,Q+\mathbf{i}P)|\mathcal{G}^{\alpha}_{\gamma,P}(q)|^{2}dP,$
(1.1)

where the parameter $\gamma$ dictates the central charge through the relation

$\displaystyle c=1+6Q^{2}\in(25,\infty),$ $\displaystyle
Q:=\frac{\gamma}{2}+\frac{2}{\gamma}\in(2,\infty).$ (1.2)

Here $C_{\gamma}(\alpha_{1},\alpha_{2},\alpha_{3})$ is the Liouville 3-point
correlation function. The exact expression of $C_{\gamma}$ as first proposed
in [DO94, ZZ96] and later proved in [KRV20]. It is worthwhile to note that the
conformal bootstrap for sphere is recently proven in [KRV20] and [Gui+21]
showed it for other geometry including torus.

In a recent work, [Gho+24, Theorem 1.1] introduced a probabilistic description
of $\mathcal{G}^{\alpha}_{\gamma,P}(q)$ using GMC measure. This probabilistic
definition of $\mathcal{G}^{\alpha}_{\gamma,P}(q)$ is shown to match with the
q-power series formula of the same (defined via Virasoro algebra in [BPZ84])
through rigorously proving the Zamolodchikov recursion (as proposed in [Zam84,
Zam87, HJS10]) for the GMC expression of $\mathcal{G}^{\alpha}_{\gamma,P}(q)$.
In fact, [Gho+24] showed that $\mathcal{G}^{\alpha}_{\gamma,P}(q)$ is defined
for $\alpha\in(-\frac{4}{\gamma},Q)$, $P\in\mathbb{R}$ and is analytic as a
function of $q\in\mathbb{C}$ in a ball $B_{r}(0)$ of radius $r$ greater than
$\frac{1}{2}$ when $\alpha\in[0,Q)$. We refer the reader to [Gho+24, Appendix
A] and [Gho+, Section 1.1] for a summary of the algebraic approach to
conformal blocks on one-point torus.

Our objective is to understand the behavior and properties of these Liouville
conformal blocks $\mathcal{G}^{\alpha}_{\gamma,P}(q)$ in the semi-classical
limit $\gamma\to 0$, shedding light on the underlying probabilistic structures
associated with conformal field theories on tori. Such a proof for all other
geometries remains an open question. Starting from the probabilistic
definition of conformal blocks on the four point sphere, one can easily adapt
our methods to study their semi-classical limits.

######  Theorem 1.1.

The semi-classical limit of the conformal block
$\mathcal{G}^{\alpha}_{\gamma,P}(q)$ exists and has the form

$\displaystyle\lim_{\gamma\rightarrow
0}\gamma^{2}\log\mathcal{G}_{\gamma,P}^{\alpha}(q)=\phi(\alpha_{0},P_{0},q)+\widetilde{f}(\alpha_{0},P_{0});$
$\displaystyle\alpha=\frac{\alpha_{0}}{\gamma},$ $\displaystyle
P=\frac{P_{0}}{\gamma},$ (1.3)

for $q=e^{\mathbf{i}\pi\tau}\in B_{r}(0)$ for some $r>0$ small,
$\alpha_{0}\in(-4,2)$ and $P_{0}\in\mathbb{R}$. Here, $\phi(.)$ is an analytic
function in $q$ and $\widetilde{f}(.)$ is a normalization constant such that

$\lim_{q\to 0}\lim_{\gamma\to
0}\gamma^{2}\log\mathcal{G}_{\gamma,P_{0}/\gamma}^{\alpha_{0}/\gamma}(q)=0.$

Moreover, $\phi(.)$ determines the accessory parameter of the Lamé equation
that can be written explicitly as

$\displaystyle\left(\partial_{z}^{2}-\frac{\alpha_{0}}{4}\left(\frac{\alpha_{0}}{4}-1\right)\wp(z)+\frac{\pi\mathbf{i}}{2}\partial_{\tau}{\phi}(\alpha_{0},P_{0},q)\right)\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)=0,$
(1.4)

where $\wp(\cdot)$ is the Weierstrass $\wp$-function (see (A.11)) and the
function $\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)$ is given by

$\displaystyle\begin{split}\widetilde{\Gamma}(z;\alpha_{0},P_{0},q):=e^{\frac{P_{0}z\pi}{2}}\exp\left(-\frac{\alpha_{0}\int^{1}_{0}\log\left(\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\right)\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx}{4\int^{1}_{0}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx}\right).\end{split}$ (1.5)

######  Remark 1.1.

The limit in (1.3) can be shown for $q\in B_{r}(0)$ for some $r>\frac{1}{2}$
when $\alpha_{0}\in[0,2)$. Furthermore, since
$\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)$ has an explicit expression in
(1.5), one can use Lamé equation to write explicit form of the accessory
parameter $\mathbf{i}\pi\partial_{\tau}\phi(\alpha_{0},P_{0},q)$ (see
Corollary 4.60). The analytic properties with respect to $\alpha_{0},P_{0},q$
of the semi-classical limit are accessible through this explicit form.

###  1.1. Zamolodchikov conjecture

The main result of this paper rigorously proves the Zamolodchikov conjecture
for the case of conformal blocks on a one-point torus. A long standing
conjecture formulated by Zamolodchikov in 1986 [Zam86, (2.30)-(2.32)]
postulates that, in the semi-classical limit ($c\to\infty$), conformal blocks
on a four-point sphere have an exponential structure. Furthermore, it is known
that, assuming such an exponential structure, the semi-classical limit of the
sphere 4-point conformal block is described by the accessory parameter of the
Heun’s equation.

The evidence for such a structure for the case of the four point sphere comes
from the analysis of the power series representation of conformal blocks
[Zam87a], from the oscillator formalism of Virasoro algebra [BDK20], and from
the combinatorial expression in terms of the Nekrasov-Okounkov functions
describing the partition functions of supersymmetric gauge theories given by
the Alday-Gaiotto-Tachikawa (AGT) correspondence [NPS18, NS10].

The semi-classical limit of conformal blocks carries significant importance in
mathematical physics as it plays a crucial role in a variety of areas from
black hole physics [AGH22, CC22] and supersymmetric gauge theories [NS10] (see
also the introduction in [Pi̧a+22]) to Teichmüller theory (see [BCP17, Tak94]
and references therein).

Of particular interest to this paper is the connection between semi-classical
conformal blocks and the Heun equation, a second-order differential equation
featuring a free parameter known as the accessory parameter. This parameter is
crucial in the semi-classical analysis as it is characterized by conformal
blocks on the sphere [Zam86]. The study of the Heun equation is interesting on
its own. See [RA95] for a bibliographical account and [Hor18] for a review on
applications to physics.

In the recent years, interest has grown in understanding the relationships
between conformal blocks, accessory parameters, and Painlevé equations [BGG22,
LN21, PP19, LN18]. Specifically, the semi-classical limit of conformal blocks
is related to the Painlevé VI equation through its Hamiltonian at special
values of the solution [Lit+14, KT13]. In turn the Hamiltonian of the Painlevé
VI equation is given by $c=1$ conformal blocks on sphere, and hence the
aforementioned relation provides a non-trivial link between the $c\to\infty$
and $c=1$ conformal blocks.

More recently, a relation between $c\to\infty$ conformal blocks and the
connection problem for the degenerations of the Heun equation was conjectured
in [Bon+23] and is dubbed ’the Trieste formula’ [LN22], with the case of the
Heun equation (without degenerations) verified in [LN22]. So, the proof of
Zamolodchikov conjecture paves the way for a rigorous treatment of the
aforementioned problems. Indeed most of above claims could be easily adapted
to the one-point torus case. In particular, the role of the Heun equation is
now played by the Lamé equation and furthermore, its accessory parameter,
under the assumption that the Zamolodchikov conjecture holds, is known to be
related to the semi-classical conformal block on the torus [Pi̧a14] and to the
elliptic form of the Painlevé VI equation with special value of the parameters
[BGG22]. In this context, we also want to highlight a similar study of the
semiclassical limit of Liouville CFT on a sphere [LRV], where they prove the
Takhtajan-Zograf theorem relating the classical Liouville action at special
points and accessory parameters of a second order Fuchsian equation of
holomorphic functions on the unit disk [TZ03].

In Theorem 1.1, we showed that, as the central charge $c$ goes to $\infty$ or
equivalently $\gamma\to 0$, the torus one point Liouville conformal block
behaves as
$\mathcal{G}_{\gamma,P}^{\alpha}(q)\sim\exp(\phi(\alpha_{0},P_{0},q)/\gamma^{2})$
where $\phi(\alpha_{0},P_{0},q)$ is the semi-classical conformal block.
Moreover, (1.4) showed that $\phi(\alpha_{0},P_{0},q)$ is related to the
accessory parameter of Lamé equation. As a result, we provide a rigorous proof
of the analogue of Zamolodchikov conjecture for one-point torus conformal
blocks. Consequently, we prove that $\phi(\ldots)$ is also related to the
Hamiltonian of the non-autonomous elliptic Calogero-Moser (NAECM) model (see
Theorem 4.2) for special values of the solution [BGG22], which in turn is
related to $c=1$ conformal blocks on a one-point torus [DDG23]. Such a proof
for all other geometries remains an open question.

###  1.2. Ideas of the proofs

Our main goal in this paper is to prove the existence and estimate the radius
of convergence for, the semi-classical limit ($\gamma\to 0$) of the conformal
block $\,\mathcal{G}_{\gamma,P}^{\alpha}(q)$. Henceforth, we refer to this as
the undeformed conformal block. The proofs of these statements involve several
steps. One of the main challenges is to show that the conformal block has
exponential structure in the semi-classical limit, i.e., as $\gamma\to 0$, the
torus 1-point conformal block $\mathcal{G}^{\alpha}_{\gamma,P}$ behaves as
$\exp\big{(}-\frac{1}{\gamma^{2}}(\phi(\alpha_{0},P_{0},q)+\tilde{f}(\alpha_{0},P_{0}))\big{)}$.
For showing this, we use the probabilistic description of
$\mathcal{G}^{\alpha}_{\gamma,P}$ given as

$\displaystyle\mathcal{G}^{\alpha}_{\gamma,P}\sim\mathbb{E}\left[\left({\sf
IGMC}^{\alpha}_{\gamma,P}(q)\right)^{-\frac{\alpha}{\gamma}}\right],\qquad{\sf
IGMC}^{\alpha}_{\gamma,P}(q):=\int^{1}_{0}\theta_{1}(x)^{-\frac{\alpha\gamma}{2}}e^{\pi\gamma
Px}e^{\gamma:Y(x;q):}dx$ (1.6)

where $:Y(x;q):$ is a renormalized log-correlated field on $[0,1]$. See
Section 2.1 for its definition. The above representation of the conformal
block was introduced by [Gho+24] and expressed in terms of the Gaussian
multiplicative chaos measure constructed using the log-correlated field
$Y(x;q)$ defined for $x\in[0,1]$. See Definition 2.2 for further discussion.
With the choice of scaling $P=\frac{P_{0}}{\gamma}$ and
$\alpha=\frac{\alpha_{0}}{\gamma}$ for $P_{0}\in\mathbb{R}$ and
$\alpha_{0}\in\mathbb{R}_{>0}$, the first difficulty is to show that
$\gamma^{2}\log\mathcal{G}^{\alpha}_{\gamma,P}$ does not blow up to infinity.
We overcome this difficulty by showing that the (deep) tail probabilities
$\gamma^{-1}\log({\sf IGMC}^{\alpha}_{\gamma,P}(q))$ behave as the tails of a
mean zero and finite variance Gaussian distribution. With replacing
$\gamma^{-1}\log({\sf IGMC}^{\alpha}_{\gamma,P}(q))$ in (1.6) by a mean zero
and finite variance Gaussian random variable $X$ yields

$\mathcal{G}^{\alpha}_{\gamma,P}\sim\mathbb{E}\Big{[}\exp\big{(}-\frac{\alpha_{0}}{\gamma}X\big{)}\Big{]}\sim\exp\Big{(}\frac{C\alpha_{0}}{\gamma^{2}}\Big{)}$

for some constant $C\in\mathbb{R}$. This shows the tightness of the sequence
$\gamma^{2}\log\mathcal{G}^{\alpha}_{\gamma,P}$ as $\gamma$ approaches $0$. We
studied the tail probabilities of $\gamma^{-1}\log({\sf
IGMC}^{\alpha}_{\gamma,P}(q))$ using the derivative GMC measure. For small
$\gamma$, mean value theorem suggests $\gamma^{-1}\log({\sf
IGMC}^{\alpha}_{\gamma,P}(q))$ behaves as ${\sf
DGMC}^{\alpha}_{\gamma^{\prime},P}(q)$ for some $\gamma^{\prime}\in(0,\gamma)$
where, for the renormalized log-correlated field $:Y(x;q):$,

${\sf
DGMC}^{\alpha}_{\gamma^{\prime},P}(q):=\frac{1}{\int^{1}_{0}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx}\int^{1}_{0}:Y(x;q):\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}e^{\gamma:Y(x;q):}dx.$

In Theorem 3.1, we show that the tail probabilities of ${\sf
DGMC}^{\alpha}_{\gamma^{\prime},P}(q)$ are like the Gaussian distribution. The
proof of Theorem 3.1 is one of the most technical part of our paper and is
built on ideas from several previous works in the study of GMC measure
including the local decomposition of log-correlated fields as sum of *-scale
invariant field and a continuous field [JSW18] and the tools based on
derivative GMC measure which are required to study the path integral of
quantum Mabuchi K-energy [lacoin2022path].

Once the tightness of $\gamma^{2}\log\mathcal{G}^{\alpha}_{\gamma,P}$ is
shown, it remains to show that all subsequential limits of
$\gamma^{2}\log\mathcal{G}^{\alpha}_{\gamma,P}$ are the same, and they are all
uniquely characterized by the function $\phi(.)$ described in Theorem 1.1 that
is related to the accessory parameter of the Lamé equation. To this end, we
use the deformation of the conformal block $\mathcal{G}^{\alpha}_{\gamma,P}$
which we will later denote as $\psi^{\alpha}_{\chi,P}(z,\tau)$ (see Definition
2.3) for $\chi\in\\{\frac{\gamma}{2},\frac{2}{\gamma}\\}$, $z\in(0,1)$ and
$\alpha,P,\tau$ are same as before. Deformed conformal block is uniquely
determined as the solution of a differential equation named the Belavin-
Polyakov-Zamolodchikov equation,

$\Big{(}\partial_{zz}-l_{\chi}(l_{\chi}+1)\wp(z)+2\mathbf{i}\pi\chi^{2}\partial_{\tau}\Big{)}\psi_{\chi,P}^{\alpha}(z,\tau)=0,\qquad\ell_{\chi}=\frac{\chi^{2}}{2}-\frac{\alpha\chi}{2}.$

For $\chi=\frac{\gamma}{2}$, we show in Theorem 4.2 that the deformed
conformal block $\psi_{\chi,P}^{\alpha}(z,\tau)$ factorizes approximately as
$\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)\times\mathcal{G}^{\alpha}_{\gamma,P}$
as $\gamma\to 0$ where $\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)$ is same as
in (1.5). Similar decomposition of deformed conformal block was previously
observed in [Lit+14, Pi̧a14]. Substituting this into the BPZ equation and
letting $\gamma\to 0$ with the scaling $\alpha=\frac{\alpha_{0}}{\gamma}$ and
$P=\frac{P_{0}}{\gamma}$ gives us the Lamé equation as shown in (1.4). Since
$\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)$ is explicit, the accessory
parameter of (1.4) which could be $\tau$-derivative of any subsequential limit
of $\gamma^{2}\log\mathcal{G}^{\alpha}_{\gamma,P}$ is now uniquely identified
which in turn shows the existence of the limit of the sequence
$\gamma^{2}\log\mathcal{G}^{\alpha}_{\gamma,P}$.

On the other hand, for $\chi=\frac{2}{\gamma}$, the BPZ equation degenerates
to the Hamilton-Jacobi equation as $\gamma\to 0$. Using this observation and
by showing the tightness of $\gamma^{2}\log\psi^{\alpha}_{\gamma,P}$ using the
tails of the associated derivative GMC measure, we demonstrate in Theorem 4.1
that the semi-classical limit of $\psi^{\alpha}_{\gamma,P}$ solves the
Hamilton-Jacobi equation for the non-autonomous elliptic Calogero Moser model
(NAECM). See Remark 4.1 for a discussion. This connection between the deformed
conformal block and Calogero-Moser model is used later in Theorem 4.2 to show
that the accessory parameter of the Lamé equation (1.4) is in fact related to
the Hamiltonian of the NAECM model.

###  1.3. Outline of the paper

This paper is organized as follows. In Section 2, we review the probabilistic
construction of the undeformed conformal block
$\,\mathcal{G}_{\gamma,P}^{\alpha}(q)$, and the deformed conformal block which
satisfies the PDE known as the Belavin-Polyakov-Zamolodchikov (BPZ) equation.

In Section 3, we introduce the main tool to analyse the semi-classical limit
of the conformal blocks, namely, the DGMC measure, which is the $\gamma$
derivative of the GMC measure. The main result of Section 3 is Theorem 3.1
which proves the tail estimates of the DGMC measure.

In Section 4, we formulate the semi-classical limit of the deformed conformal
block. In Subsection 4.1 we study the semi-classical limit of the deformed
conformal block for $\chi=\frac{2}{\gamma}$. The main result of this
Subsection is Theorem 4.1 which establishes an explicit description of such
semi-classical limit. In Subsection 4.2 we analyse the semi-classical limit of
the deformed conformal block for $\chi=\frac{\gamma}{2}$. The main result of
this Subsection is Theorem 4.2 which gives explicit description of that semi-
classical limit in terms of the solution of Lamé equation. As a consequence of
this theorem, we obtain explicit expressions for the solution and accessory
parameter of the Lamé equation.

We prove Theorem 1.1 in Section 5. We further prove that the radius of
convergence of the semi-classical conformal block is positive in Theorem 5.2.

Finally, in Section 6, we study the asymptotic behaviour of the semi-classical
limit of the conformal blocks for $\tau\to\mathbf{i}\infty$. The key component
of the analysis is the expression of the normalization constant of the
undeformed conformal block in terms of double Gamma functions. Results in this
section are used to prove results in Section 4.

###  1.4. Outlook

Our paper presents two key takeaways. Firstly, the methods we introduced in
proving the Zamolodchikov conjecture offer a road map for studying semi-
classical conformal blocks on other Riemann surfaces and for the analysis of
conformal blocks with different central charges. Secondly, the explicit
formulas for the conformal blocks and the Lamé equation obtained here provide
profound insights into the analytic and geometric aspects of conformal field
theories (CFTs) and the associated Heun-type equations. Let us now describe
some open questions of immediate interest.

  * •

A natural question that arises is the generalization of our techniques to
other geometries such as the four-point sphere and higher genus surfaces. The
key to achieving this would be to develop a systematic approach to defining
probabilistic conformal blocks on Riemann surfaces. The formulation of
probabilistic conformal blocks on a four point sphere is expected to have a
slightly varied construction [Gho+], and the irregular conformal blocks can
then be constructed through appropriate degeneration limits. The proof of the
Zamolodchikov conjecture in the aforementioned cases is expected to follow a
similar formalism to that presented here.

  * •

Semi-classical limit of Liouville CFT corresponds to taking the central charge
$c$ to $\infty$. It is natural to ask what happens when $c\to-\infty$. This
question is related to the semi-classical limit of CFTs with $c\leq 1$ (i.e.,
$\gamma$ purely imaginary). Recently, in a series of works [Wan21, Wan19,
Wan19a], Yilin Wang has investigated the large deviation problem of Schramm-
Loewner evolution ($\mathrm{SLE}_{\kappa}$) as $\kappa\to 0_{+}$ and has shown
that the large deviation rate function is equal to the Loewner energy
function. Since $\kappa\to 0_{+}$ corresponds to talking $c\to-\infty$ and SLE
is related to the correlation function of certain CFT, the results of [Wan21,
Wan19, Wan19a] describes semi-classical limit for $c\to-\infty$. In a more
recent work [PW24], Peltola and Wang characterized minimizers of the Loewner
potential using tools from real algebraic geometry which provides unique
insight about the CFT when the central charges diminishes to $-\infty$.

Besides the above questions, the explicit formulation of the semi-classical
conformal block in terms of the DGMC measure presented in this paper merits an
investigation of its own. On the other hand, one can obtain a series
representation of the sem-classical conformal block through the Zamolodchikov
recursion relations and a combinatorial relation coming from AGT
correspondence (see [FL10]). It is important to study the interrelation
between these explicit formulations. An important aspect of the explicit
representation is the dependence of the semi-classical conformal block on the
monodromy data of the NAECM model and the Lamé equation, which would lead to
interesting outcomes. We list a few a few of them.

  * •

The Trieste formula [Bon+23, (4.1.16) - (4.1.17)], [LN22, eq. 2.20b] connects
the connection formulae of the Heun’s equation with the semi-classical limit
of the fusion kernel of the 4-point sphere conformal blocks. Our description
of the semi-classical conformal block may enable the proof of an analogue to
the Trieste formula connecting the semi-classical limit of the modular kernel
(see [Gho+, (5.3)]) of one-point torus conformal block with the connection
formulae of the Lamé equation. A probabilistic description of the four-point
conformal block and its degenerations could prove the Trieste formula in full
generality.

  * •

In [LN21], the authors found a series representation of the accessory
parameter of the Heun’s equation where the coefficients satisfies some
recurrence relation. This is reminiscent of the Zamolodchkiov recurrence
relation satisfied by the coefficients of the conformal blocks. Following a
similar procedure as in [LN21], one can also derive a series representation of
the accessory parameter of Lamé equation once the monodromy data associated to
$\alpha_{0},P_{0}$ is determined. Furthermore, the accessory parameter is also
explicitly given in terms of the derivative GMC measure in (4.2), and in terms
of integrals (4.60). Such formulae would allow to study its asymptotic
properties [Nov86, XXZ21].

  * •

Through the AGT correspondence, the semi-classical conformal block on a one-
point torus determines the Nekrasov-Shatashvili (NS) limit of the
corresponding $\mathcal{N}=2^{*}$ gauge theory, or equivalently, the effective
twisted superpotential [Tes11]. The explicit form of the classical conformal
block presented in this paper would enable a rigorous proof of the Nekrasov-
Rosly-Shatashvili conjecture that relates the effective twisted superpotential
of the gauge theory to the so-called Yang-Yang function of the associated
quantum Hitchin system [NS09, NRS11, HK18].

  * •

The conformal blocks for arbitrary central charge solve certain bilinear
relations called the Nakajima-Yoshioka blow-up relations [NY03]. Such
relations for the one-point torus have been written in [Gu+20, BGG22] and it
will be interesting to see if such relations could be proved using
probabilistic conformal blocks. An interesting property of the relations is
that the blow-up relation connects the semi-classical conformal blocks with
the conformal block at central charge $c=1$ [NS10], in turn relating Painlevé
equations to Heun equations for the sphere case and the NAECM model to Lamé
equation (see also [BGG22]) for the torus case. We show the latter relation in
Theorem 4.2 based on probabilistic ideas. Showing the full blow-up relation is
beyond the scope of this paper and we leave this for a future work.

### Acknowledgements

We thank Alba Grassi, Aleksandra Korzhenkova, Oleg Lisovyi, Alex Moll,
Eveliina Peltola, Vincent Vargas for informative discussions. Part of this
work was done during the authors’ residence at the SLMath (formerly,
Mathematical Sciences Research Institute) in Berkeley, California, during the
program ’Universality and Integrability in Random Matrix Theory and
Interacting Particle Systems’. They acknowledge the support of the National
Science Foundation Grant No. DMS-1928930. H.D. also acknowledges the support
of the Okinawa Institute of Science and Technology (OIST), where she completed
part of this work during her stay for the Theoretical Sciences Visiting
Program (TSVP) "Exact Asymptotics: From Fluid Dynamics to Quantum Geometry."

The work of H.D. is partly supported by Australian Research Council Discovery
Project #DP200100210, INI-Simons fellowship, and SMRI postdoctoral fellowship.
The work of P.G. is partially supported by the NSF grant DMS-2346685. A.P. is
supported NSF MSPRF grant DMS-2103354, and RSF grant 22-11-00070.

##  2\. Probabilistic conformal blocks

In this section we recall the construction of the conformal block in [Gho+24].
We begin by defining Gaussian Multiplicative Chaos (GMC) measures in terms of
log-correlated fields, which will then be used to define the probabilistic one
point conformal block on the torus. We then introduce the notion of a deformed
conformal block which solves the PDE known as the Belavin-Polyakov-
Zamolodchikov (BPZ) equation.

###  2.1. Log-correlated fields and gaussian multiplicative chaos (GMC)
measures

For $x\in[0,1]$, the random fields $Y_{N}(x;q)$, $Y_{N}(x)$, with
$N\in\mathbb{N}$ are defined as

$\displaystyle\widetilde{Y}_{N}(x)$
$\displaystyle=\sum_{n=1}^{N}\sqrt{\frac{2}{n}}\Big{(}\widetilde{a}_{n}\cos(2\pi
nx)+\widetilde{b}_{n}\sin(2\pi nx)\Big{)},$ (2.1)
$\displaystyle\widetilde{Y}_{N}(x;q)$
$\displaystyle=\widetilde{Y}_{N}(x)+\underbrace{2\sum_{n,m=1}^{\infty}\frac{q^{nm}}{\sqrt{n}}\Big{(}\widetilde{a}_{n,m}\cos(2\pi
nx)+\widetilde{b}_{n,m}\sin(2\pi nx)\Big{)}}_{=:F(x;q)},$ (2.2)

where $\\{\widetilde{a}_{n}\\}_{n\geq 1}$, $\\{\widetilde{b}_{n}\\}_{n\geq
1}$, $\\{\widetilde{a}_{n,m}\\}_{n,m\geq 1}$,
$\\{\widetilde{b}_{n,m}\\}_{n,m\geq 1}$ are sequences of i.i.d standard real
Gaussian random variables, the modular parameter $\tau\in\mathbb{H}$, and
$q=e^{i\pi\tau}$. In what follows we use the notation

$\displaystyle Y(x):=\lim_{N\to\infty}\widetilde{Y}_{N}(x),$ $\displaystyle
Y(x;q):=\lim_{N\to\infty}\widetilde{Y}_{N}(x;q).$ (2.3)

Indeed, the above limit exists. The second limit exist in the sense of weak
convergence in the Sobolev space $H^{-s}([0,1])$ and the second limit exists
to due to the first limit and the almost convergence of infinite series of
$F(x;q)$ due to the term $q^{nm}$. The resulting random fields $Y(\cdot;q)$
and $Y(\cdot)$ are log-correlated, and the covariance kernels of (2.3) for
$\tau\in\mathbf{i}\mathbb{R}_{>0}$ are given as follows.

  * •

The field $Y(x)$ is log-correlated on $[0,1]$:

$\displaystyle\mathbb{E}[Y(x)Y(y)]=-2\log|e^{2\mathbf{i}\pi
x}-e^{2\mathbf{i}\pi y}|,$ (2.4)

which is the usual expression $\log|x-y|$ in cylindrical coordinates obtained
by the transformation $z\rightarrow e^{2\pi iz}$.

  * •

The $q$-dependent field $Y(x;q))$ is log correlated on a doubly-periodic
lattice

$\displaystyle\mathbb{E}[Y(x;q)Y(y;q)]=-2\log|\theta_{1}(x-y)|+2\log|q^{1/6}\eta(q)|.$
(2.5)

Note that for $q\rightarrow 0$ and $(x-y)\to 0$, the above correlation
functions (2.4), (2.5) coincide up to a trivial factor.

Using [Gho+24, Lemma 2.5], one finds that
$\mathbb{E}\left[F(x;q)^{2}\right]=-4\log|q^{-1/12}\eta(q)|.$ As a result, we
get

$\displaystyle\mathbb{E}\left[\widetilde{Y}_{N}(x;q)^{2}\right]=\mathbb{E}\left[\widetilde{Y}_{N}(x)^{2}\right]-4\log|q^{-1/12}\eta(q)|.$
(2.6)

With the above log-correlated fields, we now define the following random
measures.

######  Definition 2.1 (GMC measure).

For $\gamma\in(0,2)$ and $\tau\in\mathbf{i}\mathbb{R}_{>0}$, the GMC measures
$e^{\frac{\gamma}{2}Y(x)}dx$ and $e^{\frac{\gamma}{2}Y(x;q)}dx$ are defined to
be the weak limits of measures in probability

$\begin{split}e^{\frac{\gamma}{2}Y(x)}dx&:=\lim_{N\to\infty}e^{\frac{\gamma}{2}\widetilde{Y}_{N}(x)-\frac{\gamma^{2}}{8}\mathbb{E}[\widetilde{Y}_{N}(x)^{2}]}dx,\\\
e^{\frac{\gamma}{2}Y(x;q)}dx&:=\lim_{N\to\infty}e^{\frac{\gamma}{2}\widetilde{Y}_{N}(x;q)-\frac{\gamma^{2}}{8}\mathbb{E}[\widetilde{Y}_{N}(x;q)^{2}]}dx.\end{split}$
(2.7)

For a test function $f(x)$, these measures are interpreted as

$\displaystyle\int_{0}^{1}f(x)e^{\frac{\gamma}{2}Y(x;q)}dx=\lim_{N\to\infty}\int_{0}^{1}f(x)e^{\frac{\gamma}{2}\widetilde{Y}_{N}(x;q)-\frac{\gamma^{2}}{8}\mathbb{E}[\widetilde{Y}_{N}(x;q)^{2}]}dx,$
(2.8)

and a similar expression holds for the $q$-independent field $Y(x)$.

The existence of the limits above can be shown in the following way. Note that
the identity

$\displaystyle\mathbb{E}[e^{c\left(\widetilde{Y}_{k}-\widetilde{Y}_{k-1}\right)}]=e^{\frac{c^{2}}{2}},$
(2.9)

implies that the sequence in (2.8) is a martingale. Since it is positive,
almost sure convergence is guaranteed by the martingale convergence theorem,
which in turn proves the existence of the limit in (2.8).

###  2.2. One-point (undeformed) conformal block

We now recall the definition of the one-point conformal block in [Gho+24].

######  Definition 2.2.

For $\alpha\in(-\frac{4}{\gamma},Q)$, and $P\in\mathbb{R}$, we define the
probabilistic $1$-point toric conformal block as

$\mathcal{G}^{\alpha}_{\gamma,P}(q):=\frac{1}{Z^{\alpha}_{\gamma,P}}\mathbb{E}\left[\left(\int_{0}^{1}\theta_{1}(x)^{-\frac{\alpha\gamma}{2}}e^{\pi\gamma
Px}e^{\frac{\gamma}{2}Y(x;q)}dx\right)^{-\frac{\alpha}{\gamma}}\right],$
(2.10)

with the normalization

$Z^{\alpha}_{\gamma,P}(q):=q^{\frac{1}{12}(\frac{\alpha\gamma}{2}+\frac{\alpha^{2}}{2}-1)}\eta(q)^{\alpha^{2}+1-\frac{\alpha\gamma}{2}}\mathbb{E}\left[\left(\int_{0}^{1}[-2\sin(\pi
x)]^{-\frac{\alpha\gamma}{2}}e^{\pi\gamma
Px}{e^{\frac{\gamma}{2}Y(x)}dx}\right)^{-\frac{\alpha}{\gamma}}\right].$
(2.11)

Note that the normalization constant $Z^{\alpha}_{\gamma,P}$ ensures that

$\displaystyle\lim_{q\rightarrow 0}\mathcal{G}^{\alpha}_{\gamma,P}(q)=1,$
$\displaystyle\lim_{P\rightarrow\infty}\mathcal{G}^{\alpha}_{\gamma,P}(q)=q^{\frac{1}{12}}\eta(q)^{-1}.$

A key consequence of the present construction is that the analytic properties
of the conformal block (2.10) become accessible, and we recall the following
result.

######  Theorem 2.1 ( [Gho+24, Theorem 1.1]).

For $\gamma\in(0,2)$, $\alpha\in(-\frac{4}{\gamma},Q)$, and $P\in\mathbb{R}$,
the conformal block in (2.10) is analytic in a complex neighborhood of $q=0$.
Moreover, for $\alpha\in[0,Q)$, the analytic extension of (2.10) exists in a
neighborhood of $q=0$ with radius greater than $\frac{1}{2}$.

Furthermore, (2.10) can be represented a series with the coefficients
satisfying the so-called Zamolodchikov recursion [Gho+24, Proposition 2.11],
and as a combinatorial series in terms of Nekrasov-Okounkov functions [Gho+24,
Section 2.3] due to the Alday-Gaiotto-Tachikawa (AGT) correspondence [Neg16].

######  Remark 2.1.

For $-\frac{\alpha}{\gamma}=N<\frac{4}{\gamma^{2}}$ with $N\in\mathbb{N}$,
(2.10) can be expressed as a Dotsenko-Fadeev integral [MMS10, Fat+09]

$\displaystyle\mathcal{G}_{\gamma,P}^{\alpha}(q)$
$\displaystyle=\frac{1}{Z^{\alpha}_{\gamma,P}}\mathbb{E}\left[\left(\int_{0}^{1}\theta_{1}(x)^{-\frac{\alpha\gamma}{2}}e^{\gamma\pi
Px}e^{\frac{\gamma}{2}Y(x;q)}dx\right)^{-\frac{\alpha}{\gamma}}\right]$
$\displaystyle=\frac{1}{C}\int_{[0,1]^{N}}\prod_{1\leq i<j\leq
N}|\theta_{1}(x_{i}-x_{j})|^{-\frac{\gamma^{2}}{4}}\prod_{i=1}^{N}\theta_{1}(x_{i})^{\frac{N\gamma^{2}}{2}}e^{\pi\gamma
Px_{i}}\prod_{i=1}^{N}dx_{i},$

with

$\displaystyle C$ $\displaystyle=\frac{e^{\frac{N\gamma
P\pi}{2}}}{\left(\Gamma\left(1-\frac{\gamma^{2}}{8}\right)\right)^{N}}\left(\frac{\theta_{1}^{\prime}(0)}{2}\right)^{\frac{N(N+1)\gamma^{2}}{8}}\prod_{k=1}^{N}\frac{\Gamma\left(1-\frac{k\gamma^{2}}{8}\right)\Gamma\left(1+\frac{(2N+1-k)\gamma^{2}}{8}\right)}{\Gamma\left(1+\frac{k\gamma^{2}}{8}+\frac{\mathbf{i}\gamma
P}{2}\right)\Gamma\left(1+\frac{k\gamma^{2}}{8}-\frac{\mathbf{i}\gamma
P}{2}\right)}.$ (2.12)

Let us highlight the following result that will prove useful to study the
$q\to 0$ asymptotics of the semi-classical conformal blocks.

######  Proposition 2.1.

The partition function $Z$ in (2.10) can be expressed in terms of double Gamma
functions as follows [Gho+24, (2.15), Proposition 6.4]

$\displaystyle
Z^{\alpha}_{\gamma,P}=q^{\frac{1}{12}(\frac{\alpha\gamma}{2}+\frac{\alpha^{2}}{2}-1)}\eta(q)^{\alpha^{2}+1-\frac{\alpha\gamma}{2}}\mathcal{A}_{\gamma,P}(\alpha),$
(2.13)

where the $q$-independent factor is

$\displaystyle\mathcal{A}_{\gamma,P}(\alpha)=e^{\frac{i\pi\alpha^{2}}{2}}$
$\displaystyle\left(\frac{\gamma}{2}\right)^{\frac{\gamma\alpha}{4}}e^{-\frac{\pi\alpha
P}{2}}\Gamma\left(1-\frac{\gamma^{2}}{4}\right)^{\frac{\alpha}{\gamma}}$
(2.14)
$\displaystyle\times\frac{\Gamma_{\frac{\gamma}{2}}\left(Q-\frac{\alpha}{2}\right)\Gamma_{\frac{\gamma}{2}}\left(\frac{2}{\gamma}+\frac{\alpha}{2}\right)\Gamma_{\frac{\gamma}{2}}\left(Q-\frac{\alpha}{2}-\mathbf{i}P\right)\Gamma_{\frac{\gamma}{2}}\left(Q-\frac{\alpha}{2}+\mathbf{i}P\right)}{\Gamma_{\frac{\gamma}{2}}\left(\frac{2}{\gamma}\right)\Gamma_{\frac{\gamma}{2}}\left(Q-\mathbf{i}P\right)\Gamma_{\frac{\gamma}{2}}\left(Q+\mathbf{i}P\right)\Gamma_{\frac{\gamma}{2}}\left(Q-\alpha\right)},$
(2.15)

where the function $\Gamma_{\frac{\gamma}{2}}(\cdot)$ is the ratio of double
Gamma functions

$\Gamma_{\frac{\gamma}{2}}(x)=\frac{\Gamma_{2}(x|\gamma,\gamma^{-1})}{\Gamma_{2}\left({Q}/{2}|\gamma,\gamma^{-1}\right)},$

and $Q$ is given by (A.1)

###  2.3. $z$-deformed conformal block

We now define the probabilistic equivalent of inserting degenerate vertex
operator $\mathcal{V}$ inside conformal block as

$\mathcal{V}_{\chi,P}^{\alpha}(z,q):=\int_{0}^{1}\theta_{1}(z+x)^{\frac{\gamma\chi}{2}}\theta_{1}(z)^{-\frac{\gamma\chi}{2}}\theta_{1}(x)^{-\frac{\alpha\gamma}{2}}e^{\pi\gamma
Px}e^{\frac{\gamma}{2}Y(x;q)}dx\qquad\textrm{for
}z\in\mathfrak{B},\quad\chi\in\left\\{\frac{2}{\gamma},\frac{\gamma}{2}\right\\}$

with $\mathfrak{B}:=\\{z:0<\Im(z)<\frac{3}{4}\Im(\tau)\\}$. The deformed
conformal block is then in terms of $\mathcal{V}$ as follows.

######  Definition 2.3.

For $x\in(0,1)$, $q\in(0,1)$,
$\alpha\in\left(-\frac{4}{\gamma}+\chi,Q\right)$, $P\in\mathbb{R}$, the two
point toric conformal block with one degenerate insertion, known as the
$z$-deformed probabilistic conformal block is defined as

$\psi^{\alpha}_{\chi,P}(z,\tau):=\mathcal{W}(q)e^{\chi
Pz\pi}\mathbb{E}\left[\left(\mathcal{V}_{\chi,P}^{\alpha}(z,q)\right)^{-\frac{\alpha}{\gamma}+\frac{\chi}{\gamma}}\right],$
(2.16)

where

$\displaystyle\mathcal{W}(q):=q^{\frac{P^{2}}{2}+\frac{\gamma
l_{\chi}}{12\chi}-\frac{1}{6}\frac{l_{\chi}^{2}}{\chi^{2}}}\theta_{1}^{\prime}(0)^{-\frac{2l_{\chi}^{2}}{3\chi^{2}}+\frac{l_{\chi}}{3}+\frac{4l_{\chi}}{3\gamma\chi}},$
$\displaystyle l_{\chi}=\frac{\chi^{2}}{2}-\frac{\alpha\chi}{2},$ (2.17)

and $\theta_{1}^{\prime}(0)$ given by (A.6).

######  Theorem 2.2 ([Gho+24, Proposition 3.2]).

The deformed conformal block (2.16) is convergent in the domain
$D^{\alpha}_{\chi}$ where

$D^{\alpha}_{\chi}:=\\{(q,z):|q|<r_{\alpha-\chi}\textrm{ and
}z\in\mathfrak{B}\\},\qquad\mathfrak{B}_{q}:=\\{z:0<\mathrm{Im}(z)<\frac{3}{4}\mathrm{Im}(\tau)\\}$
(2.18)

Here $r_{\alpha}>0$ for $\alpha\in(-\frac{4}{\gamma},Q)$.

######  Remark 2.2.

  * 1.

The choice of the pre-factor $\mathcal{W}(q)$ in (2.16) is dictated by the BPZ
equation which will be satisfied by $\psi^{\alpha}_{\chi,P}(z,\tau)$ (see
Theorem 2.3). The expression for $\mathcal{W}(q)$ will play a major role in
fixing the monodromy of the probabilistic conformal block, and this will be
expanded upon in a future work.

  * 2.

The deformed and the undeformed conformal blocks are related as

$\displaystyle\lim_{z\to
0}\left(\theta_{1}(z)^{l_{\chi}}\psi_{\chi,P}^{\alpha}\right)=\mathcal{W}(q)Z^{\alpha-\chi}_{\gamma,P}(q)\,\mathcal{G}_{\gamma,P}^{\alpha-\chi}(q).$
(2.19)

We now state the BPZ equation for $\psi_{\chi,P}^{\alpha}(z,\tau)$.

######  Theorem 2.3 ([Gho+24, Theorem 3.5]).

The $z$-deformed toric conformal block satisfies the following BPZ equation

$\Big{(}\partial_{zz}-l_{\chi}(l_{\chi}+1)\wp(z)+2\mathbf{i}\pi\chi^{2}\partial_{\tau}\Big{)}\psi_{\chi,P}^{\alpha}(z,\tau)=0\qquad\textrm{for
}(q,z)\in D^{\alpha}_{\chi},$ (2.20)

where $\alpha\in(-\frac{4}{\gamma}+\chi,Q)$,
$\chi\in\\{\frac{\gamma}{2},\frac{2}{\gamma}\\}$, and $\wp(z)$ is Weierstrass
$\wp$-function.

##  3\. Derivative gaussian multiplicative chaos measure

The semi-classical limit of the undeformed conformal block is the $\gamma\to
0$ limit of $\mathcal{G}_{\gamma,P}^{\alpha}(q)$. From (2.10), it turns out to
be the case that the convergence of the semi-classical limit is controlled by
the convergence of the $\gamma$ derivative of the GMC measure (2.8), known as
the derivative GMC (DGMC) measure.

######  Definition 3.1 (DGMC measure).

For a bounded test function $f(x,\tau)\in L_{\infty}([0,1])$ for every
$\tau\in\mathbb{H}$ the DGMC measure is defined as

$\displaystyle({\sf DM}_{\gamma,N}f)(\tau)$
$\displaystyle:=\int_{0}^{1}\left(\frac{1}{2}Y_{N}(x;q)-\frac{\gamma}{4}\mathbb{E}[Y^{2}_{N}(x;q)]\right)e^{\frac{\gamma}{2}Y_{N}(x;q)-\frac{\gamma^{2}}{8}\mathbb{E}[Y^{2}_{N}(x;q)]}f(x,\tau)\frac{dx}{\int^{1}_{0}f(x,\tau)dx}.$
(3.1)

The $N\to\infty$ limit of the measure above is treated in the same way as
(2.7):

$\displaystyle({\sf DM}_{\gamma}f)(\tau)=\lim_{N\rightarrow\infty}({\sf
DM}_{\gamma,N}f)(\tau).$ (3.3)

######  Theorem 3.1.

For a bounded function $f$, there exist $c_{1}=c_{1}(\tau)>0$,
$c_{2}=c_{2}(\tau)>0$, $c_{3}>0$, $c_{4}>0$ such that for all $v>0$,

$\displaystyle\mathbb{P}\left[({\sf DM}_{\gamma,N}f)(\tau)\leq-v\right]\leq
c_{1}(\tau)e^{-c_{2}(\tau)v^{2}},\qquad\mathbb{P}\left[({\sf
DM}_{\gamma,N}f)(\mathbf{i}\infty)\leq-v\right]\leq c_{3}e^{-c_{4}v^{2}}.$
(3.4)

######  Remark 3.1.

It is worthwhile to note that [LRV22, Theorem 4.4] proved similar result for
the DGMC when the underlying field of the GMC satisfy the so called _smooth
white noise decomposition property_ (see Assumption 4.1 of [LRV22]). However
the log-correlated fields $Y_{\infty}$ and $Y_{\tau}$ does not satisfy all
those assumptions. Motivated by their results, we provide a self-contained
proof of Theorem 3.1 by utilizing the infinite series representation of the
fields $Y_{\infty}$ and $Y_{\tau}$ (as shown in (2.2) and (2.1)).

The remainder of this section is dedicated to proving Theorem 3.1. We restrict
to $Y_{N}(x)$ and prove the second inequality in (3.4). Because of (2.6), and
the $N$ independence of $F(x;q)$, the bound of $({\sf DM}_{\gamma,N})f(\tau)$
will coincide with the bounds of $({\sf DM}_{\gamma,N})f(\mathbf{i}\infty)$, a
fact that can be verified by repeating the statements in the proof below for
$Y_{N}(x;q)$.

We begin the proof of Theorem 3.1 by introducing the following shorthand
notation and proving some important properties of the DGMC measure. These
properties will later be used to show the tail of $({\sf
DM}_{\gamma,N})f(\mathbf{i}\infty)$.

By Theorem B of [JSW18], a log-correlated field $Y$ can be locally decomposed
as $Y=\mathcal{L}+\mathcal{R}$ where the local component $\mathcal{L}$ is
almost $*$-scale invariant field and the global component $\mathcal{R}$ is a
regular Gaussian field with Hölder continuous realizations. Moreover,
$\mathcal{L}$ and $\mathcal{R}$ are independent.

Let $\\{P_{1},\ldots,P_{m}\\}$ be a partition of the interval $[0,1]$ such
that $Y$ could be expressed as
$Y(x)=\mathcal{L}^{(\ell)}+\mathcal{R}^{(\ell)}$ for some $*$-scale invariant
field $\mathcal{L}^{(\ell)}$ and a continuous field $\mathcal{R}^{(\ell)}$. In
the following discussion, we focus on bounding the tail probabilities of
$({\sf DM}_{\gamma,N}f)(\tau)$ for any bounded function supported on one of
the partitioning intervals $\\{P_{1},\ldots,P_{m}\\}$. For notational
convenience, we denote the *-scale invariant field $\mathcal{L}^{(\ell)}$ and
$\mathcal{R}^{(\ell)}$ for the interval $P_{\ell}$ as $\mathcal{L}$ and
$\mathcal{R}$. However, the tail probability of $({\sf DM}_{\gamma,N}f)(\tau)$
for any other bounded function $f$ could be found by applying the union bound.

Let $\mathcal{K}_{\mathcal{L}}$ be the covariance kernel of the field
$\mathcal{L}$. Since $\mathcal{L}$ is a $*$-scale invariant field,
$\mathcal{K}_{\mathcal{L}}$ is translation invariant and it could be written
as

$\mathcal{K}_{\mathcal{L}}(x,y)=:\mathcal{K}_{\mathcal{L}}(x-y)=\int^{\infty}_{0}K(e^{u}(x-y))du,\quad\forall
x\neq y\in[0,1]$

where $k$ is a covariance in $\mathbb{R}$, $K(0)=1$ and
$|K(x)|\lesssim|x|^{-a}$ for some $a<1$. In particular, we can choose
$a\in(0,1)$. This choice of $a$ determines the number of intervals in the
partition $\\{P_{1},\ldots,P_{m}\\}$ of $[0,1]$. Furthermore, by Proposition
4.1 of [JSW18], there exists sequence of Gaussian fields
$\\{\mathcal{L}_{n}\\}_{k\geq 1}$ and sigma algebras
$\mathfrak{F}_{1}\subset\mathfrak{F}_{2}\subset\mathfrak{F}_{3}\subset\ldots$
such that $\mathcal{L}_{k}$ is measurable w.r.t. $\mathfrak{F}_{k}$ and,

$\displaystyle\mathbb{E}[\mathcal{L}_{n_{1}}(x)\mathcal{L}_{n_{2}}(y)]=\int^{n_{1}\wedge
n_{2}}_{0}K(e^{u}(x-y))(1-e^{-\delta u})du,$ (3.5)

where $n_{1}\wedge n_{2}$ indicates $\min\\{n_{1},n_{2}\\}$. We define
$Y_{k}:=\mathcal{L}_{k}+\mathcal{R}$ and further introduce the following
notations:

$\displaystyle\Lambda_{k}(x):=Y_{k}(x)-Y_{k-1}(x),$
$\displaystyle\Omega_{k}(x):=\frac{1}{2}\mathbb{E}\left[Y_{k}^{2}(x)\right],$
(3.6)

By the independence between $\Lambda_{k}$ and $Y_{k-1}$, we get

$\displaystyle
e^{\frac{\gamma}{2}Y_{k}(x)-\frac{\gamma^{2}}{8}\mathbb{E}[Y^{2}_{k}(x)]}\mathop{=}^{\eqref{eq:NtoN-1GFF}}e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{8}\mathbb{E}[Y^{2}_{k-1}(x)]}e^{\frac{\gamma}{2}\Lambda_{k}(x)-\frac{\gamma^{2}}{8}\mathbb{E}\left[\Lambda_{k}(x)^{2}\right]}.$
(3.7)

To begin with, we show that $(\mathcal{D}_{k},\mathfrak{F}_{k})_{k\geq 1}$ is
a martingale, where

$\displaystyle\mathcal{D}_{k}:=\int_{0}^{1}W_{k}^{0}(x)f(x)dx,$
$\displaystyle\Omega_{k}(x)=\frac{1}{2}\mathbb{E}[(\mathcal{L}_{k}(x)+\mathcal{R}(x))^{2}],$
(3.8)

and,

$\displaystyle
W_{k}^{0}(x):=\left({Y}_{k}(x)-{\gamma}\,\Omega_{k}(x)\right)e^{\frac{\gamma}{2}{Y}_{k}(x)-\frac{\gamma^{2}}{4}\Omega_{k}(x)}.$
(3.9)

The test function $f(x)$ is $\gamma$ independent, and
${Y}_{k}(x)=\mathcal{L}_{k}(x)+\mathcal{R}(x)$ for $k\in\mathbb{R}$ and
$x\in[0,1]$.

######  Proposition 3.1.

The following statements are true

  1. (1)

$(\mathcal{D}_{k},\mathfrak{F}_{k})_{k\geq 1}$ is a martingale and
$\mathcal{D}_{k}$ is bounded below uniformly in $k$.

  2. (2)

$\mathcal{D}_{k}$ converges almost surely to $\mathcal{D}_{\infty}$.

###### Proof.

(1): To prove this statement, it suffices to show that the conditional
expectation

$\mathbb{E}\Big{[}\mathcal{D}_{k}\Big{|}\mathfrak{F}_{k-1}\Big{]}=\mathcal{D}_{k-1}.$

We define $\Lambda_{k}(x):=Y_{k}(x)-Y_{k-1}(x)$ for all $x\in[0,1]$. Indeed,
we have

$\displaystyle\mathbb{E}\Big{[}\mathcal{D}_{k}\Big{|}\mathfrak{F}_{k-1}\Big{]}$
$\displaystyle=\int_{0}^{1}(Y_{k-1}(x)-{\gamma}\,\Omega_{k-1}(x))\mathbb{E}[e^{\frac{\gamma}{2}Y_{k}(x)-\frac{\gamma^{2}}{4}\Omega_{k}(x)}\Big{|}\mathfrak{F}_{k-1}]f(x)dx$
$\displaystyle+\int^{1}_{0}\mathbb{E}\left[\left(\Lambda_{k}(x)-\frac{\gamma}{2}\mathbb{E}[\Lambda^{2}_{k}(x)]\right)e^{\frac{\gamma}{2}Y_{k}(x)-\frac{\gamma^{2}}{4}\Omega_{k}}\Big{|}\mathfrak{F}_{k-1}\right]f(x)dx$
$\displaystyle=\int_{0}^{1}(Y_{k-1}(x)-{\gamma}\,\Omega_{k-1}(x))e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}}f(x)dx=\mathcal{D}_{k-1}.$

To obtain the last step, we use the identities below that follow from (3.6):

$\displaystyle\mathbb{E}[e^{\frac{\gamma}{2}Y_{k}(x)-\frac{\gamma^{2}}{4}\Omega_{k}}|\mathfrak{F}_{k-1}]$
$\displaystyle=\mathbb{E}[e^{\frac{\gamma}{2}(Y_{k-1}(x)+\Lambda_{k}(x))-\frac{\gamma^{2}}{4}\left(\Omega_{k-1}+\mathbb{E}[\Lambda^{2}_{k}(x)]\right)}\Big{|}\mathfrak{F}_{k-1}]$
$\displaystyle=e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}},$

and

$\displaystyle\int^{1}_{0}\mathbb{E}\left[\left(\Lambda_{k}(x)-\frac{\gamma}{2}\mathbb{E}[\Lambda^{2}_{k}(x)]\right)e^{\frac{\gamma}{2}Y_{k}(x)-\frac{\gamma^{2}}{4}\Omega_{k}}\Big{|}\mathfrak{F}_{k-1}\right]f(x)dx=0.$

This proves the claim that $(\mathcal{D}_{k},\mathfrak{F}_{k})_{k\geq 1}$ is a
martingale.

(2): We show that $\\{\mathcal{D}_{k}\\}_{k\geq 1}$ is a Cauchy sequence in
$L_{2}$. Thus, by Doob’s martingale convergence theorem, $\mathcal{D}_{k}$
almost surely converges to $\mathcal{D}_{\infty}$.

Let us compute the following expression for $k>n$:

$\displaystyle\mathbb{E}\left[\left(\mathcal{D}_{k}-\mathcal{D}_{n}\right)^{2}\right]$
(3.10)
$\displaystyle=\mathbb{E}\left[\mathcal{D}_{k}^{2}\right]+\mathbb{E}\left[\mathcal{D}_{n}^{2}\right]-2\mathbb{E}\left[\mathcal{D}_{k}\mathcal{D}_{n}\right]=\mathbb{E}\left[\mathcal{D}_{k}^{2}\right]+\mathbb{E}\left[\mathcal{D}_{n}^{2}\right]-2\mathbb{E}\left[\mathcal{D}_{n}\mathbb{E}[\mathcal{D}_{k}|\mathfrak{F}_{n}]\right]$
(3.11)
$\displaystyle=\mathbb{E}\left[\mathcal{D}_{k}^{2}\right]-\mathbb{E}\left[\mathcal{D}^{2}_{n}\right]$
$\displaystyle\mathop{=}^{\eqref{def:Dk}}\int_{0}^{1}\int_{0}^{1}\mathbb{E}\left[W_{k}^{0}(x)W_{k}^{0}(y)\right]f(x)f(y)dxdy-\int_{0}^{1}\int_{0}^{1}\mathbb{E}\left[W_{n}^{0}(x)W_{n}^{0}(y)\right]f(x)f(y)dxdy,$
(3.12)

where we use the tower property to obtain the last line. Note that, from the
decomposition $Y_{k}=\mathcal{L}_{k}+\mathcal{R}$, and since $K(0)=1$,

$\displaystyle\Omega_{k}(x)\mathop{=}^{\eqref{eq:expLL}}\frac{1}{2}\mathbb{E}[\mathcal{R}(x)^{2}]+\frac{1}{2}\int^{k}_{0}(1-e^{-\delta
u})du.$ (3.13)

In what follows, we denote the bound of $\Omega_{k}(x)$ by $\Omega_{k}$. Let
us now expand the second term in (3.12):

$\displaystyle\mathbb{E}\left[W_{n}^{0}(x)W_{n}^{0}(y)\right]$ (3.14)
$\displaystyle\mathop{=}^{\eqref{def:Wk0}}\mathbb{E}\left[\left(Y_{n}(x)-{\gamma}\,\Omega_{n}(x)\right)\left(Y_{n}(y)-{\gamma}\,\Omega_{n}(y)\right)e^{\frac{\gamma}{2}Y_{n}(x)-\frac{\gamma^{2}}{4}\Omega_{n}(x)}e^{\frac{\gamma}{2}Y_{n}(y)-\frac{\gamma^{2}}{4}\Omega_{n}(y)}\right]$
$\displaystyle=e^{-\frac{\gamma^{2}}{4}\left(\Omega_{n}(x)+\Omega_{n}(y)\right)}\Big{(}\mathbb{E}\left[Y_{n}(x)Y_{n}(y)e^{\frac{\gamma}{2}\left(Y_{n}(x)+Y_{n}(y)\right)}\right]+\gamma^{2}\Omega_{n}(x)\Omega_{n}(y)\mathbb{E}\left[e^{\frac{\gamma}{2}\left(Y_{n}(x)+Y_{n}(y)\right)}\right]$
$\displaystyle-\gamma\,\Omega_{n}(y)\mathbb{E}\left[Y_{n}(x)e^{\frac{\gamma}{2}\left(Y_{n}(x)+Y_{n}(y)\right)}\right]-\gamma\,\Omega_{n}(x)\mathbb{E}\left[Y_{n}(y)e^{\frac{\gamma}{2}\left(Y_{n}(x)+Y_{n}(x)\right)}\right]\Big{)}.$
(3.15)

Similarly, the first term in (3.12) can be written as

$\displaystyle\mathbb{E}\left[W_{k}^{0}(x)W_{k}^{0}(y)\right]$ (3.16)
$\displaystyle\mathop{=}^{\eqref{def:Wk0}}\mathbb{E}\Big{[}\big{(}Y_{n}(x)+\sum^{k}_{\ell=n+1}\Lambda_{\ell}(x)-{\gamma}\,\Omega_{k}\big{)}\big{(}Y_{n}(y)+\sum^{k}_{\ell=n+1}\Lambda_{\ell}(x)-{\gamma}\,\Omega_{k}\big{)}$
$\displaystyle\times
e^{\frac{\gamma}{2}(Y_{n}(y)+\sum^{k}_{\ell=n+1}\Lambda_{\ell}(y))-\frac{\gamma^{2}}{4}\Omega_{k}(y)}e^{\frac{\gamma}{2}(Y_{n}(x)+\sum^{k}_{\ell=n+1}\Lambda_{\ell}(y))-\frac{\gamma^{2}}{4}\Omega_{k}(x)}\Big{]}$
$\displaystyle=\mathbb{E}\left[W_{n}^{0}(x)W_{n}^{0}(y)\right](1+\Xi(x,y))$
(3.17)
$\displaystyle+\mathbb{E}\left[(\sum_{\ell=n+1}^{k}\Lambda_{\ell}(x)-\gamma(\Omega_{k}-\Omega_{n}))(\sum_{\ell=n+1}^{k}\Lambda_{\ell}(y)-\gamma(\Omega_{k}-\Omega_{n}))e^{\frac{\gamma}{2}\sum^{k}_{\ell=n+1}(\Lambda_{\ell}(y)+\Lambda_{\ell}(y))-\frac{\gamma^{2}}{2}(\Omega_{k}-\Omega_{n})}\right]$
$\displaystyle+\mathbb{E}\left[(Y_{n}(x)-\gamma\Omega_{n}(x))(\sum_{\ell=n+1}^{k}\Lambda_{\ell}(y)-\gamma(\Omega_{k}-\Omega_{n}))e^{\frac{\gamma}{2}(Y_{n}(x)+Y_{n}(y)+\sum^{k}_{\ell=n+1}\Lambda_{\ell}(y))-\frac{\gamma^{2}}{4}\Omega_{k}(x)}\right]$
$\displaystyle+\mathbb{E}\left[(Y_{n}(y)-\gamma\Omega_{n}(y))(\sum_{\ell=n+1}^{k}\Lambda_{\ell}(x)-\gamma(\Omega_{k}-\Omega_{n}))e^{\frac{\gamma}{2}(Y_{n}(y)+Y_{n}(x)+\sum^{k}_{\ell=n+1}\Lambda_{\ell}(x))-\frac{\gamma^{2}}{4}\Omega_{k}(y)}\right].$
(3.18)

By the tower property of the conditional expectation, the last two terms of
the expression above equal to $0$. The second term in the right hand side of
the above expression can be analysed in the following way. By Gaussian
integration by parts, we get

$\displaystyle\mathbb{E}$
$\displaystyle\left[\left(\sum_{\ell=n+1}^{k}\Lambda_{\ell}(x)-\gamma(\Omega_{k}-\Omega_{n})\right)\left(\sum_{\ell=n+1}^{k}\Lambda_{\ell}(y)-\gamma(\Omega_{k}-\Omega_{n})\right)e^{\frac{\gamma}{2}\sum^{k}_{\ell=n+1}(\Lambda_{\ell}(y)+\Lambda_{\ell}(y))-\frac{\gamma^{2}}{2}(\Omega_{k}-\Omega_{n})}\right]$
$\displaystyle=e^{\frac{\gamma^{2}}{2}\sum_{\ell=n+1}^{k}\mathbb{E}[\Lambda_{\ell}(x)\Lambda_{\ell}(y)]}\Big{[}\left(\gamma\mathbb{E}\left[\sum_{\ell=n+1}^{n}(\Lambda_{\ell}(x)+\Lambda_{\ell}(y))\Lambda_{\ell}(x)\right]-\gamma(\Omega_{k}-\Omega_{n})\right)$
$\displaystyle\times\left(\gamma\mathbb{E}\left[\sum_{\ell=n+1}^{n}(\Lambda_{\ell}(x)+\Lambda_{\ell}(y))\Lambda_{\ell}(x)\right]-\gamma(\Omega_{k}-\Omega_{n})\right)+\mathbb{E}\left[\sum_{\ell=n+1}^{n}\mathbb{E}[\Lambda_{k}(x)\Lambda_{\ell}(y)\right]\Big{]}$
$\displaystyle=e^{\frac{\gamma^{2}}{2}\sum_{\ell=n+1}^{k}\mathbb{E}[\Lambda_{\ell}(x)\Lambda_{\ell}(y)]}\left[\gamma^{2}\left(\sum_{\ell=n+1}^{k}\mathbb{E}\left[\Lambda_{\ell}(x)\Lambda_{\ell}(y)\right]\right)^{2}+\sum_{\ell=n+1}^{k}\mathbb{E}[\Lambda_{\ell}(x)\Lambda_{\ell}(y)]\right].$

For any $x\neq y\in[0,1]$,
$\sum_{\ell=n+1}^{k}\mathbb{E}[\Lambda_{\ell}(x)\Lambda_{\ell}(y)]\to 0$ as
$n,k\to\infty$ because
$\sum_{\ell=1}^{k}\mathbb{E}[\Lambda_{\ell}(x)\Lambda_{\ell}(y)]$ converges to
$\mathbb{E}[\mathcal{L}(x)\mathcal{L}(y)]$ as $k\to\infty$. This implies that
the last line of the above display converges to $0$ as $k,n\to\infty$ and
$x\neq y$. We further notice that

$\sum_{\ell=n+1}^{k}\mathbb{E}[\Lambda_{k}(x)\Lambda_{k}(y)]\leq\sum_{\ell=1}^{k}\mathbb{E}[\Lambda_{k}(x)\Lambda_{k}(y)]\leq-2\log|\sin(2\pi(x-y))|.$

Thus, the last line of the above display can be uniformly bounded by
$C|\sin(2\pi(x-y))|^{-\gamma^{2}+\delta}$ for some $\delta>0$. As a result,
the second term of (3.18) is uniformly integrable and hence, integral of that
term we obtain in (3.12) converges to $0$. Using this fact into the right hand
side of (3.12) shows that
$\mathbb{E}\big{[}(\mathcal{D}_{k}-\mathcal{D}_{n})^{2}\big{]}\to 0$ as
$k,n\to\infty$. Since $\\{(\mathcal{D}_{k},\mathfrak{F}_{k})\\}_{k}$ is a
martingale and a $L_{2}$ Cauchy sequence, $\mathcal{D}_{k}$ almost surely
converges to $\mathcal{D}_{\infty}$ by Doob’s martingale convergence theorem.

∎

We recall that the quadratic variation $\langle\cdot\rangle_{n}$ of the
martingale $\mathcal{D}$ is defined as

$\displaystyle\left\langle\mathcal{D}\right\rangle_{n}:=\sum_{k=1}^{n}\mathbb{E}\Big{[}\Big{(}\mathcal{D}_{k}-\mathcal{D}_{k-1}\Big{)}^{2}\Big{|}\mathfrak{F}_{k-1}\Big{]}.$
(3.19)

######  Proposition 3.2.

Let $\langle\cdot\rangle_{n}$ be the quadratic variation of $\mathcal{D}$.
Then we have

$\displaystyle\left\langle\mathcal{D}\right\rangle_{n}$
$\displaystyle=\sum_{k=1}^{n}\int^{1}_{0}\int^{1}_{0}\int^{k}_{k-1}K(e^{u}(x-y))(1-e^{-\delta
u})du\Big{(}1+\frac{\gamma}{2}\big{(}Y_{k-1}-\gamma\,\Omega_{k-1}\big{)}\Big{)}$
$\displaystyle\times\Big{(}1+\frac{\gamma}{2}\big{(}Y_{k-1}(y)-\gamma\,\Omega_{k-1}\big{)}\Big{)}e^{\frac{\gamma}{2}(Y_{k-1}(x)+Y_{k-1}(y))-\frac{\gamma^{2}}{4}\mathbb{E}[Y^{2}_{k-1}(x)]}f(x)f(y)dxdy+\mathcal{E}_{n},$
(3.20)

where the error term $\mathcal{E}_{n}>0$ for all $n$ positive and
$\sup_{n}\mathcal{E}_{n}<\infty$.

###### Proof.

The proof of this proposition follows along the same lines as the proof of
Proposition 3.1, with the conditional expectation instead. We begin with the
following statement

$\displaystyle\mathbb{E}\left[\left(\mathcal{D}_{k}-\mathcal{D}_{k-1}\right)^{2}|\mathfrak{F}_{k-1}\right]=\mathbb{E}\left[\mathcal{D}_{k}^{2}|\mathfrak{F}_{k-1}\right]-\mathcal{D}_{k-1}^{2}.$
(3.21)

Let us now analyze the first term in the expression above in a similar manner
to (3.18). We first compute the conditional expectation and then consider
expansion around $\gamma=0$:

$\displaystyle\mathbb{E}\left[W_{k}^{0}(x)W_{k}^{0}(y)|\mathfrak{F}_{k-1}\right]$
$\displaystyle=\mathbb{E}\Big{[}\left(Y_{k-1}(x)-\gamma\Omega_{k-1}(x)+\Lambda_{k}(x)-\gamma\left(\Omega_{k}(x)-\Omega_{k-1}(x)\right)\right)$
$\displaystyle\left(Y_{k-1}(y)-\gamma\Omega_{k-1}(y)+\Lambda_{k}(y)-\gamma\left(\Omega_{k}(y)-\Omega_{k-1}(y)\right)\right)$
$\displaystyle\times
e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)}e^{\frac{\gamma}{2}\Lambda_{k}(x)-\frac{\gamma^{2}}{4}\left(\Omega_{k}(x)-\Omega_{k-1}(x)\right)}e^{\frac{\gamma}{2}\Lambda_{k}(y)-\frac{\gamma^{2}}{4}\left(\Omega_{k}(y)-\Omega_{k-1}(y)\right)}|\mathfrak{F}_{k-1}\Big{]}$
$\displaystyle=\left(Y_{k-1}(x)-\gamma\Omega_{k-1}(x)\right)\left(Y_{k-1}(y)-\gamma\Omega_{k-1}(y)\right)e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)}$
$\displaystyle\times
e^{-\frac{\gamma^{2}}{4}\left(\Omega_{k}(x)+\Omega_{k}(y)-\Omega_{k-1}(x)-\Omega_{k-1}(y)\right)}\mathbb{E}\left[e^{\frac{\gamma}{2}\left(\Lambda_{k}(x)+\Lambda_{k}(y)\right)}|\mathfrak{F}_{k-1}\right]$
$\displaystyle+\left(Y_{k-1}(x)-\gamma\Omega_{k-1}(x)\right)e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)}e^{-\frac{\gamma^{2}}{4}\left(\Omega_{k}(x)+\Omega_{k}(y)-\Omega_{k-1}(x)-\Omega_{k-1}(y)\right)}$
$\displaystyle\times\mathbb{E}\left[\left(\Lambda_{k}(y)-\gamma\left(\Omega_{k}(y)-\Omega_{k-1}(y)\right)\right)e^{\frac{\gamma}{2}\left(\Lambda_{k}(x)+\Lambda_{k}(y)\right)}|\mathfrak{F}_{k-1}\right]$
$\displaystyle+\left(Y_{k-1}(y)-\gamma\Omega_{k-1}(y)\right)e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)}e^{-\frac{\gamma^{2}}{4}\left(\Omega_{k}(x)+\Omega_{k}(y)-\Omega_{k-1}(x)-\Omega_{k-1}(y)\right)}$
$\displaystyle\times\mathbb{E}\left[\left(\Lambda_{k}(x)-\gamma\left(\Omega_{k}(x)-\Omega_{k-1}(x)\right)\right)e^{\frac{\gamma}{2}\left(\Lambda_{k}(x)+\Lambda_{k}(y)\right)}|\mathfrak{F}_{k-1}\right]$
$\displaystyle+e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)}e^{-\frac{\gamma^{2}}{4}\left(\Omega_{k}(x)+\Omega_{k}(y)-\Omega_{k-1}(x)-\Omega_{k-1}(y)\right)}$
$\displaystyle\times\mathbb{E}\left[\left(\Lambda_{k}(x)-\gamma\left(\Omega_{k}(x)-\Omega_{k-1}(x)\right)\right)\left(\Lambda_{k}(y)-\gamma\left(\Omega_{k}(y)-\Omega_{k-1}(y)\right)\right)e^{\frac{\gamma}{2}\left(\Lambda_{k}(x)+\Lambda_{k}(y)\right)}|\mathfrak{F}_{k-1}\right]$
$\displaystyle=(\mathbf{I})+(\mathbf{II})+(\mathbf{III})+(\mathbf{IV})$ (3.22)

We first provide simplified expression of
$(\mathbf{I}),(\mathbf{II}),(\mathbf{III})$ and $(\mathbf{IV})$ which will be
followed by explanations.

$\displaystyle(\mathbf{I})=$ $\displaystyle W_{k-1}^{0}(x)W_{k-1}^{0}(y)$
$\displaystyle+\frac{\gamma^{2}}{4}(\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]+O(\gamma))\left(Y_{k-1}(x)-\gamma\Omega_{k-1}(x)\right)\left(Y_{k-1}(y)-\gamma\Omega_{k-1}(y)\right)$
(3.23) $\displaystyle\hskip 170.71652pt\times
e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)},$
(3.24) $\displaystyle(\mathbf{II})=$
$\displaystyle\frac{\gamma}{2}\,\big{(}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]+O(\gamma)\big{)}\left(Y_{k-1}(x)-\gamma\Omega_{k-1}(x)\right)e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)},$
(3.25) $\displaystyle(\mathbf{III})=$
$\displaystyle\frac{\gamma}{2}\,\big{(}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]+O(\gamma)\big{)}\left(Y_{k-1}(y)-\gamma\Omega_{k-1}(y)\right)e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)},$
(3.26) $\displaystyle(\mathbf{IV})=$
$\displaystyle\Big{(}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]+\frac{\gamma^{2}}{2}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]+O(\gamma^{3})\Big{)}e^{\frac{\gamma}{2}Y_{k-1}(x)-\frac{\gamma^{2}}{4}\Omega_{k-1}(x)}e^{\frac{\gamma}{2}Y_{k-1}(y)-\frac{\gamma^{2}}{4}\Omega_{k-1}(y)},$
(3.27)

while ignoring the subleading terms. The expressions above are obtained by
first noticing that

$\mathbb{E}\big{[}e^{\frac{\gamma}{2}(\Lambda_{k}(x)+\Lambda_{k}(y))}|\mathfrak{F}_{k-1}\big{]}=\mathbb{E}\big{[}e^{\frac{\gamma}{2}(\Lambda_{k}(x)+\Lambda_{k}(y))}\big{]}=e^{\frac{\gamma^{2}}{4}(\Omega_{k}(x)-\Omega_{k-1}(x))+\frac{\gamma^{2}}{4}(\Omega_{k}(y)-\Omega_{k-1}(y))}e^{\frac{\gamma^{2}}{4}\mathbb{E}[\Lambda_{k}(x)\Lambda_{k}(y)]}.$

Taylor expanding the term
$e^{\frac{\gamma^{2}}{2}\mathbb{E}[\Lambda_{k}(x)\Lambda_{k}(y)]}$ w.r.t
$\gamma$ yields the equality (3.24). To see the second equality (3.25), we
take the derivative w.r.t. $\gamma$ of both sides of the following identity

$\displaystyle
e^{-\frac{\gamma^{2}}{4}(\Omega_{k}(x)-\Omega_{k-1}(x))-\frac{\gamma^{2}}{4}(\Omega_{k}(y)-\Omega_{k-1}(y))}\mathbb{E}\left[e^{\frac{\gamma}{2}\left(\Lambda_{k}(x)+\Lambda_{k}(y)\right)}\right]=e^{\frac{\gamma^{2}}{4}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]},$
(3.29)

and symmetrize w.r.t. $x$ and $y$. As a result, we get the following equation

$\displaystyle\mathbb{E}\left[(\Lambda_{k}(x)-\gamma(\Omega_{k}(x)-\Omega_{k-1}(x)))e^{\frac{\gamma}{2}\left(\Lambda_{k}(x)+\Lambda_{k}(y)\right)}\right]=\frac{\gamma}{2}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]e^{\frac{\gamma^{2}}{4}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]}.$
(3.30)

Plugging the above into the expression of $(\mathbf{II})$ in (3.22) and Taylor
expanding the term
$e^{\frac{\gamma^{2}}{4}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]}$
w.r.t. $\gamma$, we obtain the second identity (3.25). The third equality
(3.26) follows by a similar argument. The fourth equality
(LABEL:eq:cond_exp_3.2) is obtained by noticing that

$\displaystyle\mathbb{E}\Big{[}$
$\displaystyle\left(\Lambda_{k}(x)-\gamma\left(\Omega_{k}(x)-\Omega_{k-1}(x)\right)\right)\left(\Lambda_{k}(y)-\gamma\left(\Omega_{k}(y)-\Omega_{k-1}(y)\right)\right)e^{\frac{\gamma}{2}\left(\Lambda_{k}(x)+\Lambda_{k}(y)\right)}|\mathfrak{F}_{k-1}\Big{]}$
(3.31)
$\displaystyle=e^{\frac{\gamma^{2}}{4}\left(\Omega_{k}(x)+\Omega_{k}(y)-\Omega_{k-1}(x)-\Omega_{k-1}(y)\right)}e^{\frac{\gamma^{2}}{4}\mathbb{E}\left[\Lambda_{k}(x)\Lambda_{k}(y)\right]}\Big{(}\gamma^{2}(\mathbb{E}\big{[}\Lambda_{k}(x)\Lambda_{k}(y)\big{]})^{2}+\mathbb{E}\big{[}\Lambda_{k}(x)\Lambda_{k}(y)\big{]}\Big{)},$
(3.32)

and

$\mathbb{E}\big{[}\Lambda_{k}(x)\Lambda_{k}(y)\big{]}=\int^{k}_{k-1}K(e^{u}(x-y))(1-e^{-\delta
u})du.$

Substituting the expressions of $(\mathbf{I}),(\mathbf{II}),(\mathbf{III})$,
$(\mathbf{IV})$ in (3.21) and rearranging the terms gives

$\displaystyle\mathbb{E}\Big{[}\Big{(}\mathcal{D}_{k}-\mathcal{D}_{k-1}\Big{)}^{2}\Big{|}\mathfrak{F}_{k-1}\Big{]}$
$\displaystyle=$
$\displaystyle\int^{1}_{0}\int^{1}_{0}\int^{k}_{k-1}K(e^{u}(x-y))(1-e^{-\delta
u})\left(1+\frac{\gamma}{2}\left(Y_{k-1}(x)-\gamma\,\Omega_{k-1}(x)\right)\right)$
$\displaystyle\times\left(1+\frac{\gamma}{2}\left(Y_{k-1}(y)-\gamma\,\Omega_{k-1}(x)\right)\right)e^{\frac{\gamma}{2}(Y_{k-1}(x)+Y_{k-1}(y))-\frac{\gamma^{2}}{2}\left(\Omega_{k-1}(x)+\Omega_{k-1}(y)\right)}f(x)f(y)dudxdy$
$\displaystyle+\gamma^{2}\mathrm{Err}_{k},$ (3.33)

where $\mathrm{Err}_{k}$ is the residual sub-leading term. In fact,
$\mathrm{Err}_{k}$ can also be written as integral w.r.t. $x$ and $y$ where
the leading term of the integrands is
$(\mathbb{E}[\Lambda_{k}(x)\Lambda_{k}(y)])^{2}$. Since the integrands of
$\mathrm{Err}_{k}$ are absolutely summable, for
$\mathcal{E}_{n}:=\sum_{k=1}^{n}\mathrm{Err}_{k}$, we have
$\sup_{n}\mathcal{E}_{n}<\infty$. This completes the proof.

∎

Now we start the proof of Theorem 3.1.

###### Proof of Theorem 3.1.

We see from the above expression (3.33) that most of the contribution to
$\left\langle\mathcal{D}\right\rangle_{n}$ is given by high values of
$Y_{k}(x)$. However, these high values should not contribute much to the
negative tail of $\left\langle\mathcal{D}\right\rangle_{n}$ since they tend to
yield high positive values of the integrand. Hence our idea is to compare
$\mathcal{D}_{k}$ with a martingale obtained by replacing $Y_{k}(x)$ by an
alternative martingale $W_{k}(x)$ which does not sum the variation of
$\mathcal{D}_{n}$ when the value of $Y_{k}(x)$ is too large. Our definition of
$\widetilde{\mathcal{D}}_{k}$ has to be carefully chosen so that it compares
well with $\mathcal{D}_{k}$ .

To analyse the quadratic variation, we begin by defining the following
quantities: the variance of $Y_{n}$ for all $n\in\mathbb{N}$, and the sequence
of stopping times

$\displaystyle T^{x}_{k}$
$\displaystyle:=\mathrm{inf}_{n}\Big{\\{}\Omega_{n}>R^{x}_{k-1}:Y_{n}(x)\geq(\gamma+\eta)\Omega_{n}+A\Big{\\}},$
(3.34) $\displaystyle R^{x}_{k}$
$\displaystyle:=\mathrm{inf}\Big{\\{}\Omega_{n}>T^{x}_{k}:Y_{n}(x)\leq\gamma\,\Omega_{n}\Big{\\}},$
(3.35)

where $\Omega_{n}$ is the bound of $\Omega_{n}(x)$, and
$R_{0}^{x}=0,A>\gamma+2$. By introducing the notation

$\displaystyle\mathcal{R}_{x}=\cup_{k=1}^{\infty}[R^{x}_{k-1},T^{x}_{k}],$
$\displaystyle:\gamma
Y_{k}(x):\,\,=\frac{\gamma}{2}Y_{k}(x)-\frac{\gamma^{2}}{4}\Omega_{k},$

we can define

$\displaystyle
W_{k}(x):=\sum_{j=1}^{k}\left(W_{j}^{0}(x)-W_{j-1}^{0}(x)\right)\mathbf{1}_{\Omega(j)\in\mathcal{R}_{x}},$
(3.36)

where $W_{j}^{0}(x)$ is given by (3.9), and its integral is denoted by

$\displaystyle\widetilde{\mathcal{D}}_{k}$
$\displaystyle:=\int^{1}_{0}W_{k}(x)f(x)dx.$ (3.37)

The function $W_{k}(x)$ has alternative representation in terms of stopping
times

$\displaystyle
W_{k}(x)=\begin{cases}\big{(}Y_{k}(x)-\gamma\,\Omega_{k}\big{)}e^{:\gamma
Y_{k}(x):}+\sum_{\ell=1}^{n}(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}+\xi_{k};&\Omega_{k}\in[R^{x}_{n},{T}^{x}_{n+1}],\\\
\\\ \sum_{\ell=1}^{n}(A+\eta T^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}+\xi_{k};&\Omega_{k}\in[{T}^{x}_{n},R^{x}_{n}],\end{cases}$
(3.38)

where $\xi_{k}$ is the sum of the residual between $W_{\ell}$ and
$(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}$ for $\ell=1,\ldots,n$. The
residual corresponding to the $\ell$-th difference is bounded by
$\max\\{0,\Lambda_{\ell}-\eta T^{x}_{\ell}-A\\}e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})T^{x}_{\ell}}$. As a result, we have

$\xi_{k}\leq\sum_{\ell=1}^{n}\max\\{0,\Lambda_{\ell}-\eta
T^{x}_{\ell}-A\\}e^{\gamma A+(\gamma\eta+\frac{\gamma^{2}}{2})T^{x}_{\ell}}.$

Repeating the calculations leading to Proposition 3.2, we find that the
quadratic variation of the function $\widetilde{\mathcal{D}}_{k}$ in (3.37) is

$\displaystyle\langle\widetilde{\mathcal{D}}\rangle_{k^{\prime}}$ (3.39)
$\displaystyle=\sum_{k=1}^{k^{\prime}}\int^{1}_{0}\int^{1}_{0}\int^{k}_{k-1}K(e^{u}(x-y))(1-e^{-\delta
u})\Big{(}1+\frac{\gamma}{2}(Y_{k-1}(x)-\gamma\,\Omega_{k-1}\big{)}\Big{)}\Big{(}1+\frac{\gamma}{2}(Y_{k-1}(y)-\gamma\,\Omega_{k-1}\big{)}\Big{)}$
$\displaystyle\times\mathbf{1}_{k\in\mathcal{R}_{x}\cap\mathcal{R}_{y}}e^{:Y_{k-1}(x):+:Y_{k-1}(y):}\mathcal{G}_{k}(x)\mathcal{G}_{k}(y)du\,dx\,dy+\widetilde{\mathcal{E}_{k^{\prime}}},$
(3.40)

for some $\mathcal{G}_{k}(x)$ which can be computed starting from (3.37) and
repeating the computations in the proof of Proposition 3.2. We also note from
Proposition 3.3 that $\widetilde{\mathcal{D}}_{k}$ is a martingale which is
bounded in $L^{2}$, and by the Doob’s martingale convergence theorem, there
exists an almost sure limit of $\widetilde{\mathcal{D}}_{k}$ we call
$\widetilde{\mathcal{D}}_{\infty}$.

Furthermore, we know that $\mathcal{D}_{k}$ in (3.8) also converges almost
surely due to Proposition 3.1, and let that limit be equal to
$\mathcal{D}_{\infty}$. From the definitions (3.8), (3.37), and noting that
the residual term $\xi_{n}\geq 0$ for all $n\in\mathbb{Z}$, it then follows
that

$\displaystyle\mathcal{D}_{\infty}\geq\widetilde{\mathcal{D}}_{\infty}-Q,$
(3.41)

where

$\displaystyle
Q:=\sum_{\ell=1}^{\infty}\int^{1}_{0}(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}f(x)dx.$ (3.42)

Note that, the definitions (LABEL:def:sfDMgN), (3.8) imply that
$\mathcal{D}_{\infty}$ is proportional to ${\sf DM}_{N}f(\mathbf{i}\infty)$.

In Propositions 3.3 and 3.4, we present sub-Gaussian tail bounds for
$\widetilde{D}_{\infty}$ and $Q$, respectively. By using these bounds in the
inequality (3.41), we obtain the following result:

$\displaystyle\mathbb{P}\left[({\sf
DM}_{\gamma,N}f)(\mathbf{i}\infty)\leq-v\right]\leq c_{3}e^{-c_{4}v^{2}},$

where the constant $c_{3}=\int_{0}^{1}f(x)dx$ comes from the normalization in
(LABEL:def:sfDMgN). Notice that

$\displaystyle\mathbb{P}\Big{(}\widetilde{\mathcal{D}}_{\infty}-Q\leq-s\Big{)}\leq\mathbb{P}\Big{(}\widetilde{\mathcal{D}}_{\infty}\leq-\frac{s}{2}\big{)}+\mathbb{P}\Big{(}Q\geq\frac{s}{2}\big{)}\leq
e^{-\lambda\frac{s}{2}}\mathbb{E}[e^{\lambda\widetilde{\mathcal{D}}_{\infty}}]+e^{-\lambda\frac{s}{2}}\mathbb{E}[e^{\lambda
Q}]$ $\displaystyle\leq e^{-\lambda s/2+c\lambda^{2}}$ $\displaystyle\leq
e^{-\frac{s^{2}}{16c}}.$

The first inequality in the above display follows by union bound, The second
inequality is obtained by applying Markov inequality. To obtain the third
inequality, we bound $\mathbb{E}[e^{\lambda\widetilde{\mathcal{D}}_{\infty}}]$
by $e^{c\lambda^{2}}$ using Proposition 3.3 and bound $\mathbb{E}[e^{\lambda
Q}]$ by $e^{c\lambda^{2}}$ using Proposition 3.4. Here $c$ is a positive
constant independent of $\lambda$. The last inequality of the above display
follows by substituting $\lambda=\frac{s}{4c}$.

If we repeat the above calculations with $Y_{N}(x;q)$ instead, the main
difference is that the constant $c$ in the sub-Gaussian tails of Propositions
3.3 and 3.4 will depend on $\tau$. We denote this $\tau$-dependent constant as
$c(\tau)$. Additionally, we define the corresponding normalization constant as
$c_{1}(\tau)=\int_{0}^{1}f(x,\tau)dx$. Consequently, we obtain the following
result:

$\displaystyle\mathbb{P}\left[({\sf DM}_{\gamma,N}f)(\tau)\leq-v\right]\leq
c_{1}(\tau)e^{-c_{2}(\tau)v^{2}},$

proving the theorem. ∎

The following reasoning leads to the proof of convergence. If a process
$M_{k}$ is a martingale and $\mathbb{E}[e^{sM_{k}}]<\infty$ for any
$s\in\mathbb{R}$, then $e^{-sM_{k}-\frac{s^{2}}{2}<M_{k}>_{k}}$ is also a
martingale, with $<.>_{k}$ representing the quadratic variation. Furthermore,
if $M_{0}=0$, then

$\displaystyle\mathbb{E}[e^{-sM_{k}-\frac{s^{2}}{2}<M_{k}>}]=1$ (3.43)

for all $k$, leading to the precise bound. We will therefore, aim to show that
${\sf DM}_{\gamma,N}f(\tau)$ is a martingale and obtain the bound through the
quadratic variation.

The convergence of $\mathcal{D}_{\infty}$ will be proved through the
convergence of the expectation of the exponential of the terms
$\widetilde{\mathcal{D}}_{\infty}$, $Q$ in Propositions 3.3, 3.4 respectively.

######  Proposition 3.3.

The following statements hold true:

  1. (1)

The quadratic variation $\langle\widetilde{\mathcal{D}}\rangle_{k^{\prime}}$
in (3.40) is $L^{2}$ bounded and ${\mathcal{D}}_{k^{\prime}}$ converges to
${\mathcal{D}}_{\infty}$ almost surely.

  2. (2)

For any $s\in\mathbb{R}_{<0}$ and $c>0$, we can establish the following upper
bound on the expectation of the exponential of
$\widetilde{\mathcal{D}}_{\infty}$, which represents the limit of
$\widetilde{\mathcal{D}}_{k}$ defined in (3.37):

$\displaystyle\mathbb{E}[e^{s\widetilde{\mathcal{D}}_{\infty}}]\leq
e^{\frac{cs^{2}}{2}}.$ (3.44)

###### Proof.

(1): To prove the first part of this proposition, it suffices to show that
$\langle\widetilde{\mathcal{D}}\rangle_{k^{\prime}}$ is bounded and
$\mathbb{E}[e^{s\widetilde{\mathcal{D}}_{\infty}}]<\infty$. We show these as
follows. To examine the bound on the quadratic variation (3.40), it is enough
to focus on $k\in[R_{k},T_{k+1}]$. By referring to (3.34), we can derive the
following expression:

$\displaystyle Y_{k-1}\leq A+\left(\gamma+\eta\right)\Omega_{k-1},$ (3.45)

and consequently,

$\displaystyle\max_{\eqref{bound:Yk-1}}\left(1+\frac{\gamma}{2}\left(Y_{k-1}(x)-\Omega_{k-1}\right)\right)$
$\displaystyle e^{\frac{\gamma}{2}Y_{k-1}-\frac{\gamma^{2}}{4}\Omega_{k-1}}$
$\displaystyle=\left(1+\frac{\gamma}{2}\left(A+\eta\,\Omega_{k-1}\right)\right)e^{\frac{\gamma
A}{2}+\frac{\gamma}{2}\left(\eta+\frac{\gamma}{2}\right)\Omega_{k-1}}.$

Substituting the above expression in (3.40) we get

$\displaystyle\langle\widetilde{\mathcal{D}}\rangle_{k^{\prime}}\leq\sum_{k=1}^{k^{\prime}}\int\int\int^{k}_{k-1}K(e^{u}(x-y))(1-e^{-\delta
u})$
$\displaystyle\left(1+\frac{\gamma}{2}\left(A+\eta\,\Omega_{k-1}\right)\right)^{2}$
(3.46) $\displaystyle\times e^{{\gamma
A}+{\gamma}\left(\eta+\frac{\gamma}{2}\right)\Omega_{k-1}}\mathbf{1}_{k\in\mathcal{R}_{x}\cap\mathcal{R}_{y}}\mathcal{G}_{k}(x)\mathcal{G}_{k}(y)du\,dx\,dy,$
(3.47)

We now claim that
$\sup_{k^{\prime}}\langle\widetilde{\mathcal{D}}\rangle_{k^{\prime}}<\infty$
for all small $\gamma>0$. Note that

$\displaystyle\Omega_{k}\mathop{=}^{\eqref{eq:Omk}}\frac{k}{2}-\frac{1-e^{-\delta
k}}{2\delta}+\frac{1}{2}\mathbb{E}[\mathcal{R}^{2}(x)].$ (3.48)

As a result $\Omega_{k}$ grows linearly with $k$. Substituting this bound into
the integrand of the above inequality yields

$\left(A+\eta\,\Omega_{k-1}\right)^{2}e^{{\gamma
A}+{\gamma}\left(\eta+\frac{\gamma}{2}\right)\Omega_{k-1}}\lesssim(\eta\,k)^{2}e^{\gamma(\eta+\frac{\gamma}{2})k}.$

Combining this bound with the inequality $K(e^{u}(x-y))\leq e^{-au}|x-y|^{-a}$
for some $a\in(0,1)$, we get

$\langle\widetilde{\mathcal{D}}\rangle_{k^{\prime}}\lesssim\int\int\sum_{k=1}^{k^{\prime}}(\eta
k)^{2}e^{-(a-\gamma(\eta+\gamma/2)k)}|x-y|^{-a}\mathbf{1}_{k\in\mathcal{R}_{x}\cap\mathcal{R}_{y}}\mathcal{G}_{k}(x)\mathcal{G}_{k}(y)du\,dx\,dy+\widetilde{\mathcal{E}_{k^{\prime}}}.$

Note that $k^{2}e^{-ak}e^{\gamma(\eta+\frac{\gamma}{2})}$ is absolutely
summable for $\gamma$ small and $|x-y|^{-a}$ is integrable. Therefore,
$\langle\widetilde{\mathcal{D}}\rangle_{k^{\prime}}$ is bounded as
$k^{\prime}$ tends to $\infty$.

(2): Now we proceed to show
$\mathbb{E}[e^{s\widetilde{\mathcal{D}}_{\infty}}]<\infty$ for any $s<0$. For
all $x$, the function $W_{k}(x)-\xi_{k}(x)$ in (3.38) has the following upper
bound

$\displaystyle
W_{k}(x)-\xi_{k}(x)\geq-\Big{(}c+|Y_{k}(x)-\gamma\,\Omega_{k}|\Big{)}e^{A+\eta\,\Omega_{k}}+\sum_{m=1}^{k}(A+\eta{T}^{x}_{m})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{m}}.$

Since $Y_{k}(x)$ is a Gaussian distribution,
$\mathbb{E}[e^{s|Y_{k}(x)|}]<\infty$ for any $s\in\mathbb{R}$, and therefore,

$\mathbb{E}\Big{[}\exp\big{(}s\widetilde{\mathcal{D}}_{k}-s\int^{1}_{0}\xi_{k}(x)f(x)dx\big{)}\Big{]}<\infty$

for any $s<0$. As a result,
$\mathbb{E}[e^{s\widetilde{\mathcal{D}}_{k}}]<\infty$ for $s<0$ since
$\xi_{k}(x)>0$ for all $x\in[0,1]$ and any integer $k$. Moreover,
$\widetilde{\mathcal{D}}_{k}$ is a martingale and its quadratic variation is
bounded. Since $\mathbb{E}[e^{s\widetilde{\mathcal{D}}_{k}}]<\infty$, We get
$\mathbb{E}[e^{s\widetilde{\mathcal{D}}_{k}}]=e^{s^{2}\langle\widetilde{\mathcal{D}}\rangle_{k}/2}$
by Wald’s identity. The identity (3.43) then suggests that,

$\displaystyle\mathbb{E}[e^{s\widetilde{\mathcal{D}}_{k}}]\leq e^{cs^{2}/2}.$

And since the bound is independent of $k$, the above inequality hold for
$\widetilde{\mathcal{D}}_{\infty}$.

∎

The final ingredient needed for the proof of Theorem 3.1 is to bound the
exponential expectation of $Q$ in (3.41).

######  Proposition 3.4.

For any $s\in\mathbb{R}$ and $c>0$, the expectation of the exponential of $Q$
defined in (3.42) has the following upper bound

$\displaystyle\mathbb{E}[e^{sQ}]\leq e^{\frac{cs^{2}}{2}}.$ (3.49)

###### Proof.

We prove this by using tools from martingale theory. Recall the definition of
$Q$ from (3.42). Recall that sigma algebras
$\mathfrak{F}_{1}\subset\mathfrak{F}_{2}\subset\mathfrak{F}_{3}\subset\ldots$
such that $\mathcal{L}_{k}$ is measurable with respect to $\mathfrak{F}_{k}$

$\displaystyle Q_{k}=\mathbb{E}\big{[}Q|\mathfrak{F}_{k}\big{]}.$ (3.50)

We now aim to prove the convergence of $Q_{k}$, for which it is sufficient to
prove the boundedness of the integrand of $Q$ in (3.42).

Consider the variable

$\displaystyle Q^{x}:=\sum^{\infty}_{\ell=1}(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}.$ (3.51)

For $T_{\ell}^{x}\in\mathbb{Z}$, the above expression satisfies the inequality

$\displaystyle
Q^{x}\leq\sum_{m=1}^{\infty}\sum_{\ell=1}^{\infty}(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}\mathbf{1}(T^{x}_{\ell}\in\left[m-1,m\right])=:\sum_{m=1}^{\infty}Q^{x,m}.$
(3.52)

The problem at hand then simplifies to proving the convergence of $Q^{x,m}$.
To compute its bound, we begin by noting that

$\displaystyle\mathbb{E}[Q^{x,m}]$ $\displaystyle=(A+\eta\,m)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})m}\sum_{\ell=1}^{\infty}\mathbb{P}\Big{(}\Big{\\{}T^{x}_{\ell}\in\left[m-1,m\right]\Big{\\}}\Big{)}$
$\displaystyle=(A+\eta m)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})m}\mathbb{E}\Big{[}\\#\Big{\\{}1\leq\ell\leq\infty:T^{x}_{\ell}\in\left[m-1,m\right]\Big{\\}}\Big{]}.$

The remainder of the proof can be divided into several steps, with each step
involving the demonstration of a lemma. The inequality (3.53) in Lemma 3.1
implies that

$\displaystyle\mathbb{E}[Q^{x,m}]\mathop{\leq}^{\eqref{eq:lemmaclaim1}}(A+\eta
m)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})m}\frac{c}{\sqrt{m-1}}e^{-\frac{(\gamma+\eta)^{2}(m-1)}{2}},$

and therefore due to (3.52),

$\displaystyle\mathbb{E}[Q^{x}]$
$\displaystyle\mathop{\leq}^{\eqref{def:Qnxm}}\sum_{m=1}^{\infty}(A+\eta
m)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})m}\frac{c}{\max\\{\sqrt{(m-1)},1\\}}e^{-\frac{(\gamma+\eta)^{2}(m-1)}{2}}$
$\displaystyle\lesssim\sum_{m=1}^{\infty}\frac{1}{\max\\{\sqrt{m-1},1\\}}(A+\eta
m)e^{-\frac{\eta^{2}m}{2}}.$

Having the above expression available, the only remaining task to obtain
(3.49) is to evaluate the quadratic variation of $Q^{x}$, which can be
obtained through Lemma 3.2. The quadratic variation of $Q_{k}$ is bounded due
to Lemma 3.2 proving this proposition. ∎

######  Lemma 3.1.

For an arbitrary constant $c$, the following inequality holds for stopping
time $T_{l}^{x}$ defined in (3.34)

$\displaystyle\mathbb{E}\Big{[}\\#\Big{\\{}1\leq\ell\leq\infty:T^{x}_{\ell}\in\left[m-1,m\right]\Big{\\}}\Big{]}$
$\displaystyle\leq\frac{c}{\max\\{\sqrt{(m-1)},1\\}}e^{-\frac{(\gamma+\eta)^{2}(m-1)}{2}}.$
(3.53)

###### Proof.

The definition of the stopping times (3.34) implies that, for
$T^{\ell}_{x}\in[m-1,m]$,

$\displaystyle\Omega_{n}\in[m-1,m],$ $\displaystyle
Y_{n}(x)\geq(\gamma+\eta)\Omega_{n}+A.$

Since $\Omega_{n}(=\int^{n}_{0}K(0)(1-e^{-\delta u}))du+c_{2}$ (see (3.48))
grows linearly in $n$, we have $c_{1}(m-1)\leq\,n\leq c_{1}m$ for some
constant $c_{1}>0$ because $\Omega_{n}\in[m-1,m]$. Under the assumption
$A>\gamma+\eta$ and recognizing that $Y_{\ell}(x)$ is the sum of independent
Gaussian random variable
$\sum_{t=1}^{\ell}(\mathcal{L}_{t}-\mathcal{L}_{t-1})+\mathcal{R}(x)$, we can
derive the following expression utilizing the maximal Bernstein’s inequality:

$\displaystyle\mathbb{P}\Big{(}\Big{\\{}\exists\ell\geq
1:T^{x}_{l}\in[m-1,m]\Big{\\}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\sup_{1\leq l\leq
c_{1}(m-1)}Y_{\ell}(x)\geq(\gamma+\eta)\Omega_{\ell}+A\Big{)}$
$\displaystyle\leq\frac{c}{\max\\{\sqrt{\Omega_{c_{1}(m-1)}},1\\}}\exp\Big{(}-(\gamma+\eta)^{2}\Omega_{c_{1}(m-1)}\Big{)}.$
(3.54)

The inequality (3.53) is then obtained as follows

$\displaystyle\mathbb{E}\Big{[}\\#\Big{\\{}1\leq\ell\leq\infty:T^{x}_{\ell}\in\left[m-1,m\right]\Big{\\}}\Big{]}$
$\displaystyle=\sum_{k=1}^{\infty}\mathbb{P}\Big{(}\\#\\{\ell:T^{x}_{\ell}\in[m-1,m]\\}\geq
k\Big{)}$
$\displaystyle\mathop{\leq}^{\eqref{eq:Concentration_k}}\sum_{k=1}^{\infty}\mathbb{P}\Big{(}\\#\\{\ell:T^{x}_{\ell}\in[m-1,m]\\}\geq
1\Big{)}\frac{1}{2^{k}}$
$\displaystyle\mathop{\leq}^{\eqref{eq:Concentration}}\frac{c}{\max\\{\sqrt{\Omega_{c_{1}(m-1)}},1\\}}\exp\Big{(}-(\gamma+\eta)^{2}\Omega_{c_{1}(m-1)}\Big{)}$
$\displaystyle\mathop{\leq}^{\eqref{eq:OmegaExpansion}}\frac{c}{\max\\{\sqrt{(m-1)},1\\}}\exp\Big{(}-(\gamma+\eta)^{2}\frac{(m-1)}{2}\Big{)}.$

The following inequality was used to obtain the result above

$\displaystyle\mathbb{P}\Big{(}\\#\\{\ell:T^{x}_{\ell}\in[m-1,m]\\}\geq
k\Big{)}\leq 2^{-k}\mathbb{P}\Big{(}\Big{\\{}\exists\ell\geq
1:T^{x}_{l}\in[m-1,m]\Big{\\}}\Big{)}.$ (3.55)

The proof of the above expression is as follows.

We begin by noting that the inequality (3.55) can be rewritten as

$\mathbb{P}\Big{(}\\#\\{\ell:T^{x}_{\ell}\in[m-1,m]\\}\geq
k\big{|}\\#\\{\ell:T^{x}_{\ell}\in[m-1,m]\\}\geq k-1\Big{)}\leq\frac{1}{2}.$

The increments of $Y_{\ell}(x)$ are independent Gaussian random variables,
i.e.,

$Y_{\ell+1}(x)-Y_{\ell}(x)\sim N\left(0,\int^{\ell+1}_{\ell}(1-e^{-\delta
u})du\right).$

Consequently, $Y_{\ell}(x)$ is regarded as the evaluation of a Brownian motion
$B_{t}$ at time $\int^{\ell}_{0}(1-e^{-\delta u})du$ and
$B_{0}:=\mathcal{R}(x)$. Recall the definition of $T_{\ell}^{x}$ in (3.34).
The fact that the increments of Brownian motion $B_{t}$ are stationary then
implies that, for $A\geq\gamma+2$,

$\mathbb{P}\Big{(}\\#\\{\ell:T^{x}_{\ell}\in[m-1,m]\\}\geq
k\big{|}\\#\\{\ell:T^{x}_{\ell}\in[m-1,m]\\}\geq
k-1\Big{)}\leq\mathbb{P}\Big{(}\inf_{t\in[0,1]}B_{t}\leq\gamma-A\Big{)}\leq\frac{1}{2}.$

This completes the proof of the lemma.

∎

######  Lemma 3.2.

The quadratic variation of $Q_{k}$ defined in (3.50) can be expressed as

$\displaystyle\langle
Q_{k}\rangle_{k}=\sum_{\ell=1}^{k}\int^{1}_{0}\int^{1}_{0}\mathbb{E}\Big{[}\mathfrak{A}^{x}_{\ell}\mathfrak{A}^{y}_{\ell}\Big{|}\mathfrak{F}_{\ell}\Big{]}f(x)f(y)dxdy,$
(3.56)

for some random function $\mathfrak{A}^{x}_{\ell}$ (defined in (3.65)) such
that

$\displaystyle|\mathfrak{A}^{x}_{\ell}|\leq
C\Big{(}1+\Omega_{\ell}\Big{)}e^{(\gamma\eta+\frac{\gamma^{2}}{2})\Omega_{\ell}}.$
(3.57)

The constant $C>0$ is independent of $x\in[0,1]$ and the integer $\ell$.

###### Proof.

Recall that $Q_{k}$ in (3.50) is a martingale. Taking into account the
definition of the integrand of $Q$ given by (3.51):

$\displaystyle Q^{x}=\sum_{\ell=1}^{\infty}(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}},$

we define the martingale

$\displaystyle
Q^{x}_{\ell}:=\mathbb{E}\left[Q^{x}|\mathfrak{F}_{\ell}\right],\quad\ell\in\mathbb{N}.$
(3.58)

With these notations, the quadratic variation of $Q_{k}$ is given by

$\displaystyle\langle Q_{k}\rangle_{k}$
$\displaystyle=\sum_{\ell=1}^{k}\mathbb{E}\Big{[}\big{(}Q_{\ell+1}-Q_{\ell}\big{)}^{2}\Big{|}\mathfrak{F}_{\ell}\Big{]}$
$\displaystyle=\sum_{\ell=1}^{k}\int\int\mathbb{E}\Big{[}(Q^{x}_{\ell+1}-Q^{x}_{\ell})(Q^{y}_{\ell+1}-Q^{y}_{\ell})\Big{|}\mathfrak{F}_{\ell}\Big{]}f(x)f(y)dxdy.$
(3.59)

In what follows, we rewrite $Q^{x}_{\ell+1}-Q^{x}_{\ell}$ in terms a new set
of stopping time and use that expression to find almost sure bound on
$Q^{x}_{\ell+1}-Q^{x}_{\ell}$.

Let $k^{\prime\prime}<k^{\prime}$ represent the number of distinct values of
$T^{x}_{\ell}$ between $0$ and $\Omega_{k^{\prime}}$. We divide the discussion
into two cases:

  1. (1)

The first case arises when
$\Omega_{k^{\prime}}\in[T^{x}_{k^{\prime\prime}},R^{x}_{k^{\prime\prime}}]$.

  2. (2)

The second cases occurs when
$\Omega_{k^{\prime}}\in[R^{x}_{k^{\prime\prime}},T^{x}_{k^{\prime\prime}+1}]$.

In the first case, we write

$\displaystyle Q^{x}_{k^{\prime}}$
$\displaystyle=\sum_{\ell=1}^{k^{\prime\prime}}(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}+\mathbb{E}\left[\sum_{\ell=1}^{\infty}\mathbf{1}_{\left\\{\widehat{T}_{\ell}^{x}<+\infty\right\\}}(A+\eta\widehat{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{\ell}+\Omega_{k^{\prime}}\right)}\right],$

where the set of stopping times $\widehat{T}^{x}_{\ell}$ are defined in the
following way. Starting with $\widehat{T}^{x}_{0}:=0$, for $\ell\geq 1$,

$\displaystyle\widehat{R}_{\ell}^{x}$
$\displaystyle:=\textrm{inf}\left\\{\Omega_{s+k^{\prime}}\geq\widehat{T}_{\ell-1}^{x}:H_{s+k^{\prime}}\leq\eta\,\Omega_{\ell}\right\\},$
(3.60) $\displaystyle\widehat{T}_{\ell}^{x}$
$\displaystyle:=\textrm{inf}\left\\{\Omega_{s+k^{\prime}}\geq\widehat{R}_{\ell}^{x}:H_{s+k^{\prime}}\geq(\eta+\gamma)\Omega_{\ell}+A\right\\},$
(3.61)

where

$\displaystyle
H_{s}(x):=Y_{k^{\prime}}(x)+\sum_{n=k^{\prime}+1}^{s}(\mathcal{L}_{n}(x)-\mathcal{L}_{n-1}(x)),\quad
s>k^{\prime}.$ (3.62)

Similarly, in the second case, we can express the value of
$Q^{x}_{k^{\prime}}$ as

$\displaystyle Q^{x}_{k^{\prime}}$
$\displaystyle=\sum_{\ell=1}^{k^{\prime\prime}}(A+\eta{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{\ell}}+\mathbb{E}\left[\sum_{\ell=1}^{\infty}\mathbf{1}_{\left\\{\widetilde{T}_{\ell}^{x}<+\infty\right\\}}(A+\eta\widetilde{T}^{x}_{\ell})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widetilde{T}^{x}_{\ell}+\Omega_{k^{\prime}}\right)}\right].$

Here we define $\widetilde{R}^{x}_{0}:=0$, and for $\ell\geq 1$, the stopping
times are defined as

$\displaystyle\widetilde{T}_{\ell}^{x}$
$\displaystyle:=\textrm{inf}\left\\{\Omega_{s+k^{\prime}}\geq\widetilde{R}_{\ell-1}^{x}:H_{s+k^{\prime}}\geq(\eta+\gamma)\Omega_{\ell}+A\right\\},$
(3.63) $\displaystyle\widetilde{R}_{\ell}^{x}$
$\displaystyle:=\textrm{inf}\left\\{\Omega_{s+k^{\prime}}\geq\widetilde{T}_{\ell}^{x}:H_{s+k^{\prime}}\leq\gamma\,\Omega_{\ell}\right\\},$
(3.64)

with the definition (3.62) of $H_{s}$.

With the above notation, the integrand of the quadratic variation (3.59) has
the expression

$\displaystyle\left(Q_{k^{\prime}+1}^{x}-Q_{k^{\prime}}^{x}\right)$
$\displaystyle=:\mathfrak{A}_{k^{\prime}}^{x}$
$\displaystyle=(A+\eta{T}^{x}_{k^{\prime\prime}+1})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{k^{\prime\prime}_{k^{\prime}+1}}}{\mathbf{1}(k^{\prime\prime}_{k^{\prime}+1}\neq
k^{\prime\prime}_{k^{\prime}})}$
$\displaystyle+\mathbb{E}_{Y_{k^{\prime}+1}}\left[\sum_{\ell=1}^{\infty}\mathbf{1}_{\left\\{\widehat{T}_{\ell}^{x}<+\infty\right\\}}\left(A+\eta\widehat{T}^{x}_{\ell}+\eta\,\Omega_{k^{\prime}+1}\right)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{\ell}+\Omega_{k^{\prime}+1}\right)}\right]$
$\displaystyle-\mathbb{E}_{Y_{k^{\prime}}}\left[\sum_{\ell=1}^{\infty}\mathbf{1}_{\left\\{\widehat{T}_{\ell}^{x}<+\infty\right\\}}\left(A+\eta\widehat{T}^{x}_{\ell}+\eta\,\Omega_{k^{\prime}}\right)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{\ell}+\Omega_{k^{\prime}}\right)}\right],$
(3.65)

and substituting the above in (3.59), we obtain (3.56). We now proceed to
complete showing (3.57). We seek to show uniform bound on
$\mathfrak{A}^{x}_{k^{\prime}}$. To this end, we introduce the following two
events:

$\displaystyle\mathbf{1}^{(1)}_{\mathsf{Good}}:=\mathbf{1}\Big{(}Y_{k^{\prime}}(x)\geq\gamma\,\Omega_{k^{\prime}}\Big{)},\qquad\mathbf{1}^{(2)}_{\mathsf{Good}}:=\mathbf{1}\Big{(}Y_{k^{\prime}}(x)\leq(\gamma+\eta)\Omega_{k^{\prime}}\Big{)}.$
(3.66)

We now complete proof of (3.57) using the following two claims whose proofs
are deferred to the end of this section. The Claims 3.1, 3.2 therefore imply
that $\mathfrak{A}_{k^{\prime}}^{x}$ is uniformly bounded.

∎

######  Claim 3.1.

There exists $C_{1}=C_{1}(A,\gamma,\eta)>0$ such that

$\displaystyle\Big{|}\mathfrak{A}^{x}_{k^{\prime}}\mathbf{1}^{(1)}_{\sf
Good}\Big{|}$ $\displaystyle\leq
C_{1}\Big{(}1+\,\Omega_{k^{\prime}}\Big{)}e^{(\gamma\eta+\frac{\gamma^{2}}{2})\,\Omega_{k^{\prime}}}.$

###### Proof.

Let us rewrite (3.65) as

$\displaystyle\mathfrak{A}^{x}_{k^{\prime}}=I_{1}+I_{2}-I_{3}$ (3.67)

where

$\displaystyle I_{1}:=$
$\displaystyle(A+\eta{T}^{x}_{k^{\prime\prime}_{k^{\prime}+1}})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2}){T}^{x}_{k^{\prime\prime}_{k^{\prime}+1}}}{\mathbf{1}(k^{\prime\prime}_{k^{\prime}+1}\neq
k^{\prime\prime}_{k^{\prime}})},$ (3.68) $\displaystyle I_{2}:=$
$\displaystyle\mathbb{E}_{Y_{k^{\prime}+1}}\left[\sum_{\ell=1}^{\infty}\mathbf{1}_{\left\\{\widehat{T}_{\ell}^{x}(k^{\prime}+1)<+\infty\right\\}}\left(A+\eta\widehat{T}^{x}_{\ell}+\eta\,\Omega_{k^{\prime}+1}\right)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{\ell}+\Omega_{k^{\prime}+1}\right)}\right],$
(3.69) $\displaystyle I_{3}:=$
$\displaystyle\mathbb{E}_{Y_{k^{\prime}}}\left[\sum_{\ell=1}^{\infty}\mathbf{1}_{\left\\{\widehat{T}_{\ell}^{x}(k^{\prime})<+\infty\right\\}}\left(A+\eta\widehat{T}^{x}_{\ell}+\eta\,\,\Omega_{k^{\prime}}\right)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{\ell}+\,\Omega_{k^{\prime}}\right)}\right].$
(3.70)

Our aim in the following analysis is to illustrate that

$\displaystyle|I_{1}|\mathbf{1}^{(1)}_{\sf Good}\leq
C(1+\Omega_{k^{\prime}})e^{(\gamma\eta+\frac{\gamma^{2}}{2})\Omega_{k^{\prime}}},\quad|I_{2}-I_{3}|\mathbf{1}^{(1)}_{\sf
Good}\leq
C\Big{(}1+\Omega_{k^{\prime}}\Big{)}e^{(\gamma\eta+\frac{\gamma^{2}}{2})\,\Omega_{k^{\prime}}}.$
(3.71)

To prove the first inequality in (3.71), we begin by recalling the expressions
(3.66), (3.68). When the conditions

$\displaystyle k^{\prime\prime}_{k^{\prime}+1}\neq
k^{\prime\prime}_{k^{\prime}},$ $\displaystyle\mathbf{1}^{(1)}_{\sf
Good}=\mathbf{1}\Big{(}Y_{k^{\prime}}(x)\geq\gamma\,\Omega_{k^{\prime}}\Big{)},$

hold simultaneously, the definitions (3.60), (3.61) imply that

$\displaystyle\Omega_{k^{\prime}+1}=R^{x}_{k^{\prime\prime}},$
$\displaystyle\Omega_{k^{\prime}+1}\geq
T^{x}_{k^{\prime\prime}_{k^{\prime}+1}}.$

Resultantly,

$\displaystyle
T^{x}_{k^{\prime\prime}_{k^{\prime}+1}}\leq\Omega_{k^{\prime}+1}\leq(1+o(1))\,\Omega_{k^{\prime}}.$
(3.72)

Substituting the above bound in the expression for $I_{1}$ (3.68), we obtain
the first inequality in (3.71).

Continuing with the proof of the second inequality, we divide the discussion
into two cases. First, we consider the case when
$\Omega_{k^{\prime}},\Omega_{k^{\prime}+1}$ lie in the same interval,
$\Omega_{k^{\prime}},\Omega_{k^{\prime}+1}\in[T^{x}_{k^{\prime\prime}_{k^{\prime}}},R^{x}_{k^{\prime\prime}_{k^{\prime}}}]$.
In this scenario, from (3.60), (3.61), observe that

$\big{\\{}\widehat{T}^{x}_{\ell}(k^{\prime}),\widehat{R}^{x}_{\ell}(k^{\prime})\big{\\}}=\left\\{\widehat{T}^{x}_{\ell}(k^{\prime}+1)+(\Omega_{k^{\prime}+1}-\Omega_{k^{\prime}}),\quad\widehat{R}^{x}_{\ell}(k^{\prime}+1)+(\Omega_{k^{\prime}+1}-\Omega_{k^{\prime}})\right\\},$

and as a result $I_{2}$, $I_{3}$ in (3.69), (3.70) coincide with each other.
The second case is when $\Omega_{k^{\prime}},\Omega_{k^{\prime}+1}$ belong to
subsequent intervals,

$\Omega_{k^{\prime}}\in\left[T^{x}_{k^{\prime\prime}_{k^{\prime}}},R^{x}_{k^{\prime\prime}_{k^{\prime}}}\right],\qquad\Omega_{k^{\prime}+1}\in\left[R^{x}_{k^{\prime\prime}_{k^{\prime}}},T^{x}_{k^{\prime\prime}_{k^{\prime}}+1}\right].$

In this case, we observe from the expressions for $I_{2}$, $I_{3}$ in (3.69),
$\eqref{def:I3}$ that

$\displaystyle
I_{2}-I_{3}=-\mathbb{E}_{Y_{k^{\prime}}}\left[\mathbf{1}_{\left\\{\widehat{T}_{1}^{x}<+\infty\right\\}}\left(A+\eta\widehat{T}^{x}_{1}+\eta\,\Omega_{k^{\prime}}\right)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{1}+\Omega_{k^{\prime}}\right)}\right].$
(3.73)

We may further rewrite the right hand side of the above equation as

$\displaystyle\text{r.h.s. of
\eqref{eq:I2I3}}=-\mathbb{E}_{Y_{k^{\prime}}}\left[\sum_{\ell=1}^{\infty}\mathbf{1}\big{(}\widehat{T}^{x}_{1}\in[\ell-1,\ell]\big{)}\left(A+\eta\widehat{T}^{x}_{1}+\eta\,\Omega_{k^{\prime}}\right)e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{1}+\Omega_{k^{\prime}}\right)}\right].$

Starting from the expression above, we obtain the following inequality

$\displaystyle|I_{2}-I_{3}|$
$\displaystyle\leq\sum_{\ell=1}^{\infty}\left(A+\eta\ell+\eta\,\Omega_{k^{\prime}}\right)\mathbb{E}\Big{[}\mathbf{1}\big{(}\widehat{T}^{x}_{1}\in[\ell-1,\ell]\big{)}e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widehat{T}^{x}_{1}+\Omega_{k^{\prime}}\right)}\Big{]}$
$\displaystyle\leq\sum_{\ell=1}^{\infty}(A+\eta\ell+\eta\,\Omega_{k^{\prime}})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\ell+\Omega_{k^{\prime}}\right)}\mathbb{E}\Big{[}\\#\big{(}\widehat{T}^{x}_{1}\in[\ell-1,\ell]\big{)}\Big{]}.$
(3.74)

By employing a similar argument as in Lemma 3.1, we arrive at the following
result:

$\mathbb{E}\Big{[}\\#\big{(}\widehat{T}^{x}_{1}\in[\ell-1,\ell]\big{)}\Big{]}\leq\frac{C}{\max\\{\sqrt{\ell-1},1\\}}e^{-\frac{(\gamma+\eta)^{2}}{2}(\ell-1)}.$

Finally, plugging the above inequality into the right hand side (3.74) shows
that

$\displaystyle|I_{2}-I_{3}|$
$\displaystyle\leq\sum_{\ell=1}^{\infty}(A+\eta\ell+\eta\,\Omega_{k^{\prime}})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\ell+\Omega_{k^{\prime}}\right)}\frac{C}{\max\\{\sqrt{\ell-1},1\\}}e^{-\frac{(\gamma+\eta)^{2}}{2}(\ell-1)}$
$\displaystyle\leq C\Big{(}1+\eta\,\Omega_{k^{\prime}}\Big{)}e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\Omega_{k^{\prime}}},$ (3.75)

which implies the second inequality in (3.71). This completes the proof of
Claim 3.1. ∎

######  Claim 3.2.

There exists $C_{2}=C_{2}(A,\gamma,\eta)>0$ such that

$\displaystyle\Big{|}\mathfrak{A}^{x}_{k^{\prime}}\mathbf{1}^{(2)}_{\sf
Good}\Big{|}$ $\displaystyle\leq
C_{2}\Big{(}1+\Omega_{k^{\prime}}\Big{)}e^{(\gamma\eta+\frac{\gamma^{2}}{2})\Omega_{k^{\prime}}}.$

###### Proof of Claim 3.2.

Let us recall the notations of $I_{1},I_{2},I_{3}$ from the proof of Claim
3.1. Similar to the previous result, we demonstrate that

$\displaystyle|I_{1}|\mathbf{1}^{(2)}_{\sf Good}\leq
C(1+\Omega_{k^{\prime}})e^{(\gamma\eta+\frac{\gamma^{2}}{2})\Omega_{k^{\prime}}},\quad|I_{2}-I_{3}|\mathbf{1}^{(2)}_{\sf
Good}\leq
C\Big{(}1+\Omega_{k^{\prime}}\Big{)}e^{(\gamma\eta+\frac{\gamma^{2}}{2})\Omega_{k^{\prime}}}.$
(3.76)

The first inequality can be obtained by noting that the conditions
$k^{\prime\prime}_{k^{\prime}+1}\neq k^{\prime\prime}_{k^{\prime}}$ and
$\mathbf{1}^{(2)}_{\sf Good}=1$, imply that
$\Omega_{k^{\prime}+1}=T^{x}_{k^{\prime\prime}_{k^{\prime}+1}}$. Consequently,
following a similar approach to (3.72), we have

$T^{x}_{k^{\prime\prime}_{k^{\prime}+1}}\leq(1+o(1))\Omega_{k^{\prime}}$

which yields the first inequality in the above display by a direct
substitution of the above inequality in $I_{1}$ (3.68).

To prove the second inequality, we repeat the arguments used in the proof of
Claim 3.1. Once again we consider the following two cases. In the first case,
$\Omega_{k^{\prime}},\Omega_{k^{\prime}+1}\in\left[R^{x}_{k^{\prime\prime}_{k^{\prime}}},T^{x}_{k^{\prime\prime}_{k^{\prime}}+1}\right]$,
and from (3.64), (3.63), we have

$\big{\\{}\widetilde{T}^{x}_{\ell}(k^{\prime}),\widetilde{R}^{x}_{\ell}(k^{\prime})\big{\\}}=\left\\{\widetilde{T}^{x}_{\ell}(k^{\prime}+1)+(\Omega_{k^{\prime}+1}-\Omega_{k^{\prime}}),\widetilde{R}^{x}_{\ell}(k^{\prime}+1)+(\Omega_{k^{\prime}+1}-\Omega_{k^{\prime}})\right\\},$

which shows that $I_{2}-I_{3}=0$. In the second case where

$\Omega_{k^{\prime}}\in\left[R^{x}_{k^{\prime\prime}_{k^{\prime}}},T^{x}_{k^{\prime\prime}_{k^{\prime}}+1}\right],\qquad\textrm{and}\qquad\Omega_{k^{\prime}}\in\left[T^{x}_{k^{\prime\prime}_{k^{\prime}}+1},R^{x}_{k^{\prime\prime}_{k^{\prime}}+1}\right],$

we obtain the expression below from (3.69), (3.70):

$I_{2}-I_{3}=-\mathbb{E}_{Y_{k^{\prime}}}\left[\mathbf{1}_{\left\\{\widetilde{T}_{1}^{x}<+\infty\right\\}}(A+\eta\widetilde{T}^{x}_{1}+\eta\,\Omega_{k^{\prime}})e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\left(\widetilde{T}^{x}_{1}+\Omega_{k^{\prime}}\right)}\right].$

Now, following the similar argument as in (3.75), we arrive at

$|I_{2}-I_{3}|\leq C\Big{(}1+\eta\,\Omega_{k^{\prime}}\Big{)}e^{\gamma
A+(\gamma\eta+\frac{\gamma^{2}}{2})\Omega_{k^{\prime}}}.$

Combining the bounds on $|I_{2}-I_{3}|$ in the above two cases yields the
second inequality in (3.76). This completes the proof of Claim 3.2. ∎

##  4\. Semi-classical limits for deformed conformal blocks

In this section, we carry out the analysis for the semi-classical limit of
deformed conformal blocks (2.3) in two cases, for $\chi=\frac{\gamma}{2}$ and
$\chi=\frac{2}{\gamma}$. It is notable that the structure of the conformal
blocks in each of these cases is different, and in the limit $\gamma\to 0$,
the BPZ equation satisfied by the deformed conformal block reduces to the
following equations:

  * •

For $\chi=\frac{2}{\gamma}$, it reduces to the Hamilton-Jacobi equation of the
non-autonomous elliptic Calogero-Moser model, (see Theorem 4.1).

  * •

For $\chi=\frac{\gamma}{2}$, it reduces to the Lamé equation, (see Theorem
4.2).

As a result of these two reductions, the accessory parameter of the Lamé
equation [BGG22] and the action from the Hamilton-Jacobi equation can be
related by using the semi-classical analogue of the connection (see (2.19))
between the undeformed conformal block and deformed conformal blocks for
different values of $\chi$.

The above results, while interesting on their own, are vital to prove the
existence of the semi-classical limit of the undeformed conformal block, which
we study in Section 5. Furthermore, the relation between the semi-classical
limits of the deformed and undeformed conformal blocks as revealed in Theorem
4.2 and relation (4.56) lets us prove the structure of the conformal block
(1.3) in Theorem 1.1.

Before proceeding to the discussion of main results in this section, we
introduce few notations which will be used throughout the rest of the paper.
In the remainder of the paper when we write $\alpha$ and $P$ we mean
$\frac{\alpha_{0}}{\gamma}$ and $\frac{P_{0}}{\gamma}$. We also define the
DGMC measures (LABEL:def:sfDMgN)

$\displaystyle{\sf DM}_{\gamma,N}\Theta(\tau):={\sf
DM}_{\gamma,N}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi P_{0}x},$
$\displaystyle{\sf DM}_{\gamma,N}\Theta(\mathbf{i}\infty)={\sf
DM}_{\gamma,N}\big{(}\sin(\pi x)\big{)}^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x},$ (4.1)

that arise in the semi-classical limit of the undeformed conformal block
(2.10), and the measure

$\displaystyle{\sf DM}_{\gamma,N}\widetilde{\Theta}(z,\tau;\chi):={\sf
DM}_{\gamma,N}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x},$ (4.2)

will be useful in obtaining the semi-classical limit of the deformed conformal
block (2.3) for $\chi=\frac{2}{\gamma}$. We recall the notation (3.3) from the
previous section:

$\displaystyle{\sf DM}_{\bullet}=\lim_{N\to\infty}{\sf DM}_{\bullet,N}.$

which means ${\sf DM}_{\gamma}\Theta(\tau)$, ${\sf
DM}_{\gamma}\Theta(\mathbf{i}\infty)$ and ${\sf
DM}_{\gamma}\widetilde{\Theta}(z,\tau;\chi)$ are the large $N$ limits of ${\sf
DM}_{\gamma,N}\Theta(\tau)$, ${\sf DM}_{\gamma,N}\Theta(\mathbf{i}\infty)$ and
${\sf DM}_{\gamma,N}\widetilde{\Theta}(z,\tau;\chi)$.

###  4.1. Semi-classical limit of the deformed conformal blocks for
$\chi=\frac{2}{\gamma}$ and the Hamilton-Jacobi equation

In this section, we study the semi-classical limit of the deformed block
$\psi^{\alpha}_{\chi}(z,q)$ when the deformation parameter $\chi$ is equal to
$\frac{2}{\gamma}$. We show that in this case, the semi-classical limit is
given by the solution of a Hamilton-Jacobi equation with the potential given
by the Weierstrass $\wp$-function as described in physics literature [Lit+14,
BGG22]. Our proof heavily relies on the BPZ equation satisfied by the
$\psi^{\alpha}_{\chi}(z,q)$ and the tail probabilities of the derivative GMC
which we found in Section 3. In what follows, we first state the main result
(Theorem 4.1) of this subsection. Before proceeding to the proof, we state and
prove two main tools in Proposition 4.1 and Proposition 4.2. These two
propositions claim and prove the boundedness and the tightness of the sequence
of the deformed conformal block $\psi^{\alpha}_{2/\gamma}(z,q)$ as $\gamma\to
0$. The proof of Theorem 4.1 will be detailed in Section 4.1.1.

######  Theorem 4.1.

For $\chi=\frac{2}{\gamma}$, $\alpha_{0}\in(-2,2)$, $P_{0}\in\mathbb{R}$,
$q\in(0,1)$, the semi-classical limit of deformed conformal blocks exists and
assumes the form

$\displaystyle\lim_{\gamma\rightarrow
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)=\widetilde{\phi}(z,q),$
(4.3)

where

$\displaystyle\widetilde{\phi}(z,q)$
$\displaystyle:=\frac{-\alpha_{0}^{2}+\alpha_{0}+6P_{0}^{2}+2}{12}\log(-2\pi)+\frac{-\alpha_{0}^{2}+\alpha_{0}+6P_{0}^{2}+2}{12}\log
q+2P_{0}z$
$\displaystyle-(\alpha_{0}-2)\left(\log\int^{1}_{0}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\theta(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx\right)$ (4.4) $\displaystyle+\lim_{\gamma\to
0}\gamma^{2}\log\mathbb{E}\left[\exp\left(-\frac{\alpha_{0}-2}{\gamma}{\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}\left(z,\tau;\frac{2}{\gamma}\right)\right)\right].$
(4.5)

Furthermore, $\widetilde{\phi}(z,q)$ satisfies the following the partial
differential equation which arises as the semi-classical limit of the BPZ
equation (2.20):

$\displaystyle(\partial_{z}\widetilde{\phi}(z,\tau))^{2}-\left({2-\alpha_{0}}\right)^{2}\wp(z)+2\pi\mathbf{i}\partial_{\tau}\widetilde{\phi}(z,\tau)=0,$
(4.6)

which is the Hamilton-Jacobi equation for the non-autonomous Calogero Moser
model (see below) with $\widetilde{\phi}(z,\tau)$ describing the action.

######  Remark 4.1.

The non-autonomous elliptic Calogero-Moser model is defined by the equations
of motion [Man98]

$\displaystyle v(\tau)=2\pi\mathbf{i}\frac{du(\tau)}{d\tau},$ $\displaystyle
2\pi\mathbf{i}\frac{dv(\tau)}{d\tau}=m^{2}\wp^{\prime}(2u(\tau)),$ (4.7)

of the Hamiltonian

$\displaystyle H(\tau)=v(\tau)^{2}-m^{2}\wp(2u(\tau)|\tau).$ (4.8)

The solution $u(\tau)$ depends on $\tau$, $m$, and the initial conditions or
equivalently, on the monodromy data through the Riemann-Hilbert correspondence
[Bon+20, DDG23]. The equation (4.6) is then the Hamilton-Jacobi equation of
the system above with the identification

$\displaystyle z=2u(\tau),$ $\displaystyle
m^{2}=\left(2-\alpha_{0}\right)^{2}.$ (4.9)

Given initial condition at $\tau\to\mathbf{i}\infty$ the solution of the
Hamilton-Jacobi equation is unique and coincides with the classical action
[Arn13] corresponding to the Hamiltonian above (see [BGG22], [Bus80, p. 258])

$\displaystyle\widetilde{\phi}(2u(\tau),\tau)=\int_{\mathbf{i}\infty}^{\tau}\left(v(\tau)^{2}+m^{2}\wp(2u(\tau)|\tau)\right)\frac{d\tau}{2\pi\mathbf{i}}.$
(4.10)

######  Proposition 4.1.

For $\chi=\frac{2}{\gamma}$, $q\in(0,1)$ and with the parameters
$\alpha_{0}\in(-2,2)$, $P_{0}\in\mathbb{R}$, the sequence of the semi-
classical limit of the deformed conformal block (2.16) is bounded

$\displaystyle\limsup_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{\chi}(z;q)<\infty,\quad\liminf_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{\chi}(z;q)>-\infty.$ (4.11)

###### Proof.

Recall the expression of the deformed conformal block (4.39) reads:

$\psi^{\alpha_{0}/\gamma}_{\chi}(z,\tau)=\mathcal{W}(q)e^{\chi
P_{0}z\pi/\gamma}\mathbb{E}\left[\mathcal{V}_{\chi,P}^{\alpha_{0}/\gamma}(z,q)^{-\frac{\alpha_{0}}{\gamma^{2}}+\frac{\chi}{\gamma}}\right],$
(4.12)

where

$\mathcal{V}^{\alpha}_{\chi,P}(z,q):=\int^{1}_{0}\theta_{1}(z+x)^{\frac{\gamma\chi}{2}}\theta_{1}(z)^{-\frac{\gamma\chi}{2}}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}e^{\frac{\gamma}{2}Y(x;q)}dx.$

We analyse each term of the above expression. The parameter $l_{\chi}$ for
$\chi=2/\gamma$ simplifies as

$\displaystyle
l_{\chi}=\frac{\chi^{2}}{2}-\frac{\alpha\chi}{2}=\frac{(2-\alpha_{0})}{\gamma^{2}}.$
(4.13)

The pre-factor of (4.12) defined in (2.17) then assumes the form

$\displaystyle\mathcal{W}(q)e^{\chi Pz\pi}$
$\displaystyle=(-2\pi)^{\frac{1}{24}\left(2-\alpha_{0}\right)+\frac{-\alpha_{0}^{2}+\alpha_{0}+6P_{0}^{2}+2}{12\gamma^{2}}}q^{\frac{1}{24}\left(2-\alpha_{0}\right)+\frac{-\alpha_{0}^{2}+\alpha_{0}+6P_{0}^{2}+2}{12\gamma^{2}}}e^{\frac{2P_{0}z\pi}{\gamma^{2}}}+\mathcal{O}(q^{\gamma^{2}}).$
(4.14)

Note that the above expression consists of terms that are dependent on both
$z$ and $\gamma$.

We now consider the term $\mathcal{V}^{\alpha}_{\chi,P}(z,q)$ and focus on
obtaining uniform bounds of the expression
$\gamma^{2}\log\mathbb{E}\left[\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{(2-\alpha_{0})/\gamma^{2}}\right]$
as functions of $z$ and $q$ as they vary on a compact set. To this end, we
note that

$\displaystyle\mathbb{E}\left[\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{\frac{(2-\alpha_{0})}{\gamma^{2}}}\right]$
$\displaystyle=\left(\int^{1}_{0}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\theta(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx\right)^{-\left(\frac{\alpha_{0}-2}{\gamma^{2}}\right)}$
$\displaystyle\times$
$\displaystyle\mathbb{E}\left[\exp\left(-\frac{\alpha_{0}-2}{\gamma^{2}}\left(\log\mathcal{V}^{\alpha}_{\chi,P}(z,q)-\log\int^{1}_{0}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\theta(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx\right)\right)\right].$ (4.15)

The last term in the expression above can be simplified as

$\displaystyle\mathbb{E}$
$\displaystyle\left[\exp\left(-\frac{\alpha_{0}-2}{\gamma^{2}}\left(\log\mathcal{V}^{\alpha}_{\chi,P}(z,q)-\log\int^{1}_{0}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}\right)\right)\right]$
$\displaystyle\mathop{=}^{\eqref{def:DGMCtilTheta}}\mathbb{E}\left[\exp\left(-\frac{\alpha_{0}-2}{\gamma^{2}}\int^{\gamma}_{0}{\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}\big{(}z,\tau;\frac{2}{\gamma}\big{)}d\gamma^{\prime}\right)\right].$
(4.17)

By the Jensen’s inequality, the right hand side of the above display is
bounded by

$\displaystyle\frac{1}{\gamma}\int^{\gamma}_{0}\mathbb{E}\left[\exp\left(-\frac{\alpha_{0}-2}{\gamma}{\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}\big{(}z,\tau;\frac{2}{\gamma^{\prime}}\big{)}\right)\right]d\gamma^{\prime}.$
(4.18)

We first now investigate the growth of the above expression when
$\alpha_{0}\geq 2$ and then, show the growth in the case when $\alpha_{0}<2$.
The first case is for theoretical interests since the actual range of
$\alpha_{0}$ for any fixed $\gamma>0$ is in between
$(-2,2+\frac{\gamma^{2}}{2})$. The boundedness of the above expression follows
from the property of $z$-deformed DGMC provided by Lemma 4.1:

$\displaystyle\mathbb{P}\left({\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}\big{(}z,\tau;\frac{2}{\gamma}\big{)}\leq-v\right)\leq\widetilde{c}_{1}e^{-\widetilde{c}_{2}v^{2}}.$
(4.19)

for any $v>0$. We then obtain the inequality

$\displaystyle\mathbb{E}\big{[}e^{-\frac{\alpha_{0}-2}{\gamma}{\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}(z,\tau;\frac{2}{\gamma})}\big{]}$
$\displaystyle\leq
1+\int^{\infty}_{0}e^{\frac{(\alpha_{0}-2)}{\gamma}v}\mathbb{P}\left({\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}(z,\tau;\frac{2}{\gamma})\leq-v\right)dv$
$\displaystyle\leq
1+\widetilde{c}_{1}e^{\frac{(\alpha_{0}-2)^{2}}{2\widetilde{c}_{2}\gamma^{2}}}$

for any $\alpha_{0}\geq 2$. Substituting this above bound into (4.18) shows
that $\limsup_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,\tau)<\infty$ when
$\alpha_{0}\geq 2$.

Now we proceed to prove the same claim when $\alpha_{0}<2$. For any positive
integer $N$, we can write
$\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{N}\big{]}$ as a Selberg-
type integral

$\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{N}\big{]}=\int_{[0,1]^{N}}\ldots\int^{1}_{0}\prod_{i<j}^{N}|\theta_{1}(x_{i}-x_{j})|^{-\frac{\gamma^{2}}{2}}\prod_{i=1}^{N}\theta_{1}(z+x_{i})^{\frac{\gamma\chi}{2}}\theta_{1}(z)^{-\frac{\gamma\chi}{2}}\theta_{1}(x_{i})^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x_{i}}dx_{i}.$

Notice that $\theta(z)=\mathfrak{p}(z;\tau)\sin(\pi z)$ where
$c_{1}\leq\mathfrak{p}(z;\tau)\leq c_{2}$ as $z$ varies in $(0,1)$. As a
result, we can prove that the right hand side of the above integral is bounded
by above and below by some constant multiple of

$\int_{[0,1]^{N}}\prod_{i<j}^{N}|\sin(\pi(x_{i}-x_{j}))|^{-\frac{\gamma^{2}}{2}}\prod_{i=1}^{N}\sin(x_{i})^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x_{i}}dx_{i}.$

The above integral has an explicit expression as stated in (2.12) of Remark
2.1. As $N$ grows, the ratios of gamma function in (2.12) grows as
$\exp(cN^{2}\gamma^{2})$. When $\alpha_{0}<2$,
$\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{(2-\alpha_{0})/\gamma^{2}}\big{]}$
can be bounded by
$\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{N}\big{]}$ for some
integer $N=\lfloor(2-\alpha_{0})/\gamma^{2}\rfloor$. Now the large $N$
asymptotics of the above integral (obtained through the asymptotics of the
ratios of gamma function in (2.12)) shows that
$\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{(2-\alpha_{0})/\gamma^{2}}\big{]}$
is bounded by $\exp(c/\gamma^{2})$ for some constant $c=c(\alpha_{0})>0$ when
$\alpha_{0}<2$. As a result, we get $\limsup_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,\tau)<\infty$ as
$\gamma\to 0$.

It remains now to show that $\liminf_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{\chi}(z,\tau)>-\infty$. For any
$\alpha_{0}\in(-2,2)$, notice that $h(x)=x^{(2-\alpha_{0})/\gamma^{2}}$ is a
convex function of $x$ for all small $\gamma$. Hence, by Jensen’s inequality,
we have

$\displaystyle\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{(2-\alpha_{0})/\gamma^{2}}\big{]}\geq\big{(}\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)\Big{]}\big{)}^{(2-\alpha_{0})/\gamma^{2}}.$
(4.20)

With the explicit form of $\mathcal{V}^{\alpha}_{\chi,P}(z,q)$,

$\big{(}\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)\big{]}\big{)}^{(2-\alpha_{0})/\gamma^{2}}=\Big{(}\int^{1}_{0}\theta_{1}(z+x)\theta_{1}(z)^{-1}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}\Big{)}^{\frac{2-\alpha_{0}}{\gamma^{2}}},$

we see that $\liminf_{\gamma\to
0}\gamma^{2}\log\mathbb{E}\big{[}\mathcal{V}^{\alpha}_{\chi,P}(z,q)^{(2-\alpha_{0})/\gamma^{2}}\big{]}>-\infty$
due to the above inequality. Therefore, along with the expression (4.14), we
have $\liminf_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{\chi}(z,\tau)>-\infty$ for the
deformed conformal block (4.12). This completes the proof of this proposition.

∎

######  Proposition 4.2.

As $\gamma\downarrow 0$,
$\\{\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)\\}_{\gamma>0}$ is a
tight sequence of functions in $C^{2}([-M,M]\times[0,q_{0}])$ for any $M>0$
and $q_{0}\in(0,1)$. The parameters $\alpha_{0}\in(-2,2)$,
$P_{0}\in\mathbb{R}$.

###### Proof.

Let us start by rewriting the deformed conformal block (2.16) as (see also eq.
(3.5) in [Gho+24])

$\displaystyle\psi_{\chi}^{\alpha}(z,q)=q^{\left(\frac{P^{2}}{2}+\frac{1}{6\chi^{2}}l_{\chi}(l_{\chi}+1)\right)}\widehat{\psi}_{\chi}^{\alpha}(z,q).$
(4.21)

To prove this proposition, it is then sufficient to show that the limits
$\gamma\to 0$ and $q\to 0$ commute for $\widehat{\psi}_{\chi}^{\alpha}(z,q)$.
The key to proving the commutativity is to express the function
$\widehat{\psi}_{\chi}^{\alpha}(z,q)$ such that the integrand inside the
expectation is $q$ independent. Such a structure can be obtained via
Girsanov’s theorem (see [Gho+24, Theorem C.5]) that lets us re-express the
conformal block in terms of the $q$-independent measure
$e^{\frac{\gamma}{2}Y(x)}$, resulting in the following expression for the
function $\widehat{\psi}_{\chi}^{\alpha}(z,q)$ in the equation above:

$\displaystyle\widehat{\psi}_{\chi}^{\alpha}(z,q)=C(q)e^{\chi
Pz\pi}\theta_{1}(z)^{-l_{\chi}}C_{1}(q)\left(-\mathbf{i}e^{-\mathbf{i}\pi
z}q^{1/6}\eta(q)\right)^{l_{\chi}}$
$\displaystyle\times\mathbb{E}\left[e^{\frac{\alpha}{2}F_{\tau}(0)}\mathcal{Q}(q)e^{\chi\mathcal{Y}(z,q)}e^{\chi\mathcal{X}(z,q)-\frac{\chi^{2}}{2}\mathbb{E}\left[\mathcal{X}(z,q)^{2}\right]}\left(\int_{0}^{1}dxe^{\frac{\gamma}{2}Y(x)}e^{\pi\gamma
Px}\left(2\sin(\pi
x)\right)^{-\frac{\alpha\gamma}{2}}\right)^{-\frac{\alpha}{\gamma}+\frac{\chi}{\gamma}}\right],$
(4.22)

where

$\displaystyle\begin{split}C(q)&:=q^{\frac{\gamma
l_{\chi}}{12\chi}-\frac{1}{6}\frac{l_{\chi}^{2}}{\chi^{2}}-\frac{1}{6\chi^{2}}l_{\chi}(l_{\chi}+1)}\theta_{1}^{\prime}(0)^{-\frac{2l_{\chi}^{2}}{3\chi^{2}}+\frac{l_{\chi}}{3}+\frac{4l_{\chi}}{3\gamma\chi}}e^{-\frac{1}{2}\mathbf{i}\pi\alpha\gamma\left(-\frac{\alpha}{\gamma}+\frac{\chi}{\gamma}\right)},\\\
C_{1}(q)&:=\left(q^{1/6}\eta(q)\right)^{\alpha(\alpha-\chi)/2}e^{\left(\frac{\alpha\gamma}{8}-\frac{\gamma\chi}{8}-\frac{\alpha^{2}}{8}\right)\mathbb{E}\left[F_{\tau}(0)^{2}\right]},\\\
\mathcal{Q}(q)&:=\exp\left[\sqrt{2}\sum_{m,n=1}^{\infty}q^{nm}\left(\alpha_{n,m}\alpha_{n}+\beta_{n,m}\beta_{n}\right)-\sum_{n=1}^{\infty}\left(\sum_{m=1}^{\infty}q^{nm}\alpha_{m,n}\right)^{2}-\sum_{n=1}^{\infty}\left(\sum_{m=1}^{\infty}q^{nm}\beta_{m,n}\right)^{2}\right],\\\
F_{\tau}(0)&:=2\sum_{n,m\geq 1}\frac{q^{nm}}{\sqrt{n}}\alpha_{n,m}\\\
\mathcal{X}(z,q)&:=-\sqrt{2}\sum_{n,m=1}^{\infty}\frac{q^{(2m-1)n}}{\sqrt{n}}\left(\cos\left(2\pi(z-\frac{\tau}{2})n\right)\alpha_{n}-\sin\left(2\pi(z-\frac{\tau}{2})n\right)\beta_{n}\right),\\\
\mathcal{Y}(z,q)&:=2\sum_{m,n,k\geq
1}\alpha_{n,m}\frac{q^{(2k-1+m)n}}{\sqrt{n}}\cos\left(2\pi(z-\frac{\tau}{2})n\right)-2\sum_{m,n,k\geq
1}\beta_{n,m}\frac{q^{(2k-1+m)n}}{\sqrt{n}}\sin\left(2\pi(z-\frac{\tau}{2})n\right),\end{split}$
(4.24)

and for convenience we define the $\gamma$-independent entity

$\displaystyle\widetilde{\mathcal{X}}(z,q):=\frac{\alpha_{0}}{2}F_{\tau}(0)+2(\mathcal{X}(z,q)+\mathcal{Y}(z,q)).$
(4.25)

With all the above definitions, the equation (LABEL:eq:hatpsigir) simplifies
as

$\displaystyle\widehat{\psi}_{\chi}^{\alpha}(z,q)=C(q)e^{\chi
Pz\pi}\theta_{1}(z)^{-l_{\chi}}C_{1}(q)\left(-\mathbf{i}e^{-\mathbf{i}\pi
z}q^{1/6}\eta(q)\right)^{\frac{\chi}{2}(\chi-\alpha)}e^{-\frac{\chi^{2}}{2}\mathbb{E}\left[\mathcal{X}(z,q)^{2}\right]}$
$\displaystyle\times\mathbb{E}\left[e^{\frac{1}{\gamma}\widetilde{\mathcal{X}}(z,q)}\mathcal{Q}(q)\left(\int_{0}^{1}dxe^{\frac{\gamma}{2}Y(x)}e^{\pi\gamma
Px}\left(2\sin(\pi
x)\right)^{-\frac{\alpha\gamma}{2}}\right)^{-\frac{\alpha}{\gamma}+\frac{\chi}{\gamma}}\right].$
(4.26)

Restricting to $\chi=\frac{2}{\gamma}$, we observe that the term

$C(q)e^{\chi
Pz\pi}\theta_{1}(z)^{-l_{\chi}}C_{1}(q)\left(-\mathbf{i}e^{-\mathbf{i}\pi
z}q^{1/6}\eta(q)\right)^{\frac{\chi}{2}(\chi-\alpha)}$

is independent of $q$. Therefore, the first term in (4.1) has the following
limit

$\displaystyle\lim_{q\to 0}\lim_{\gamma\to 0}\gamma^{2}\log\left[C(q)e^{\chi
Pz\pi}\theta_{1}(z)^{-l_{\chi}}C_{1}(q)\left(-\mathbf{i}e^{-\mathbf{i}\pi
z}q^{1/6}\eta(q)\right)^{\frac{\chi}{2}(\chi-\alpha)}e^{-\frac{\chi^{2}}{2}\mathbb{E}\left[\mathcal{X}(z,q)^{2}\right]}\right]$
$\displaystyle=-2\lim_{q\to 0}\mathbb{E}\left[\mathcal{X}(z,q)^{2}\right].$
(4.27)

As for the second term of (4.1), the facts that $\lim_{q\to
0}\mathcal{Q}(q)=1$, and the term containing the integral is $q$-independent
allow us to factorize the terms inside the expectation as follows:

$\displaystyle\lim_{q\to 0}\lim_{\gamma\to
0}\gamma^{2}\log\mathbb{E}\left[e^{\frac{1}{\gamma}\widetilde{\mathcal{X}}(z,q)}\mathcal{Q}(q)\left(\int_{0}^{1}dxe^{\frac{\gamma}{2}Y(x)}e^{\pi\gamma
Px}\left(2\sin(\pi
x)\right)^{-\frac{\alpha\gamma}{2}}\right)^{-\frac{\alpha}{\gamma}+\frac{\chi}{\gamma}}\right]$
$\displaystyle=\lim_{q\to 0}\lim_{\gamma\to
0}\gamma^{2}\log\mathbb{E}\left[e^{\frac{1}{\gamma}\widetilde{\mathcal{X}}(z,q)}\right]$
(4.28) $\displaystyle+\lim_{\gamma\to
0}\gamma^{2}\log\mathbb{E}\left[\left(\int_{0}^{1}dxe^{\frac{\gamma}{2}Y(x)}e^{\pi\gamma
Px}\left(2\sin(\pi
x)\right)^{-\frac{\alpha\gamma}{2}}\right)^{-\frac{\alpha}{\gamma}+\frac{\chi}{\gamma}}\right].$
(4.29)

The second term in the expression above is a constant

$\displaystyle C_{2}(\alpha_{0},P_{0}):=\lim_{\gamma\to
0}\gamma^{2}\log\mathbb{E}\left[\left(\int_{0}^{1}dxe^{\frac{\gamma}{2}Y(x)}e^{\pi\gamma
Px}\left(2\sin(\pi
x)\right)^{-\frac{\alpha\gamma}{2}}\right)^{-\frac{\alpha}{\gamma}+\frac{\chi}{\gamma}}\right],$

while the first term of the expression (4.1) simplifies as follows due to the
identity
$\mathbb{E}[e^{\frac{1}{\gamma}\widetilde{\mathcal{X}}(z,q)}]=e^{\frac{1}{2\gamma^{2}}\mathbb{E}[(\widetilde{\mathcal{X}}(z,q))^{2}]}$:

$\displaystyle\lim_{q\to 0}\lim_{\gamma\to
0}\gamma^{2}\log\mathbb{E}[e^{\frac{1}{\gamma}\widetilde{\mathcal{X}}(z,q)}]=\lim_{q\to
0}\frac{1}{2}\mathbb{E}[(\widetilde{\mathcal{X}}(z,q))^{2}].$ (4.30)

With (4.1), (4.1), (4.30), the limit of the regularized $z$-deformed conformal
block (4.1) is therefore

$\displaystyle\lim_{q\to 0}\lim_{\gamma\to
0}\gamma^{2}\log\widehat{\psi}_{2/\gamma}^{\alpha_{0}/\gamma}$
$\displaystyle=-2\lim_{q\to
0}\mathbb{E}\left[\mathcal{X}(z,q)^{2}\right]+\lim_{q\to
0}\frac{1}{2}\mathbb{E}[(\widetilde{\mathcal{X}}(z,q))^{2}]+C_{2}(\alpha_{0},P_{0})$
$\displaystyle\mathop{=}^{\eqref{def:tilX}}\lim_{q\to
0}\frac{1}{2}\mathbb{E}\left[\frac{\alpha_{0}^{2}}{4}F_{\tau}(0)^{2}+2\left(2\mathcal{Y}+\alpha_{0}F_{\tau}(0)\right)\left(\mathcal{X}+\mathcal{Y}\right)+4\mathcal{X}\mathcal{Y}\right]+C_{2}(\alpha_{0},P_{0}),$

which is an analytic function in $q$ in some open ball $|q|<r_{0}$ for all
$z\in(0,1)$. This shows that the above convergence is uniform on the ball
$|q|<r_{0}-\epsilon$ for any $\epsilon>0$. The limits $\gamma\to 0$, $q\to 0$
are therefore commutative, proving this proposition. ∎

The following lemmas which are simple corollary of the results from Section 3
played important role in the proofs of Proposition 4.1 and Proposition 4.2.

######  Lemma 4.1.

There exist $\widetilde{c}_{1}=\widetilde{c}_{1}(z,\tau)$,
$\widetilde{c}_{2}=\widetilde{c}_{2}(\tau)$ such that for all $v>0$, the DGMC
measure (4.2) satisfies the following inequality

$\displaystyle\mathbb{P}\left[{\sf
DM}_{\gamma,N}\widetilde{\Theta}(z,\tau;\chi)\leq-v\right]\leq\widetilde{c}_{1}e^{-\widetilde{c}_{2}v^{2}}.$
(4.31)

###### Proof.

Note that (4.2) is (LABEL:def:sfDMgN) with
$f(z,x,\tau)=\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi P_{0}x}$, and
$f(.,z,\tau)\in L_{1}([0,1])$. Moreover, from the proof of Theorem 3.1, we can
observe that the exponent in (3.4) arises from the explicit form of $Y(x;q)$
whereas the proportionality constant is given by the normalization factor
$\int_{0}^{1}f(z,x,\tau)$, resulting in (4.31). ∎

######  Lemma 4.2.

The semi-classical limits of the following $z$ and $\tau$ derivatives of the
conformal block are bounded

$\displaystyle\limsup_{\gamma\to
0}\gamma^{2}\partial_{z}\log\big{(}\psi^{\alpha/\gamma}_{\chi}(z,q)\big{)}<\infty,$
$\displaystyle\quad\liminf_{\gamma\to
0}\gamma^{2}\partial_{z}\log\big{(}\psi^{\alpha/\gamma}_{\chi}(z,q)\big{)}>-\infty$
(4.32) $\displaystyle\limsup_{\gamma\to
0}\gamma^{2}\partial_{\tau}\log\big{(}\psi^{\alpha/\gamma}_{\chi}(z,q)\big{)}<\infty,$
$\displaystyle\quad\liminf_{\gamma\to
0}\gamma^{2}\partial_{\tau}\log\big{(}\psi^{\alpha/\gamma}_{\chi}(z,q)\big{)}>-\infty$
(4.33)

###### Proof.

Using the form of the conformal block (4.21),(LABEL:eq:hatpsigir) where the
$z$ and $\tau$ dependence is separated from the GMC measure, we can prove the
boundedness of the $z$ and $\tau$ derivatives
$\partial_{z}\log\big{(}\psi^{\alpha/\gamma}_{\chi}(z,q)\big{)}$,
$\partial_{\tau}\log\big{(}\psi^{\alpha/\gamma}_{\chi}(z,q)\big{)}$ in the
following way. Repeating the arguments of the proof of Proposition 4.1, we
observe that the expressions for the $z$, $\tau$ derivative above is
determined by the DGMC measure (LABEL:def:sfDMgN) with appropriate test
functions. The proof of this lemma then follows from the tail of the measure
given by the second expression in (3.4) with suitable constant functions.

∎

####  4.1.1. Proof of Theorem 4.1

The explicit form of the semi-classical conformal block can be obtained in the
following manner. We begin with the following term in the conformal block
(4.12):

$\displaystyle\mathbb{E}\left[\mathcal{V}^{\alpha}_{\gamma,P}(z,q)^{-\frac{(2-\alpha_{0})}{\gamma^{2}}}\right]$
$\displaystyle\mathop{=}^{\eqref{asymp:term2},\eqref{asymp:term3}}\left(\int^{1}_{0}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\theta(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx\right)^{-\left(\frac{\alpha_{0}-2}{\gamma^{2}}\right)}$
$\displaystyle\mathbb{E}\left[\exp\left(-\frac{\alpha_{0}-2}{\gamma}{\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}(z,\tau;\frac{2}{\gamma})\right)\right].$

Along with the expression (4.14), (4.12) reads

$\displaystyle\psi^{\alpha}_{2/\gamma}(z,\tau)$
$\displaystyle=(-2\pi)^{\frac{1}{24}\left(2-\alpha_{0}\right)+\frac{-\alpha_{0}^{2}+\alpha_{0}+6P_{0}^{2}+2}{12\gamma^{2}}}q^{\frac{1}{24}\left(2-\alpha_{0}\right)+\frac{-\alpha_{0}^{2}+\alpha_{0}+6P_{0}^{2}+2}{12\gamma^{2}}}e^{\frac{2P_{0}z\pi}{\gamma^{2}}}$
$\displaystyle\times\left(\int^{1}_{0}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\theta(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx\right)^{-\left(\frac{\alpha_{0}-2}{\gamma^{2}}\right)}\mathbb{E}\left[\exp\left(-\frac{\alpha_{0}-2}{\gamma}{\sf
DM}_{\gamma^{\prime}}\widetilde{\Theta}(z,\tau;\frac{2}{\gamma})\right)\right]+\mathcal{O}(\gamma^{2}).$

Proposition 4.2 implies that
$\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)$, is a tight sequence
of functions in $C^{2}([-M,M]\times[0,q_{0}])$ for any $M>0$ and
$q_{0}\in(0,1)$. As a result, any subsequence of
$\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)$ has a further
converging subsequence. Now we show that all subsequential limits of
$\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)$ are same. To this
end, let us fix a subsequential limit

$\displaystyle\lim_{\gamma\rightarrow
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)=\widetilde{\phi}(z,q).$
(4.34)

Furthermore, due to Lemma 4.2,
$\gamma^{2}\partial_{z}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)$,
$\gamma^{2}\partial_{\tau}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)$ are
also tight sequences of functions in $C^{2}([-M,M]\times[0,q_{0}])$ for any
$M>0$ and $q_{0}\in(0,1)$. Consequently, with (4.34), the BPZ equation (2.20)
in the limit $\gamma\to 0$ reads

$\displaystyle(\partial_{z}\widetilde{\phi}(z,q))^{2}-\left(2-\alpha_{0}\right)^{2}\wp(z)+2\pi\mathbf{i}\partial_{\tau}\widetilde{\phi}(z,q)=0.$

Note that the above equation is the Hamilton-Jacobi equation corresponding to
the non-autonomous elliptic Calogero-Moser model, see Remark 4.1.

From all of the above, any sub-sequential limit of
$\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z,q)$ will solve (4.6).
Note that the commutativity of limits $\gamma\to 0$ and $q\to 0$ uniquely
determines the initial condition of Hamilton-Jacobi equation (4.6) through
Proposition 6.3. Then, by the uniqueness of the solution of the Hamilton-
Jacobi equation (see Remark 4.1) and the fact that all sub-sequential limits
of $\widetilde{\phi}(z,q)$ satisfy (4.6), we obtain that all the sub-
sequential limits of $\widetilde{\phi}(z,q)$ are same. As a result,
$\lim_{\gamma\to 0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{2/\gamma}(z;q)$
exists and coincides with (4.1).

###  4.2. Semi-classical limit of the deformed conformal blocks for
$\chi=\frac{\gamma}{2}$ and the Lamé equation

As in the previous section, we study the semi-classical limit of the deformed
conformal block for the deformation parameter $\chi=\frac{\gamma}{2}$. We show
that the semi-classical limit of the deformed conformal block solves Lamé
equation as noted in [Pi̧a14]. Furthermore, we derive relation between the
accessory parameter of the Lamé equation and the Hamiltonian mentioned in
(4.8) (see also [BGG22]). We first state this semi-classical limit in the
following result (Theorem 4.2). We prove this result after stating and proving
Proposition 4.3 which shows the boundedness of the sequence of deformed
conformal block $\psi^{\alpha}_{\gamma/2}(z,q)$ as $\gamma\to 0$.

######  Theorem 4.2.

Consider $\chi=\frac{\gamma}{2}$, $q\in(0,1)$, and the constants
$\alpha_{0}\in(-4,2)$, $P_{0}\in\mathbb{R}$. Then, the following limit exists

$\displaystyle\lim_{\gamma\rightarrow
0}\psi^{\alpha_{0}/\gamma}_{\gamma/2}(z,q)e^{-\phi(q)/\gamma^{2}}=:\widetilde{\Gamma}(z;\alpha_{0},P_{0},q),$

where $\phi(q)$ is given by

$\displaystyle\phi(q)$
$\displaystyle:=-\frac{\alpha_{0}\left(\alpha_{0}+4\right)}{6}\log(-2\pi)+\frac{6P_{0}^{2}-\alpha_{0}\left(\alpha_{0}+2\right)}{12}\log
q-\alpha_{0}\left(\log\int^{1}_{0}\theta(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx\right)$ $\displaystyle+\lim_{\gamma\to
0}\gamma^{2}\log\mathbb{E}\left[\exp\left(-\frac{\alpha_{0}-2\gamma^{2}}{\gamma}{\sf
DM}_{\gamma^{\prime}}\Theta(\tau)\right)\right].$ (4.35)

Moreover, $\phi(q)$ has the following properties:

  1. (1)

$\partial_{\tau}\phi(q)$ is assuming the role of the accessory parameter of
the Lamé equation

$\displaystyle\left(\partial_{z}^{2}-\frac{\alpha_{0}}{4}\left(\frac{\alpha_{0}}{4}-1\right)\wp(z)+\frac{\pi\mathbf{i}}{2}\partial_{\tau}{\phi}(q)\right)\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)=0.$
(4.36)

where the solution of the above equation
$\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)$ is given by

$\displaystyle\widetilde{\Gamma}(z;\alpha_{0},P_{0},q)=e^{\frac{P_{0}z\pi}{2}}\exp\left(-\frac{\alpha_{0}\int^{1}_{0}\log\left(\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\right)\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx}{4\int^{1}_{0}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}dx}\right).$ (4.37)

  2. (2)

Furthermore, $\partial_{\tau}\phi(q)$ is also described by the Hamiltonian in
(4.8) as:

$\displaystyle
2\pi\mathbf{i}\partial_{\tau_{\star}}\phi(q_{*})=H(\tau_{\star}),$
$\displaystyle q_{*}=e^{\mathbf{i}\pi\tau_{*}}$ (4.38)

for all $\tau_{*}$ such that $u(\tau_{*})=0$ (see Remark 4.1 for the
definition of $u(\tau)$).

Let us now detail the propositions needed to prove the above theorem.

######  Proposition 4.3.

For $\chi=\frac{\gamma}{2}$, $q\in(0,1)$, and the constants
$\alpha_{0}\in(-4,2)$, $P_{0}\in\mathbb{R}$, the sequence of the deformed
conformal blocks (2.16) is bounded

$\displaystyle\limsup_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{\gamma/2}(z,q)<\infty,\quad\liminf_{\gamma\to
0}\gamma^{2}\log\psi^{\alpha_{0}/\gamma}_{\gamma/2}(z,q)>-\infty.$

###### Proof.

We begin by recalling the expression for the deformed conformal block (2.16)
with the scaling $\alpha=\alpha_{0}/\gamma$, $P=P_{0}/\gamma$:

$\psi^{\alpha_{0}/\gamma}_{\chi}(z,\tau)=\mathcal{W}(q)e^{\chi\frac{P_{0}}{\gamma}z\pi}\mathbb{E}\left[\mathcal{V}_{\chi,P_{0}/\gamma}^{\alpha_{0}/\gamma}(z,q)^{-\frac{\alpha_{0}}{\gamma^{2}}+\frac{\chi}{\gamma}}\right],$
(4.39)

where

$\displaystyle\mathcal{V}^{\alpha_{0}/\gamma}_{\chi,P_{0}/\gamma}(z,q):=\int^{1}_{0}\theta_{1}(z+x)^{\frac{\gamma\chi}{2}}\theta_{1}(z)^{-\frac{\gamma\chi}{2}}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}e^{\frac{\gamma}{2}Y(x;q)}dx.$ (4.40)

Let us examine each of the terms above separately, beginning with the
parameter $l_{\chi}$ for $\chi=\gamma/2$:

$\displaystyle
l_{\chi}=\frac{\chi^{2}}{2}-\frac{\alpha\chi}{2}=\frac{\gamma^{2}}{8}-\frac{\alpha_{0}}{4}.$

The pre-factor in (4.39) that is defined by (2.17), simplifies as follows in
the limit $\gamma\to 0$:

$\displaystyle\mathcal{W}(q)e^{\chi\frac{P_{0}}{\gamma}z\pi}$
$\displaystyle\mathop{=}^{\eqref{def:Wq}}q^{\frac{P_{0}^{2}}{2\gamma^{2}}+\frac{\gamma
l_{\chi}}{12\chi}-\frac{1}{6}\frac{l_{\chi}^{2}}{\chi^{2}}}\theta_{1}^{\prime}(0)^{-\frac{2l_{\chi}^{2}}{3\chi^{2}}+\frac{l_{\chi}}{3}+\frac{4l_{\chi}}{3\gamma\chi}}e^{P_{0}z\pi/2}$
$\displaystyle\mathop{=}^{\eqref{def:theta1p0}}(-2\pi)^{-\frac{\alpha_{0}\left(\alpha_{0}+4\right)}{6\gamma^{2}}+\frac{1}{12}(\alpha_{0}+4)}q^{\frac{1}{48}\left(\alpha_{0}+4\right)+\frac{6P_{0}^{2}-\alpha_{0}\left(\alpha_{0}+2\right)}{12\gamma^{2}}}e^{\frac{P_{0}z\pi}{2}}+\mathcal{O}(q^{\gamma^{2}}).$
(4.41)

It is worth noting that the term above can be divided into two components: one
dependent on $z$ and the other dependent on $\gamma$. Such a structure will be
important in the subsequent analysis.

Next we observe that

$\theta_{1}(z+x)^{\gamma^{2}/4}\theta_{1}(z)^{-\gamma^{2}/4}=e^{\gamma^{2}/4\log(\theta_{1}(z+x)/\theta_{1}(z))}=1+\frac{\gamma^{2}}{4}\log\big{(}\frac{\theta_{1}(z+x)}{\theta_{1}(z)}\big{)}+O(\gamma^{3}).$

Substituting the above expression in (4.40), we obtain the following:

$\displaystyle\mathcal{V}^{\alpha_{0}/\gamma}_{\chi,P_{0}/\gamma}(z,q)$
$\displaystyle=\int^{1}_{0}\theta_{1}(x)^{-\frac{\alpha_{0}}{2}}e^{\pi
P_{0}x}e^{\frac{\gamma}{2}Y(x;q)}dx$