# A model for dynamical solvent control of molecular junction electronic
properties

Maxim F. Gelin111E-mail<EMAIL_ADDRESS>School of Sciences, Hangzhou Dianzi
University, 310018 Hangzhou, China  Daniel S. Kosov222E-mail:
<EMAIL_ADDRESS>College of Science and Engineering, James Cook
University, Townsville, QLD, 4811, Australia

###### Abstract

Experimental measurements of electron transport properties of molecular
junctions are often performed in solvents. Solvent-molecule coupling and
physical properties of the solvent can be used as the external stimulus to
control electric current through a molecule. In this paper, we propose a
model, which includes dynamical effects of solvent-molecule interaction in the
non-equilibrium Green’s function calculations of electric current. The solvent
is considered as a macroscopic dipole moment that reorients stochastically and
interacts with the electrons tunnelling through the molecular junction. The
Keldysh-Kadanoff-Baym equations for electronic Green’s functions are solved in
time-domain with subsequent averaging over random realisations of rotational
variables using Furutsu-Novikov method for exact closure of infinite hierarchy
of stochastic correlation functions. The developed theory requires the use of
wide-band approximation as well as classical treatment of solvent degrees of
freedom. The theory is applied to a model molecular junction. It is
demonstrated that not only electrostatic interaction between molecular
junction and solvent but also solvent viscosity can be used to control
electrical properties of the junction. Aligning of the rotating dipole moment
breaks particle-hole symmetry of the transmission favouring either hole or
electron transport channels depending upon the aligning potential.

##  I Introduction

The single-molecule electrical experiments now go way beyond the initial
current as a function of applied voltage measurements. The environment is no
longer playing a passive role but used as experimental means to control
electronic properties of molecular electronic junction.Xiang et al. (2016)
Mechanical stretching, $\pi$ stacking,Wu et al. (2008) hydrogen bondingNishino
et al. (2013) and supramolecular interactions Bui et al. (2013) have been used
to change electrical properties of molecular junctions. Recently, there is a
growing interest in the use of solvent to control transport of electrons in
single-molecule junction. Kuznetsov (2007); Milan et al. (2016); Dzhioev and
Kosov (2012); Leary et al. (2009); Stipe et al. (1997); Luka-Guth et al.
(2016); Kotiuga et al. (2015); Fatemi et al. (2011); Ghane et al. (2015);
Kuznetsov (2007); Choi et al. (2016); Kornyshev et al. (2006)

The scope of theoretical studies of solvated molecular electronic junctions is
somewhat limited. On the computational side, density functional theory
nonequilibrium Green’s function based calculations take into account a few
surrounding solvent molecules kept in the fixed optimised geometry to mimic
the effect of molecule-solvent interaction.Leary et al. (2009); Milan et al.
(2016); Ghane et al. (2015) Several simple models were developed to treat
electrolyte environment of molecular electronic junctions via Poisson-
Boltzmann equations incorporating effect of the solvent into voltage drop
across the junction with subsequent rate equation calculations of electric
current.Kornyshev et al. (2006); Zhang et al. (2008) Born solvation model was
combined in the self-consistent manner with static nonequilibrium Green’s
functions calculations electric current calculations to treat the
problem.Luka-Guth et al. (2016)

These theoretical works provide an important insight on the role of the
solvent, however, they all assumed that the time-scales for electrons and
solvent motion can be completely separated. In other words, all time-dependent
effects from solvent response and rearrangement due to the dynamical charging
and discharging of the molecular junction by current-carrying electrons have
been neglected. In this paper, we developed the model which takes into account
solvent dynamics. The solvent is modelled as a macroscopic, stochastically
rotating dipole moment which interacts with the electrons tunnelling through
the molecular junction. This interaction makes the Hamiltonian for quantum
transport problem explicitly time-dependent which is handled exactly by
nonequilibrium Green’s functions calculations. The evaluation of observables
obtained from electronic Green’s functions requires the averaging over
realisations of Gaussian stochastic process, which is performed using Furutsu-
Novikov method for stochastic calculus.Klyatskin (2005) As far as time-
evolution of the system is concerned, quantum electronic dynamics are treated
on an equal footing with classical solvent dynamics in our method.

The paper is organised as follows. Section II describes the theory. It
introduces model Hamiltonian for molecular junctions, main definitions for
Green’s functions and self-energies, as well as describes solution of the
Keldysh-Kadanoff-Baym equations coupled to the stochastic rotor dynamics.
Section II also provides expression for electronic current averaged over
stochastic realisations of solvent dynamics and details the use of Furutsu-
Novikov method. Section III illustrates the theory by calculations on model
molecular systems and discusses the main physical observations. The main
results of the paper are summarised in section IV. The technical details of
the derivations and numerical algorithm description are relegated to the
appendices.

##  II Theory

###  II.1 Model Hamiltonian

In this section we discuss the physical model for molecular electronic
junction coupled to solvent. The system Hamiltonian consists of five parts

$H(t)=H_{\text{molecule}}+H_{\text{leads}}+H_{\text{solvent}}(t)\\\
+H_{\text{leads-molecule}}+H_{\text{solvent-molecule}}(t).$ (1)

Here $H_{\text{molecule}}$ is the molecular Hamiltonian, $H_{\text{leads}}$
describes electron reservoirs in the right and left leads. Coupling between
the molecule and the leads is given by $H_{\text{leads-molecule}}$. The
solvent is described by $H_{\text{solvent}}$ and the coupling between molecule
and the solvent is denoted by $H_{\text{solvent-molecule}}$, these parts of
the Hamiltonian are time-dependent.

The molecule is modelled as a single resonant-level, which can be occupied by
the zero or one spin-less electrons (the electron spin will not be considered
explicitly in our derivations but we will recreate the factor of two due to
spin degeneracy in the final expressions for electric current). The
corresponding Hamiltonian is

$H_{\text{molecule}}=\epsilon d^{\dagger}d,$ (2)

where $d^{\dagger}$ ($d$) creates (annihilates) electron on molecular orbital
with energy $\epsilon$.

The left and right leads are modelled as macroscopic reservoirs of non-
interacting electrons

$H_{\text{leads}}=\sum_{k\alpha}\epsilon_{k\alpha}d^{\dagger}_{k\alpha}d_{k\alpha},$
(3)

where $d^{\dagger}_{k\alpha}$ and $d_{k\alpha}$ are the creation and
annihilation operators for a single particle state of energies
$\epsilon_{k\alpha}$ for left $\alpha=L$ or right $\alpha=R$ leads. The
couplings between leads and molecule are described by the tunnelling
interaction

$H_{\text{leads-
molecule}}=\sum_{k\alpha}(t_{k\alpha}d^{\dagger}_{k\alpha}d+t^{*}_{k\alpha}d^{\dagger}d_{k\alpha}),$
(4)

where $t_{k\alpha}$ is the tunnelling amplitude between molecule and leads
states.

The solvent, which is modelled as classical rigid rotating dipole moment
$\mathbf{\mu}$ with moment of inertia $I$, is described by the time-dependent
Hamiltonian

$H_{\text{solvent}}(t)=\frac{1}{2I}\left(\frac{d\phi}{dt}\right)^{2},$ (5)

where $\phi$ is the orientation angle of the dipole moment relative to the
molecule. This level of description of solute-solvent interactions is common
in theory of rotational relaxation in polar solvents and constitutes the
essence of stochastic cage model Coffey et al. (1996), where the solvent shell
surrounding the molecule is described as an effective quasiparticle which
interacts with the molecule by appropriate anisotropic potential and
stochastically interacts with the balk solvent. We thus assume that the
orientation angle $\phi$ and the conjugated rotational frequency $\Omega$ obey
the stochastic evolution equations

$\frac{d\phi(t)}{dt}=\Omega(t),$ (6)

and

$\frac{d\Omega(t)}{dt}=G(\phi(t))-\xi\Omega(t)+F(t).$ (7)

Here $U(\phi)$ is an external potential,

$G(\phi)=-\frac{dU(\phi)}{d\phi}$ (8)

is the torque acting on the rotator, and $F(t)$ is a $\delta$-correlated
stochastic Gaussian process,

$\langle F(t)\rangle=0,\,\,\,\langle
F(t)F(t^{\prime})\rangle=2\xi\delta(t-t^{\prime}),$ (9)

where $\xi$ is the rotational friction. The fluctuation-dissipation theorem
relation between the noise and viscosity is assumed here, which implies that
the solvent is maintained in thermodynamic equilibrium. We use the
dimensionless variables, in which $\Omega$ and $\xi$ are expressed in units of
$\sqrt{k_{B}T/I}$, where $k_{B}$ is Boltzmann’s constant and $T$ is a
temperature.

The interaction between the molecule and the solvent is given by point charge-
dipole interaction

$H_{\text{solvent-
molecule}}(t)=\frac{1}{4\pi\epsilon_{0}}\frac{ed^{\dagger}d\;\mu\cos(\phi)}{r^{2}},$
(10)

where the instantaneous charge of the molecule due to tunnelling of electrons
is given by electron number operator $d^{\dagger}d$ multiplied by electron
charge $e$, $\mu$ is magnitude of the classical dipole moment and $r$ is the
distance between molecule and solvent cage centre of mass, and $\epsilon_{0}$
is electrical permittivity. As typical in electron transport calculation we
assume that $e$ carries the negative sign ($e=-|e|$). This part of the
Hamiltonian is explicitly time-dependent due to angle $\phi$ being time-
dependent variable undergoing stochastic fluctuations. Of course, such a
treatment of solvent-molecule interaction is an oversimplification, but it
allows us to conveniently rewrite Eq. (10) as

$H_{\text{solvent-molecule}}(t)=-\lambda\cos(\phi),$ (11)

where the parameter

$\lambda=-\frac{1}{4\pi\epsilon_{0}}\frac{e\mu}{r^{2}}.$ (12)

can be interpreted as an effective solvent-molecule coupling strength. Eqs.
(5)-(11) describe a variant of the stochastic cage model which is called the
itinerant oscillator model Coffey et al. (1996). Despite its simplicity, this
model captures essential physics. It was successfully applied to the
description of molecular rotation in polar liquids and solvents.

It is convenient to include solvent-molecule interaction into the molecule
Hamiltonian by introducing time-dependent energy level

$h(\phi(t))=\epsilon-\lambda\text{cos}(\phi(t)).$ (13)

Then combining two terms together we get

$H_{\text{molecule}}+H_{\text{solvent-molecule}}=h(\phi(t))d^{\dagger}d.$ (14)

###  II.2 Green’s functions and self-energies

In this section, Green’s functions and self-energies definitions are
introduced and basic notation is established; they will be used throughout the
paper. The exact and non-adiabatic (computed along a given angular trajectory
$\phi(t)$) retarded, advanced and lesser Green’s functions in the molecular
space are defined as Haug and Jauho (2010)

$\mathcal{G}^{R}(t,t^{\prime})=-i\theta(t-t^{\prime})\langle\\{d(t),d^{\dagger}(t^{\prime})\\}\rangle,$
(15) $\mathcal{G}^{A}(t,t^{\prime})=\Big{(}{\cal
G}^{R}(t^{\prime},t)\Big{)}^{*}$ (16)

and

$\mathcal{G}^{<}(t,t^{\prime})=i\langle d^{\dagger}(t^{\prime})d(t)\rangle.$
(17)

The retarded, advanced and lesser self-energies for $\alpha$ lead are given by
the standard expressions

${\Sigma}_{\alpha}^{R}(t,t^{\prime})=-i{{}\theta(t-t^{\prime})}\sum_{k}t^{*}_{k\alpha}e^{-i\epsilon_{k}(t-t^{\prime})}t_{k\alpha},$
(18)

${\Sigma}_{\alpha}^{A}(t,t^{\prime})=\Big{(}{\Sigma}_{\alpha}^{R}(t^{\prime},t)\Big{)}^{*},$
(19) ${\Sigma}_{\alpha}^{<}(t,t^{\prime})=2\pi
i\sum_{k}t^{*}_{k\alpha}f_{\alpha}(\epsilon_{k})e^{-i\epsilon_{k}(t-t^{\prime})}t_{k\alpha}.$
(20)

Here $f_{\alpha}$ is the Fermi-Dirac distribution. The total self-energies are
the sum of contributions from the left and right leads as

${\Sigma}^{R,A,<}(t,t^{\prime})={\Sigma}^{R,A,<}_{L}(t,t^{\prime})+{\Sigma}_{R}^{R,A,<}(t,t^{\prime}).$
(21)

The self-energies in energy-domain are defined as a Fourier transformation

${\Sigma}^{R,A,<}(\omega)=\int
d(t-t^{\prime})e^{i\omega(t-t^{\prime})}{\Sigma}^{R,A,<}(t,t^{\prime}).$ (22)

The wide-band approximation will be used in our derivations and calculations:
within this approximation the time-dependence of retarded and advanced self-
energies is reduced to delta-function

$\Sigma_{\alpha}^{R}(t,t^{\prime})=-\frac{i}{2}\Gamma_{\alpha}\delta(t-t^{\prime}),\;\;\;\Sigma_{\alpha}^{A}(t,t^{\prime})=\frac{i}{2}\Gamma_{\alpha}\delta(t-t^{\prime}),$
(23)

where the level-broadening function $\Gamma_{\alpha}$ is time-independent
parameter which describes the strength of molecule-lead coupling. The wide-
band approximation means the assumption that both the leads’ density of states
and all tunneling amplitudes between leads’ and molecular states are energy-
independent constants. Our approach depends critically on the use of the wind-
band approximation to solve analytically Keldysh-Kadanoff-Baym equations.

Notice that we set $\hbar=1$ in all definitions of Green’s functions. This
implies that all quantum mechanical energy related quantities $\epsilon$,
$\lambda$, and $\Gamma_{\alpha}$ are scaled by $\hbar$ and measured in the
units of frequency (1/time).

###  II.3 Solution of Keldysh-Kadanoff-Baym equations and expression for the
current

Suppose that a stochastic trajectory of the dipole rotation angle $\phi(t)$ is
known from the solution of the Langevin equation. Our goal in this section is
to solve exactly Keldysh-Kadanoff-Baym equations for the Green’s functions and
consequently obtain a concise expression for electric current averaged over
the stochastic trajectory $\phi(t)$.

We begin with the general expression for electric current from $\alpha$ lead
in time-dependent systemHaug and Jauho (2010)

$J_{\alpha}(t)=2e\text{ Re
}\int_{-\infty}^{+\infty}dt^{\prime}\Big{[}\mathcal{G}^{<}(t,t^{\prime})\Sigma_{\alpha}^{A}(t^{\prime},t)\\\
+\mathcal{G}^{R}(t,t^{\prime})\Sigma_{\alpha}^{<}(t^{\prime},t)\big{]}.$ (24)

Next, we transform self-energies to the energy domain whilst leaving molecular
Green’s functions time-dependent

$J_{\alpha}(t)=2e\text{ Re
}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int_{-\infty}^{+\infty}dt^{\prime}e^{-i\omega(t^{\prime}-t)}\\\
\times\Big{[}\mathcal{G}^{<}(t,t^{\prime})\Sigma_{\alpha}^{A}(\omega)+\mathcal{G}^{R}(t,t^{\prime})\Sigma_{\alpha}^{<}(\omega)\big{]}.$
(25)

Electric current should satisfy the continuity equation at each time moment
$t$

$\frac{dN}{dt}=J_{L}(t)+J_{R}(t),$ (26)

where $N$ is the total number of electrons in the molecule at time $t$. Using
the continuity equation and rearranging the terms, we eliminate lesser Green’s
function from the expression and get

$J_{L}(t)=\frac{\Gamma_{L}}{\Gamma}\Big{\\{}\frac{dN}{dt}-2e\Gamma_{R}\text{
Im
}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int_{-\infty}^{+\infty}dt^{\prime}e^{-i\omega(t^{\prime}-t)}\mathcal{G}^{R}(t,t^{\prime})[f_{L}(\omega)-f_{R}(\omega)]\Big{\\}}.$
(27)

Let us average the above equation over the stochastic realisation of the
dipole moment rotation process. The average

$\langle\frac{dN}{dt}\rangle=0$ (28)

obviously disappears, otherwise the molecule would either accumulate or loose
charge continuously. The averaged current becomes

$\langle J_{L}\rangle=-2e\frac{\Gamma_{L}\Gamma_{R}}{\Gamma}\\\ \text{ Im
}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int_{-\infty}^{+\infty}dt^{\prime}e^{-i\omega(t^{\prime}-t)}\langle\mathcal{G}^{R}(t,t^{\prime})\rangle[f_{L}(\omega)-f_{R}(\omega)].$
(29)

Notice, that the averaged electric current does not depend on time, since the
retarded Green’s function will depend on relative time only once averaged over
the stochastic realisations.

Our next goal is to find the explicit expression for the retarded Green’s
function to enter it into the expression to electric current. We begin with
the Keldysh-Kadanoff-Baym equation for the retarded Green’s functionHaug and
Jauho (2010)

$(i\partial_{t}-h(t))\mathcal{G}^{R}(t,t^{\prime})\\\
-\int_{-\infty}^{+\infty}dt_{1}\Sigma^{R}(t,t_{1})\mathcal{G}^{R}(t_{1},t^{\prime})=\delta(t-t^{\prime}).$
(30)

It is reduced in the wide-band approximation to

$(i\partial_{t}-h(t))\mathcal{G}^{R}(t,t^{\prime})-\frac{i}{2}\Gamma\mathcal{G}^{R}(t_{1},t^{\prime})=\delta(t-t^{\prime}),$
(31)

where $h(t)$ denotes $h(\phi(t))$ given by equation (13) and

$\Gamma=\Gamma_{L}+\Gamma_{R}$ (32)

is the total level-broadening function. This differential equation can be
solved analytically and yields

$\mathcal{G}^{R}(t,t^{\prime})=-i{{}\theta(t-t^{\prime})}e^{-\frac{1}{2}\Gamma(t-t^{\prime})}e^{-i\int^{t}_{t^{\prime}}dt_{1}h(t_{1})}$
(33)

Notice that analytical expression (33) requires the wide-band approximation
treatment for leads’ self-energies. Substituting retarded Green’s function
(33) into the expression for current (29) gives

$\langle J_{L}\rangle=2e\frac{\Gamma_{L}\Gamma_{R}}{\Gamma}\text{ Re
}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int_{-\infty}^{t}dt^{\prime}e^{-i(\omega-\frac{i}{2}\Gamma)(t-t^{\prime})}\langle
e^{-i\int^{t}_{t^{\prime}}dt_{1}h(t_{1})}\rangle[f_{L}(\omega)-f_{R}(\omega)].$
(34)

Changing variables of integration to $\tau=t-t^{\prime}$ and taking into
account that quantity $\langle
e^{-i\int^{t}_{t^{\prime}}dt_{1}h(t_{1})}\rangle$ depends on relative time
only once averaged over realisations of the stochastic process we arrive to

$\langle J_{L}\rangle=2e\frac{\Gamma_{L}\Gamma_{R}}{\Gamma}\text{ Re
}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int^{\infty}_{0}d\tau
e^{-i(\omega-\frac{i}{2}\Gamma)\tau)}R(\tau)[f_{L}(\omega)-f_{R}(\omega)]$
(35)

where

$R(\tau)=\langle e^{-i\int^{\tau}_{0}dt_{1}h(t_{1})}\rangle$ (36)

Eq. (35) can be rewritten in the Landauer form

$\langle
J_{L}\rangle=e\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}T(\omega)[f_{L}(\omega)-f_{R}(\omega)]$
(37)

where we have introduced ”transmission coefficient”

$T(\omega)=2\frac{\Gamma_{L}\Gamma_{R}}{\Gamma}\text{ Re
}\int^{\infty}_{0}d\tau e^{-i(\omega-\frac{i}{2}\Gamma)\tau)}R(\tau).$ (38)

Notice that although being called ”transmission coefficient”, $T(\omega)$
given by (38) should not be assigned the meaning of probability for electron
with energy $\omega$ to tunnel across the molecule, since the inelastic
processes due to the dynamical coupling of tunnelling electron with rigid
rotator are included into our consideration.

###  II.4 Averaging over the stochastic solvent dynamics

The aim here is to develop a method to compute $R(t)$ of Eq. (36) (for
convenience of notation, $t$ will be used as a variable in $R(t)$) where
averaging $\langle...\rangle$ is performed over realisations of stochastic
variable $\phi$ and time-evolution of $\phi(t)$ is given by stochastic
differential equations (6) and (7).

Let us define the quantity

$A_{t}([\phi(\tau)])=e^{-i\int_{0}^{t}dt_{1}h(\phi(t_{1}))}.$ (39)

Hereafter, the notation $A_{t}[\phi(\tau)]$ means that $A$ is a function of
$t$ and a functional of the stochastic process $\phi(\tau)$. Evidently,

$R(t)=\langle A_{t}([\phi(\tau)])\rangle.$ (40)

Let us now introduce the stochastic functional

$R(\phi,\Omega,t)=\delta(\phi-\phi(t))\delta(\Omega-\Omega(t))A_{t}[\phi(\tau)]$
(41)

where $\phi(t)$ and $\Omega(t)$ are certain realisations of the stochastic
processes (6) and (7). Averaging Eq. (41) over all realisations of these
processes yields

$\rho(\phi,\Omega,t)=\left\langle R(\phi,\Omega,t)\right\rangle,$ (42)

which can be interpreted as a probability density of $A_{t}[\phi(\tau)]$ to
have a certain value given $\phi(t)$ and $\Omega(t)$ are equaled to $\phi$ and
$\Omega$ at a time moment $t$. Then, according to van Kampen lemma,Van Kampen
(2007) we obtain

$R(t)=\int_{-\infty}^{\infty}d\Omega\int_{0}^{2\pi}d\phi\rho(\phi,\Omega,t).$
(43)

We are in the position now to derive a closed-form Fokker-Planck equation for
$\rho(\phi,\Omega,t)$. Formally differentiating $R(\phi,\Omega,t)$ with
respect to time, using stochastic equations (6) and (7) and taking the
average, we obtain

$\partial_{t}\rho(\phi,\Omega,t)=\Big{(}-ih(\phi)-\Omega\partial_{\phi}-G(\phi)\partial_{\Omega}+\xi\partial_{\Omega}\Omega\Big{)}\rho(\phi,\Omega,t)\\\
-\partial_{\Omega}\Omega\left\langle F(t)R(\phi,\Omega,t)\right\rangle.$ (44)

Employing the Furutsu-Novikov formulaKlyatskin (2005), we get

$\left\langle
F(t)R(\phi,\Omega,t)\right\rangle=\intop_{0}^{t}dt^{\prime}\left\langle
F(t)F(t^{\prime})\right\rangle\left\langle\frac{\delta
R(\phi,\Omega,t)}{\delta F(t^{\prime})}\right\rangle.$ (45)

To evaluate the functional derivative, we follow the method of Ref. Klyatskin
(2005):

$\frac{\delta R(\phi,\Omega,t)}{\delta
F(t^{\prime})}=\Big{(}-\partial_{\phi}\frac{\delta\phi(t)}{\delta
F(t^{\prime})}-\partial_{\Omega}\frac{\delta\Omega(t)}{\delta
F(t^{\prime})}\\\ -i\int_{0}^{t}dt^{\prime\prime}\frac{\delta
h(\phi(t^{\prime\prime}))}{\delta
F(t^{\prime})}\Big{)}\delta(\phi-\phi(t))\delta(\Omega-\Omega(t))A_{t}[\phi(\tau)]\\\
=\Big{(}-\partial_{\phi}\frac{\delta\phi(t)}{\delta
F(t^{\prime})}-\partial_{\Omega}\frac{\delta\Omega(t)}{\delta
F(t^{\prime})}\\\
+i\lambda\int_{0}^{t}dt^{\prime\prime}\sin(\phi(t^{\prime\prime}))\frac{\delta\phi(t^{\prime\prime})}{\delta
F(t^{\prime})}\Big{)}\delta(\phi-\phi(t))\delta(\Omega-\Omega(t))A_{t}[\phi(\tau)].$
(46)

The stochastic equations (6) and (7) can be rewritten in the integral form as
follows:

$\phi(t)=\phi(0)+\intop_{0}^{t}d\tau\int_{-\infty}^{\infty}d\Omega\Omega\delta(\Omega-\Omega(\tau)),$
(47)

and

$\Omega(t)=\Omega(0)+\intop_{0}^{t}d\tau\left(G(\phi(\tau))+\xi\Omega(\tau)+F(\tau)\right)=\\\
\Omega(0)+\intop_{0}^{t}d\tau\int_{-\infty}^{\infty}d\Omega\int_{0}^{2\pi}d\phi\left(G(\phi)+\xi\Omega+F(\tau)\right)\delta(\phi-\phi(\tau))\delta(\Omega-\Omega(\tau)).$
(48)

Differentiating (47) and (48) with respect to $F(t^{\prime})$, we obtain:

$\frac{\delta\phi(t)}{\delta
F(t^{\prime})}=-\intop_{t^{\prime}}^{t}d\tau\int_{-\infty}^{\infty}d\Omega\Omega\partial_{\Omega}\delta(\Omega-\Omega(\tau))\frac{\delta\Omega(\tau)}{\delta
F(t^{\prime})},$ (49) $\frac{\delta\Omega(t)}{\delta
F(t^{\prime})}=\intop_{0}^{t}d\tau\int_{-\infty}^{\infty}d\Omega\int_{0}^{2\pi}d\phi\left[\frac{\delta
F(t)}{\delta
F(t^{\prime})}-\left(G(\phi)+\xi\Omega+F(\tau)\right)\left(\partial_{\phi}\frac{\delta\phi(t)}{\delta
F(t^{\prime})}+\partial_{\Omega}\frac{\delta\phi(t)}{\delta
F(t^{\prime})}\right)\right]\delta(\phi-\phi(\tau))\delta(\Omega-\Omega(\tau))\\\
=1-\intop_{t^{\prime}}^{t}d\tau\int_{-\infty}^{\infty}d\Omega\int_{0}^{2\pi}d\phi\left[\left(G(\phi)+\xi\Omega+F(\tau)\right)\left(\partial_{\phi}\frac{\delta\phi(t)}{\delta
F(t^{\prime})}+\partial_{\Omega}\frac{\delta\phi(t)}{\delta
F(t^{\prime})}\right)\right]\delta(\phi-\phi(\tau))\delta(\Omega-\Omega(\tau)).$
(50)

Due to causality, we have replaced

$\intop_{0}^{t}d\tau\rightarrow\intop_{t^{\prime}}^{t}d\tau$ (51)

because

$\frac{\delta\phi(t)}{\delta F(t^{\prime})}=\frac{\delta\Omega(t)}{\delta
F(t^{\prime})}=0\,\,\,\mathrm{for}\,\,\tau<t^{\prime}.$ (52)

Since the stochastic torque $F(t)$ is delta-correlated (Eq. (9)), we need to
evaluate the functional derivative at $t^{\prime}=t.$ Putting $t^{\prime}=t$
in Eqs. (49) and (50), we obtain

$\frac{\delta\phi(t)}{\delta F(t)}=0,\,\,\,\frac{\delta\Omega(t)}{\delta
F(t)}=1.$ (53)

Formula (52) allows us to make the replacement
$\int_{0}^{t}dt^{\prime\prime}\rightarrow\int_{t^{\prime}}^{t}dt^{\prime\prime}$
in Eq. (46), which insures that this integral vanishes at $t^{\prime}=t$.
Hence

$\frac{\delta R(\phi,\Omega,t)}{\delta F(t)}=-\partial_{\Omega}\Omega
R(\phi,\Omega,t).$ (54)

Finally, evaluating the integral in Eq. (45) through the explicit expression
(9) for $\left\langle F(t)F(t^{\prime})\right\rangle,$ we arrive at the
Fokker-Planck equation

$\partial_{t}\rho(\phi,\Omega,t)=\Big{(}-ih(\phi)-\Omega\partial_{\phi}-G(\phi)\partial_{\Omega}\\\
+\xi\left(\partial_{\Omega}\Omega+\partial_{\Omega}^{2}\right)\Big{)}\rho(\phi,\Omega,t).$
(55)

The first three terms in Eq. (55) describe evolution of $\rho(\phi,\Omega,t)$
without action of the solvent, while the impact of the solvent is taken care
of by the addition of the Fokker-Planck dissipation operator. A simple and
intuitively clear additive structure of Eq. (55) is caused by the Markovianity
of the stochastic process $\\{\phi(t),\Omega(t)\\}$. Eq. (55) should be solved
with the initial condition

$\rho(\phi,\Omega,0)=\rho_{B}(\Omega)\rho_{B}(\phi)$ (56)

where

$\rho_{B}(\Omega)=\frac{1}{\sqrt{2\pi}}e^{-\Omega^{2}/2},\;\;\;\rho_{B}(\phi)=Z^{-1}e^{-U(\phi)}$
(57)

and $Z$ is the partition function.

To summarise, the differential equation (55) is the main result of section
IID. The solution of equation (55) is used to compute $R(t)$ via (43), then
the electronic transmission (38) is computed with help of $R(t)$ with
subsequent calculations of electric current (37).

##  III Results

In this section, numerical and analytical results are presented to illustrate
the proposed theory. The rotational Langevin equations (6) and (7) were
conveniently written in terms of dimensionless units in which time is measured
in terms of $\sqrt{I/(k_{B}T)}$. This means that rotational friction $\xi$
proportional to the solvent viscosity, which is an important parameter for the
discussion in the present section, is measured in units of thermal rotational
frequency $\sqrt{k_{B}T/I}$. Whilst defining the Green’s functions and solving
Keldysh-Kadanoff-Baym equations we set $\hbar=1$, which means that level
broadening $\Gamma$, molecular orbital energy $\epsilon$, and electron-
rotational coupling $\lambda$ will be also given in units of
$\sqrt{k_{B}T/I}$. The Fermi energy $E_{F}$ of the leads is set to zero, hence
molecular orbital energy $\epsilon$ should be understood as the energy with
respect to $E_{F}$. Since $\epsilon$ merely determines position of the
transmission maximum, we set $\epsilon$ throughout this section. Without
restricting the generality of the model this choice makes discussions and
analytical expressions more lucid and transparent for physical interpretation.

We begin with the consideration of free rotation, where the influence of the
external alignment potential

$U(\phi)=\alpha\cos(\phi-\phi_{0})$ (58)

can be neglected ($\alpha=0$). Fig. 1 shows how the transmission scales with
electron-rotational coupling $\lambda$. The transmission is computed
numerically using algorithm detailed in Appendix A. For $\lambda=0$,
electronic energy of the molecule is totally decoupled from its rotational
motion. Hence the transmission is simply a Lorentzian centered at
$\omega=\epsilon$ with a width $\sim\Gamma$, as described by Eq. (85). If
$\lambda$ increases, then Eq. (86) predicts that the resonant value of the
electronic transmission decreases, $T(\omega=\epsilon)\sim\lambda^{-3/2}$,
while the transmission starts to exhibit two symmetric maxima at
$\omega=\epsilon\pm\omega_{m}$. Positions of these maxima obey the inequality
$\omega_{m}=\sqrt{\lambda^{2}-\Gamma^{2}}<\lambda$ (see Eq. (87)) while the
intensity of the transmission in the maxima scales as
$T(\omega=\epsilon\pm\omega_{m})\sim\lambda^{-1/2}$. This behavior of the
transmission can be rationalized by the consideration of the problem in the
time domain: In the underdamped limit, according to Eq. (80), $R(t)\sim
J_{0}(\lambda t)$ and one can anticipate that the oscillation frequency
$\approx\lambda$ should manifest itself in the transmission for
$\lambda\gg\Gamma_{L},\Gamma_{R}$.

In terms of physical observation Fig. 1 tells us that in the resonant regime,
when molecular orbital energy is aligned with leads Fermi energy, the coupling
to the surrounding solvent always suppresses the conductivity of the junction.
However, quite oppositely, if the electron transport is dominated by the
molecular orbital which is shifted below or above Fermi energy, the solvent
may considerably increase the conductance of the system.

Figure 1: Electronic transmissions $T(\omega)$ computed for various values of
electron-rotational coupling $\lambda$. Parameters used in calculations:
$\epsilon=0$, $\Gamma_{L}=\Gamma_{R}=\xi=1$. The case of free rotation
($\alpha=0$) is considered here.  Figure 2: Electronic transmissions
$T(\omega)$ computed for various values of rotational friction $\xi$ in the
case of free rotation ($\alpha=0$). Parameters used in calculations:
$\epsilon=0$, $\Gamma_{L}=\Gamma_{R}=0.3$, $\lambda=1$.

There are three characteristic timescales in the system. One is associated
with the time of electron tunneling across the molecule, $\Gamma^{-1}$. The
other is the relaxation timescale for solvent dynamics, $\xi^{-1}$ in the
underdamped limit and $\xi=D^{-1}$ in the overdamped limit ($D$ being the
diffusion coefficient). The third, $\lambda^{-1}$, is associated with the
electron-rotational coupling and, consequently, with the rate of energy
exchange between tunneling electrons and external rotator. Fig. 2 illustrates
how $T(\omega)$ is affected by the these timescales. As $\xi$ increases
relative to $\Gamma$, $T(\omega)$ exhibits a transformation from a single-peak
to a double-peak structure. This somewhat counterintuitive behavior can be
understood from the following considerations. In the limit $\lambda\gg 1$ and
$\xi<\lambda$, the transmission is quite accurately described by the bath-free
formula of Eq. (83), which has been analyzed above. For $\lambda<1$ and
$\xi<1$ (underdamped limit) $T(\omega)$ is also rather insensitive to $\xi$:
the rotational friction quenches slightly the amplitude of oscillations in
$R(t)\sim J_{0}(\lambda t)$, while the shape of $T(\omega)$ is largely
determined by the coupling to the leads, through the exponentially decaying
factor $\exp(-\Gamma\tau)$ as given in Eq. (38). The situation changes in the
overdamped limit ($\xi\gg\lambda$). In this case, $T(\omega)$ can be evaluated
analytically (see Appendix B). Eq. (78) reveals then: the higher the friction,
the more pronounced the peaks of $T(\omega)$. For $\lambda\ll 1$, for example,
Eq. (78) reduces to

$\tilde{\rho}_{0}(s)\approx\frac{s+\xi^{-1}}{s^{2}+s\xi^{-1}+\lambda^{2}/2}.$
(59)

The expression on the right has two simple poles at
$(-\xi^{-1}\pm\sqrt{\xi^{-2}-2\lambda^{2}})/2$. For $\lambda>/(\sqrt{2}\xi)$
the poles become complex. Their imaginary part, which determines the
$T(\omega)$ maxima $\omega_{m}=\sqrt{2\lambda^{2}-\xi^{-2}}/2$, increases with
friction, reaching a value of $\omega_{m}=\lambda/\sqrt{2}<\lambda$ in the
overdamped limit $\xi\rightarrow\infty$. Then, if the broadening of the
transmission induced by the coupling to the leads is not too high
($\Gamma<\omega_{m}$) or, equivalently, the electron tunneling time is not too
short, $T(\omega)$ exhibits the double-peak structure. In the context of the
possibility to manipulate molecular conductivity, Fig. 2 reveals that the
viscosity of the solvent can be used as effective control parameter for
electron transport properties of molecular junction.

The effect of the external potential is illustrated by Fig. 3, which shows
$T(\omega)$ for different $\lambda$ in the presence of the aligning potential
(58) with $\alpha=1$ and $\phi_{0}=0$. Evidently, $U(\phi)$ breaks the
rotational symmetry of the transmission, making the angles of $\pm\phi$
inequivalent. Hence the transmission develops particle-hole asymmetry, and
$T(\omega)$ has very different behavior for hole ($\omega<E_{F}$) and electron
($\omega>E_{F}$) transport. The particle-hole asymmetry becomes more profound
as the strength of electron-rotation coupling $\lambda$ increases, producing a
pronounced maximum at positive $\omega$. As in the case of free rotation shown
in Fig. 1, the shift of the peak relative to the molecular orbital energy is
determined by the electron-rotational coupling $\lambda$. The asymmetry
depends on the aligning angle $\phi_{0}$: If $\phi=\pi$ is chosen, the peak of
$T(\omega)$ is shifted to hole rather than electron region.

Figure 3: Electronic transmissions $T(\omega)$ computed for various values of
electron-rotational coupling $\lambda$ in the presence of the external
potential $U(\phi)$ with $\alpha=1$ and $\phi_{0}=0$. The remaining
parameters: $\epsilon=0$, $\Gamma_{L}=\Gamma_{R}=1$, $\xi=1$.

##  IV conclusions

In this paper, we have developed a quantum transport theory which includes
dynamical effects of solvent-molecule interaction in the non-equilibrium
Green’s function calculations of electric current. The solvent is considered
as a classical macroscopic dipole moment which reorients stochastically and
interacts with the electrons tunneling through the molecular junctions. This
dynamical electron-rotation interaction makes the quantum mechanical
Hamiltonian explicitly time-dependent. The Keldysh-Kadanoff-Baym equations are
solved in time-domain. The obtained expression for electric current requires
the averaging over realisations of stochastic process related to the dipole
moment rotation. The averaging over random realisations of rotational
variables is performed using Furutsu-Novikov method to close hierarchy of
equations for stochastic correlation functions.

We applied the theory for a model molecular junction - a single molecular
orbital coupled to classical rotating dipole via electrostatic interaction.
Our calculations shows that, in the resonant regime when molecular orbital
energy is aligned with the leads Fermi energy, coupling to the surrounding
solvent always suppresses the conductivity of the junction. However, if the
electron transport is controlled by the molecular orbital which is shifted
below or above Fermi energy, the solvent may noticeably increase the
conductance of the system. It is found that the viscosity of the solvent can
be used as very effective stimulus to control electron transport properties of
molecular junction. If the rotation of the dipole moment is subjected to
external aligning potential, the solvent breaks particle-hole symmetry of
electron transport favouring holes or electron transport channel depending
upon the aligning angle.

Acknowledgment M.F.G. acknowledges support of Hangzhou Dianzi University
through the startup funding.

DATA AVAILABILITY

The data that supports the findings of this study are available within the
article.

##  Appendix A Numerical method

For obtaining a matrix form of (55), we use Hermit polynomials $H_{n}(\Omega)$
to expand $\rho(\phi,\Omega,t)$ in a series as

$\rho(\phi,\Omega,t)=\sum_{k=-\infty}^{\infty}\,\sum_{n=0}^{\infty}e^{-ik\phi}g_{n}(\Omega)\rho_{kn}(t).$
(60)

where

$g_{n}(\Omega)=\frac{1}{n!}H_{n}(\Omega)\rho_{B}(\Omega),$ (61)

and

$\int_{-\infty}^{\infty}d\Omega H_{m}(\Omega)g_{n}(\Omega)=\delta_{mn}.$ (62)

Then:

$\int_{-\infty}^{\infty}d\Omega
H_{m}(\Omega)\partial_{\Omega}g_{n}(\Omega)=-m\delta_{m-1,n}.$ (63)
$\int_{-\infty}^{\infty}d\Omega H_{m}(\Omega)\Omega
g_{n}(\Omega)=m\delta_{m-1,n}+\delta_{m+1,n}.$ (64)
$\int_{-\infty}^{\infty}d\Omega
H_{m}(\Omega)\left(\partial_{\Omega}\Omega+\partial_{\Omega}^{2}\right)g_{n}(\Omega)=-m\delta_{m,n}.$
(65)

Hence:

$\partial_{t}\rho_{kn}(t)=-i\epsilon\rho_{kn}(t)+i\frac{\lambda}{2}\left[\rho_{k+1,n}(t)+\rho_{k-1,n}(t)\right]\\\
+ik\left[\rho_{k,n+1}(t)+n\rho_{k,n-1}(t)\right]\\\
+i\frac{\alpha}{2}n\left[e^{-i\phi_{0}}\rho_{k+1,n-1}(t)-e^{i\phi_{0}}\rho_{k-1,n-1}(t)\right]-\xi
n\rho_{kn}(t).$ (66)

It should be solved with the initial condition

$\rho_{kn}(0)=\rho_{k}^{(0)}\delta_{n0}$ (67)

where

$\rho_{k}^{(0)}=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi e^{ik\phi}\rho_{B}(\phi)$
(68)

Evidently,

$R(t)=\rho_{00}(t).$ (69)

##  Appendix B Overdamped limit: Analytical expression for electronic
transmission

In the overdamped limit ($\xi\gg 1$) the Fokker-Planck equation (55) reduces
to the rotational diffusion equation

$\partial_{t}\rho(\phi,t)=\left(-ih(\phi)+\xi^{-1}\left(\partial_{\phi}^{2}-\partial_{\phi}G(\phi)\right)\right)\rho(\phi,t).$
(70)

In the case of free rotation ($G(\phi)=0$, $\alpha=0$) this equation can be
solved analytically. Expanding $\rho(\phi,t)$ in Fourier series,

$\rho(\phi,t)=\sum_{k=-\infty}^{\infty}e^{-ik\phi}\rho_{k}(t),$ (71)

we obtain

$\partial_{t}\rho_{k}(t)=-i\epsilon\rho_{k}(t)+i\frac{\lambda}{2}\left[\rho_{k+1}(t)+\rho_{k-1}(t)\right]-\xi^{-1}k^{2}\rho_{k}(t),$
(72) $\rho_{k}(0)=\delta_{0k}.$ (73)

We wish to evaluate

$R(t)=\rho_{0}(t).$ (74)

By introducing the Laplace transform

$\tilde{\rho}_{k}(s)=\int_{0}^{\infty}dte^{-st}\rho_{k}(t),$ (75)

we can rewrite Eq. (72) through the three-term recurrence relations

$-\delta_{0k}=a_{k}\tilde{\rho}_{k}(s)+b_{k}\tilde{\rho}_{k+1}(s)+c_{k}\tilde{\rho}_{k-1}(s)$
(76)

where

$a_{k}=-(s_{\epsilon}+\xi^{-1}k^{2}),\,\,b_{k}=c_{k}=i\frac{\lambda}{2},\,\,s_{\epsilon}\equiv
s+i\epsilon.$ (77)

These recurrence relations can be solved through the continued fraction:

$\tilde{\rho}_{0}(s)=\cfrac{1}{s_{\epsilon}+\cfrac{\lambda^{2}/2}{s_{\epsilon}+\xi^{-1}+\cfrac{\lambda^{2}/4}{s_{\epsilon}+4\xi^{-1}+\cfrac{\lambda^{2}/4}{s_{\epsilon}+9\xi^{-1}+...}}}}.$
(78)

Then,

$T(\omega)=\frac{1}{\pi}\frac{\Gamma_{L}\Gamma_{R}}{\Gamma_{L}+\Gamma_{R}}\mathrm{Re}\tilde{\rho}_{0}\left(i(\omega+\epsilon)+\Gamma\right).$
(79)

##  Appendix C Frictionless rigid rotor: analytical results for electronic
transmission

If $\xi=0$, Eq. (36) can be evaluated analytically:

$R(t)=e^{-i\epsilon
t}\int_{-\infty}^{\infty}d\Omega\rho_{B}(\Omega)\int_{0}^{2\pi}\frac{d\phi}{2\pi}e^{i\lambda\int_{0}^{t}dt_{1}\cos(\phi+\Omega
t_{1})}\\\ =e^{-i\epsilon t}\int_{0}^{2\pi}\frac{d\phi}{2\pi}e^{i\lambda
t\cos(\phi)}=e^{-i\epsilon t}J_{0}(\lambda t),$ (80)

where $J_{0}(x)$ is the Bessel function of the first kind. Since

$J_{0}(\lambda t)\approx\sqrt{\frac{2}{\pi\lambda t}}\cos(\lambda t-\pi/4)$
(81)

for large $t$, $R(t)$ exhibits algebraically damped oscillations with a period
$2\pi/\lambda$ (for $\epsilon=0$).

The transmission of Eq. (38) can be evaluated as follows:

$T(\omega)=\mathrm{Re}\int_{0}^{\infty}dte^{-\Gamma
t}e^{-i(\epsilon-\omega)t}\int_{0}^{2\pi}d\phi e^{i\lambda t\cos(\phi)}\\\
=\mathrm{Re}\int_{0}^{2\pi}d\phi\frac{1}{\Gamma+i(\epsilon-\omega-\lambda\cos(\phi))}\\\
=\frac{2\pi}{\Gamma}\mathrm{Re}\frac{1}{\sqrt{\left[\Gamma+i(\epsilon-\omega)\right]^{2}+\lambda^{2}}}.$
(82)

Explicitly,

$T(\omega)=\frac{2\pi}{\Gamma}\sqrt{\frac{2\Gamma^{2}-z+\sqrt{z^{2}+4\lambda^{2}\Gamma^{2}}}{2(z^{2}+4\lambda^{2}\Gamma^{2})}}.$
(83)

where

$z=(\epsilon-\omega)^{2}+\Gamma^{2}-\lambda^{2}.$ (84)

In two particular cases, much simpler and easier-to-grasp formulas are
deducible from Eq. (83). If $\Gamma\gg\lambda$, the transmission is described
by the Lorentzian centered at $\omega=\epsilon$,

$T(\omega)\sim\frac{\Gamma}{(\epsilon-\omega)^{2}+\Gamma^{2}}.$ (85)

In the opoosite limit $\lambda\gg\Gamma$, the transmission exhibits two
symmetric maxima positioned at $\epsilon-\omega=\pm\omega_{m}$:

$T(\omega)\sim\sqrt{\frac{\lambda\Gamma}{((\epsilon-\omega)^{2}-\omega_{m}^{2})^{2}+4\lambda^{2}\Gamma^{2}}}$
(86)

with

$\omega_{m}=\sqrt{\lambda^{2}-\Gamma^{2}}.$ (87)

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#  Large deviations of radial SLE∞

###### Abstract

We derive the large deviation principle for radial Schramm-Loewner evolution
($\operatorname{SLE}$) on the unit disk with parameter
$\kappa\rightarrow\infty$. Restricting to the time interval $[0,1]$, the good
rate function is finite only on a certain family of Loewner chains driven by
absolutely continuous probability measures
$\\{\phi_{t}^{2}(\zeta)\,d\zeta\\}_{t\in[0,1]}$ on the unit circle and equals
$\int_{0}^{1}\int_{S^{1}}|\phi_{t}^{\prime}|^{2}/2\,d\zeta\,dt$. Our proof
relies on the large deviation principle for the long-time average of the
Brownian occupation measure by Donsker and Varadhan.

##  1 Introduction

The Schramm-Loewner evolution is a one parameter family of random fractal
curves (denoted as $\operatorname{SLE}_{\kappa}$ with parameter $\kappa>0$).
It was introduced by Oded Schramm [Sch00] and has been a central topic in the
two dimensional random conformal geometry. A version of such curves starting
from a fixed boundary point to a fixed interior point on some two-dimensional
simply connected domain $D$ are called _radial_ $\operatorname{SLE}$s. Let us
recall briefly the definition. The radial $\operatorname{SLE}_{\kappa}$ on the
unit disk $\mathbb{D}=\\{\zeta\in\mathbb{C}:\left\lvert\zeta\right\rvert=1\\}$
targeted at $0$ is the random curve associated to the radial Loewner chain,
whose _driving function_ $t\mapsto\zeta_{t}$ is given by a Brownian motion on
the unit circle
$S^{1}=\\{\zeta\in\mathbb{C}:\left\lvert\zeta\right\rvert=1\\}$ with variance
$\kappa$. That is,

$\zeta_{t}:=B^{\kappa}_{t}:=e^{iW_{\kappa t}},$ (1.1)

where $W_{t}$ is a standard linear Brownian motion. More precisely, we
consider the Loewner ODE for all $z\in\mathbb{D}$

$\partial_{t}g_{t}(z)=-g_{t}(z)\frac{g_{t}(z)+\zeta_{t}}{g_{t}(z)-\zeta_{t}},$
(1.2)

or equivalently, the Loewner PDE satisfied by $f_{t}:=g_{t}^{-1}$

$\partial_{t}f_{t}(z)=zf^{\prime}_{t}(z)\frac{z+\zeta_{t}}{z-\zeta_{t}},\quad
z\in\mathbb{D}$ (1.3)

with the initial condition $f_{0}(z)=g_{0}(z)=z$. For a given $t>0$, $f_{t}$
is a conformal map from $\mathbb{D}$ onto a simply connected domain
$D_{t}\subset\mathbb{D}$ (and $s\mapsto g_{s}(z)$ is a well-defined solution
of (1.2) up to time $t$ if and only if $z\in D_{t}$) such that $f_{t}(0)=0$
and $f_{t}^{\prime}(0)=e^{-t}$. The family of conformal maps
$\\{f_{t}\\}_{t\geq 0}$ is called the _capacity parametrized radial Loewner
chain_ or _normalized subordination chain_ driven by $t\mapsto\zeta_{t}$.
$\operatorname{SLE}_{\kappa}$ is the curve $t\mapsto\gamma_{t}$ which is
defined as $\gamma_{t}:=\lim_{r\to 1-}f_{t}(r\zeta_{t})$, see [RS05]. In
particular, the curve starts at $\gamma_{0}=1$. The radial
$\operatorname{SLE}_{\kappa}$ on an arbitrary simply connected domain $D$ is
defined via the unique conformal map from $\mathbb{D}$ to $D$ respecting the
starting and target points. It is well-known that
$\operatorname{SLE}_{\kappa}$ exhibits phase transitions as $\kappa$ varies.
Larger values of $\kappa$ correspond in some sense to “wilder”
$\operatorname{SLE}_{\kappa}$ curves; in the $\kappa\geq 8$ regime the curve
is space-filling.

In this work, we study the $\kappa\to\infty$ asymptotic behavior of radial
$\operatorname{SLE}$. To simplify notation we consider
$\operatorname{SLE}_{\kappa}$ run on the time interval $[0,1]$ throughout the
paper, but our results are easily generalized to arbitrarily bounded time
intervals. Hence we denote by $\\{\cdot\\}$ the family
$\\{\cdot\\}_{t\in[0,1]}$ to avoid repeating indices.

Our first result (Proposition 1.1) characterizes the limit as
$\kappa\to\infty$ of the time-evolution of the $\operatorname{SLE}_{\kappa}$
hulls. We argue heuristically as follows. We view the time-dependent vector
field $\\{-z(z+\zeta_{t})/(z-\zeta_{t})\\}$ which generates the flow
$\\{g_{t}\\}$ as
$\\{\int_{S^{1}}-z(z+\zeta)/(z-\zeta)\delta_{B^{\kappa}_{t}}(\zeta)\\}$, where
$\delta_{B^{\kappa}_{t}}$ is the Dirac mass at $B^{\kappa}_{t}$. During a
short time interval where the flow is well-defined for the point $z$, we have
$g_{t}(z)\approx g_{t+\Delta t}(z)$ and hence

$\displaystyle\Delta g_{t}(z)$ $\displaystyle\approx\int_{t}^{t+\Delta
t}\int_{S^{1}}-g_{t}(z)(g_{t}(z)+\zeta)/(g_{t}(z)-\zeta)\delta_{B^{\kappa}_{s}}(\zeta)ds$
$\displaystyle=\int_{S^{1}}-g_{t}(z)(g_{t}(z)+\zeta)/(g_{t}(z)-\zeta)d(L^{\kappa}_{t+\Delta
t}(\zeta)-L^{\kappa}_{t}(\zeta)),$

where $L^{\kappa}_{t}$ is the _occupation measure_ (or _local time_) on
$S^{1}$ of $B^{\kappa}$ up to time $t$. We show that as $\kappa\to\infty$, the
driving function oscillates so quickly that its local time in $[t,t+\Delta t]$
is almost uniform on $S^{1}$, so in the limit we get a _measure-driven Loewner
chain_ with driving measure uniform on $S^{1}$. That is,

$\partial_{t}g_{t}(z)=\frac{1}{2\pi}\int_{S^{1}}-g_{t}(z)\frac{g_{t}(z)+\zeta}{g_{t}(z)-\zeta}d\zeta,$

where $d\zeta$ denotes the Lebesgue measure. This implies
$\partial_{t}g_{t}(z)=g_{t}(z)$, that is, $g_{t}(z)=e^{t}z$ or equivalently
$f_{t}(z)=e^{-t}z$. See Section 2 for more details on the measure-driven
Loewner chain. We show in Section 3.2:

######  Proposition 1.1.

As $\kappa\to\infty$, the Loewner chain $\\{f_{t}\\}$ driven by
$\\{\zeta_{t}\\}$ (defined in (1.1)) converges to $\\{z\mapsto e^{-t}z\\}$
almost surely, with respect to the uniform Carathéodory topology.

We shall mention that Loewner chains are also used in the study of the
Hastings-Levitov model of randomly aggregating particles and similar small-
particle limits have been studied, see [JVST12] and references therein.

The heuristic argument above suggests that the large deviations of
$\operatorname{SLE}_{\kappa}$ boil down to the large deviations of the
Brownian occupation measure, which we now describe.

For any metric space $X$, let $\mathcal{M}_{1}(X)$ denote the set of Borel
probability measures equipped with the Prokhorov topology (the topology of
weak convergence). Let

$\mathcal{N}=\\{\rho\in\mathcal{M}_{1}(S^{1}\times[0,1]):\rho(S^{1}\times
I)=|I|\text{ for all intervals }I\subset[0,1]\\}.$ (1.4)

The condition imposed here allows us to write $\rho\in\mathcal{N}$ as a
disintegration $\\{\rho_{t}\\}$ over the time interval $[0,1]$ (with
$\rho_{t}\in\mathcal{M}_{1}(S^{1})$ for a.e. $t$); see (2.1). We identify
$\rho$ and the time-indexed family $\\{\rho_{t}\\}$. The second result we show
is:

######  Theorem 1.2.

The process of measures $\\{\delta_{B^{\kappa}_{t}}\\}\in\mathcal{N}$
satisfies the large deviation principle with good rate function
$\mathcal{E}(\rho):=\int_{0}^{1}I(\rho_{t})\,dt$ for $\rho\in\mathcal{N}$,
where $I(\mu)$ is defined for each $\mu\in\mathcal{M}_{1}(S^{1})$ as

$I(\mu):=\frac{1}{2}\int_{S^{1}}|\phi^{\prime}(\zeta)|^{2}d\zeta$ (1.5)

if $\mu(d\zeta)=\phi^{2}(\zeta)\,d\zeta$ and $\phi$ is absolutely continuous,
and $I(\mu):=\infty$ otherwise. That is, for every closed set $C$ and open set
$G$ of $\mathcal{N}$,

$\displaystyle\limsup_{\kappa\to\infty}$
$\displaystyle\frac{1}{\kappa}\log\mathbb{P}\left[\\{\delta_{B^{\kappa}_{t}}\\}\in
C\right]\leq-\operatorname*{\vphantom{p}inf}_{\rho\in C}\mathcal{E}(\rho);$
$\displaystyle\operatorname*{\vphantom{p}lim\,inf}_{\kappa\to\infty}$
$\displaystyle\frac{1}{\kappa}\log\mathbb{P}\left[\\{\delta_{B^{\kappa}_{t}}\\}\in
G\right]\geq-\operatorname*{\vphantom{p}inf}_{\rho\in G}\mathcal{E}(\rho);$

and the sub-level set $\\{\rho\in\mathcal{N}:\mathcal{E}(\rho)\leq c\\}$ is
compact for all $c>0$.

Our proof is based on a result by Donsker and Varadhan [DV75] on large
deviations of the Brownian occupation measure (see Sections 3.3–3.4). The
$\kappa\to\infty$ large deviations of $\operatorname{SLE}$ then follows
immediately from the continuity of the Loewner transform (Theorem 2.2) and the
contraction principle [DZ10, Theorem 4.2.1].

######  Corollary 1.3.

The family of $\operatorname{SLE}_{\kappa}$ satisfies the $\kappa\to\infty$
large deviation principle with the good rate function

$I_{\operatorname{SLE}_{\infty}}(\\{K_{t}\\}):=\mathcal{E}(\rho),$

where $\\{\rho_{t}\\}$ is the driving measure whose Loewner transform is
$\\{K_{t}\\}$.

Let us conclude the introduction with two comments.

The study of large deviations of $\operatorname{SLE}$, while of inherent
interest, is also motivated by problems from complex analysis and geometric
function theory. In a forthcoming work [VW], Viklund and the third author
investigate the _duality_ between the rate functions of
$\operatorname{SLE}_{0+}$ (termed as the _Loewner energy_ introduced in
[Wan19a, RW19]) and $\operatorname{SLE}_{\infty}$ that is reminiscent of the
$\operatorname{SLE}$ duality [Dub09, Zha08] which couples
$\operatorname{SLE}_{\kappa}$ to the outer boundary of
$\operatorname{SLE}_{16/\kappa}$ for $\kappa<4$. Note that $\mathcal{E}(\rho)$
attains its minimum if and only if $\\{D_{t}\\}$ are concentric disks, and
$\\{\partial D_{t}\\}$ are circles which also have the minimal Loewner energy.

It is also natural to consider the large deviations of _chordal_
$\operatorname{SLE}_{\infty}$ (say, in $\mathbb{H}$ targeted at $\infty$).
However, in contrast with the radial case, the family indexed by $\kappa$ of
random measures $\\{\delta_{W_{\kappa t}}\\}$ on $\mathbb{R}\times[0,1]$ is
not tight and the corresponding Loewner flow converges to the identity map for
any fixed time $t$. To obtain a non-trivial limit, one needs to renormalize
appropriately (see e.g., Beffara’s thesis [Bef03, Sec.5.2] for a non-conformal
normalization) and consider generalized Stieltjes transformation of measures
for the large deviations. Therefore, for simplicity we choose to study the
radial case and suggest the large deviations of chordal
$\operatorname{SLE}_{\infty}$ as an interesting question. We will show a
simulation of large-$\kappa$ chordal SLEs and discuss some other questions at
the end of the paper.

The paper is organized as follows: In Section 2, we explain the measure-driven
radial Loewner evolution. In Section 3 we prove the main results of our paper.
In Section 4 we present some comments, observations and questions.

Acknowledgements:  We thank Wendelin Werner for helpful comments on the
presentation of our paper and Fredrik Viklund and Scott Sheffield for
inspiring discussions. We also thank an anonymous referee for comments on the
first version of this paper. M.A. and M.P. were partially supported by NSF
Award DMS 1712862. Y.W. was partially supported by the NSF Award DMS 1953945.

##  2 Measure-driven radial Loewner evolution

In this section we collect some known facts on the measure-driven Loewner
evolution (also known as Loewner-Kufarev evolution) that are essential to our
proofs. Recall that

$\mathcal{N}=\\{\rho\in\mathcal{M}_{1}(S^{1}\times[0,1]):\rho(S^{1}\times
I)=|I|\text{ for all intervals }I\subset[0,1]\\}$

endowed with the Prokhorov topology. From the disintegration theorem (see e.g.
[Bil95, Theorem 33.3]), for each measure $\rho\in\mathcal{N}$ there exists a
Borel measurable map $t\mapsto\rho_{t}$ (sending
$[0,1]\to\mathcal{M}_{1}(S^{1})$) such that for every measurable function
$\varphi:S^{1}\times[0,1]\rightarrow\mathbb{R}$ we have

$\int_{S^{1}\times[0,1]}\varphi(\zeta,t)\,d\rho=\int_{0}^{1}\int_{S^{1}}\varphi(\zeta,t)\,\rho_{t}(d\zeta)\,dt.$
(2.1)

We say $\\{\rho_{t}\\}$ is a _disintegration_ of $\rho$; it is unique in the
sense that any two disintegrations $\\{\rho_{t}\\},\\{\widetilde{\rho}_{t}\\}$
of $\rho$ must satisfy $\rho_{t}=\widetilde{\rho}_{t}$ for a.e. $t$. We always
denote by $\\{\rho_{t}\\}$ one such disintegration of $\rho\in\mathcal{N}$.

The Loewner chain driven by a measure $\rho\in\mathcal{N}$ is defined
similarly to (1.2). For $z\in\mathbb{D}$, consider the _Loewner-Kufarev ODE_

$\partial_{t}g_{t}(z)=-g_{t}(z)\int_{S^{1}}\frac{g_{t}(z)+\zeta}{g_{t}(z)-\zeta}\,\rho_{t}(d\zeta)$

with the initial condition $g_{0}(z)=z$. Let $T_{z}$ be the supremum of all
$t$ such that the solution is well-defined up to time $t$ with
$g_{t}(z)\in\mathbb{D}$, and $D_{t}:=\\{z\in\mathbb{D}:T_{z}>t\\}$ is a simply
connected open set containing $0$. We define the _hull_
$K_{t}:=\mathbb{D}\setminus D_{t}$ associated with the Loewner chain. Note
that when $\kappa\geq 8$, the family $\\{\gamma[0,t]\\}$ of radial
$\operatorname{SLE}_{\kappa}$ is exactly the family of hulls $\\{K_{t}\\}$
driven by the measure $\\{\delta_{B^{\kappa}_{t}}\\}$.

The function $g_{t}$ defined above is the unique conformal map of $D_{t}$ onto
$\mathbb{D}$ such that $g_{t}(0)=0$ and $g_{t}^{\prime}(0)>0$; moreover
$g_{t}^{\prime}(0)=e^{t}$ (i.e. $D_{t}$ has conformal radius $e^{-t}$ seen
from $0$) since $\partial_{t}\log g_{t}^{\prime}(0)=|\rho_{t}|=1$ (see e.g.
[Law05, Thm. 4.13]).

If $g_{t}$ is the solution of a Loewner-Kufarev ODE then its inverse
$f_{t}=g_{t}^{-1}$ satisfies the _Loewner-Kufarev PDE_ :

$\partial_{t}f_{t}(z)=zf_{t}^{\prime}(z)\int_{S^{1}}\frac{z+\zeta}{z-\zeta}\rho_{t}(d\zeta),$

and vice versa. Note that $f_{t}(0)=0$, $f_{t}^{\prime}(0)=e^{-t}$, and
$f_{t}(\mathbb{D})=D_{t}\subset f_{s}(\mathbb{D})$ for $s\leq t$. Such a time-
indexed family $\\{f_{t}\\}$ is called _a normalized chain of subordinations_.
We write $\mathcal{S}$ for the set of normalized chains of subordinations
$\\{f_{t}\\}$ on $[0,1]$. An element of $\mathcal{S}$ can be equivalently
represented by either $\\{f_{t}\\}$ or the process of hulls $\\{K_{t}\\}$. The
map $\mathcal{L}:\rho\mapsto\\{f_{t}\\}$ (or interchangeably
$\mathcal{L}:\rho\mapsto\\{K_{t}\\}$) is called the _Loewner transform_. In
fact, $\mathcal{L}$ is a bijection:

######  Theorem 2.1 (Bijectivity of the Loewner transform [Pom65, Satz 4]).

The family $(f_{t})_{t\in[0,1]}$ is a normalized chain of subordination over
$[0,1]$ if and only if

  * •

$t\mapsto f_{t}(z)$ is absolutely continuous in $[0,1]$ and for all $r<1$,
there is $K(r)$ > 0 such that $|f_{t}(z)-f_{s}(z)|\leq K(r)|t-s|$ for all
$z\in r\mathbb{D}$;

  * •

and there is a ($t$-a.e. unique) function $h(z,t)$ that is analytic in $z$,
measurable in $t$ with $h(0,t)=1$ and $\operatorname{Re}h(z,t)>0$, so that for
$t$-a.e. we have

$\partial_{t}f_{t}(z)=-zf_{t}^{\prime}(z)h(z,t).$

From the Herglotz representation of $h(\cdot,t)$, there exists a unique
$\rho_{t}\in\mathcal{M}_{1}(S^{1})$ such that

$h(z,t)=\int_{S^{1}}\frac{\zeta+z}{\zeta-z}\rho_{t}(d\zeta),\quad\forall
z\in\mathbb{D}.$

Therefore $\\{f_{t}\\}$ satisfies the Loewner PDE driven by the (a.e. uniquely
determined) measurable function $t\mapsto\rho_{t}$.

We now equip $\mathcal{S}$ with a topology. View $\mathcal{S}$ as the set of
normalized chains of subordinations $\\{f_{t}\\}$ on $[0,1]$, and change
notation by writing $f(z,t)=f_{t}(z)$. We endow $\mathcal{S}$ with the
topology of uniform convergence of $f$ on compact subsets of
$\mathbb{D}\times[0,1]$. (Equivalently, if we view $\mathcal{S}$ as the set of
processes of hulls $\\{K_{t}\\}$, this is the topology of _uniform
Carathéodory convergence_.) The continuity of $\mathcal{L}$ has been, e.g.,
derived in [MS16b, Proposition 6.1] (see also [JVST12]).

######  Theorem 2.2 (Continuity).

The Loewner transform $\mathcal{L}:\mathcal{N}\to\mathcal{S}$ is a
homeomorphism.

##  3 Proofs of the main results

In this section, we study the random measure
$\\{\delta_{B^{\kappa}_{t}}\\}\in\mathcal{N}$. In Section 3.1 we approximate
$\mathcal{N}$ by spaces of time-averaged measures. In Section 3.2 we verify
that $\\{\delta_{B^{\kappa}_{t}}\\}\in\mathcal{N}$ converges almost surely as
$\kappa\to\infty$ to the uniform measure on $S^{1}\times[0,1]$; this yields
Proposition 1.1. In Section 3.3, we review the large deviation principle for
the circular Brownian motion occupation measure, which is a special case of
seminal work of Donsker and Varadhan [DV75]. Finally, in Section 3.4 we prove
Theorem 1.2, the large deviation principle for
$\\{\delta_{B^{\kappa}_{t}}\\}\in\mathcal{N}$.

###  3.1 Time-discretized approximations of measures

We emphasize that the results of this section are wholly deterministic.

For $n\geq 0$, let $\mathcal{I}_{n}:=\\{0,1,2,\cdots,2^{n}-1\\}$ be an index
set, and define
$\mathcal{Y}_{n}:=\left(\mathcal{M}_{1}(S^{1})\right)^{\mathcal{I}_{n}}.$ We
note that $\mathcal{Y}_{n}$ is endowed with the product topology. For each
$i\in\mathcal{I}_{n}$ we define a function
$P_{n}^{i}:\mathcal{N}\to\mathcal{M}_{1}(S^{1})$ via

$P_{n}^{i}(\rho):=2^{n}\int_{i/2^{n}}^{(i+1)/2^{n}}\rho_{t}\ dt,$ (3.1)

where here $\\{\rho_{t}\\}$ is a disintegration of $\rho$ with respect to $t$,
as in (2.1). We define also the map $P_{n}:\mathcal{N}\to\mathcal{Y}_{n}$ via
$P_{n}=(P_{n}^{i})_{i\in\mathcal{I}_{n}}$. That is, $P_{n}$ averages $\rho$
along each $2^{-n}$-time interval, and outputs the $2^{n}$-tuple of these
$2^{n}$ time-averages.

We consider $\mathcal{Y}_{n}$ to be the space of time-discretized
approximations of $\mathcal{N}$, in the following sense. Define a map
$F_{n}:\mathcal{Y}_{n}\to\mathcal{N}$ via

$F_{n}\left((\mu_{i})_{i\in\mathcal{I}_{n}}\right):=\sum_{i\in\mathcal{I}_{n}}\mu_{i}\otimes\operatorname{Leb}_{[i/2^{n},(i+1)/2^{n}]}.$

Then one can view $F_{n}(P_{n}(\rho))$ as a “level-$n$ approximation” of
$\rho$ (see Lemma 3.1).

We have provided a way of projecting an element of $\mathcal{N}$ to the space
of level-$n$ approximations $\mathcal{Y}_{n}$. Now we write down a map
$P_{n,n+1}:\mathcal{Y}_{n+1}\to\mathcal{Y}_{n}$ which takes in a finer
approximation and outputs a coarser approximation:

$P_{n,n+1}\left((\mu_{i})_{i\in\mathcal{I}_{n+1}}\right):=\left(\frac{\mu_{0}+\mu_{1}}{2},\dots,\frac{\mu_{2^{n+1}-2}+\mu_{2^{n+1}-1}}{2}\right).$

That is, we average pairs of components of $\mathcal{Y}_{n+1}$. It is clear
that

$P_{n}=P_{n,n+1}\circ P_{n+1}.$ (3.2)

The convergence of $P_{n}(\rho_{j})\xrightarrow{j\to\infty}P_{n}(\rho)$ in
$\mathcal{Y}_{n}$ is equivalent to the convergence
$\rho_{j}(f)\xrightarrow{j\to\infty}\rho(f)$ for the functions $f$ which are
piecewise constant in time for each time interval $(i/2^{n},(i+1)/2^{n})$. For
each fixed $n$, this is a coarser topology than that of $\mathcal{N}$. The
following lemma shows that the $n\to\infty$ topology agrees with that of
$\mathcal{N}$.

######  Lemma 3.1.

We have $\mathcal{N}=\varprojlim\mathcal{Y}_{n}.$ That is, as topological
spaces, $\mathcal{N}$ is the projective (inverse) limit of $\mathcal{Y}_{n}$
as $n\to\infty$.

###### Proof.

Let $\mathcal{Y}=\varprojlim\mathcal{Y}_{n}$; this is the subset of
$\prod_{j=0}^{\infty}\mathcal{Y}_{j}$ comprising elements
$(y^{0},y^{1},\dots)$ such that $P_{n,n+1}(y^{n+1})=y^{n}$ for all $n\geq 0$.
The topology on $\mathcal{Y}$ is inherited from
$\prod_{j=0}^{\infty}\mathcal{Y}_{j}$. Because of the coherence relation
(3.2), we can define a map $P:\mathcal{N}\to\mathcal{Y}$ by
$P(\rho):=(P_{j}(\rho))_{j\geq 0}.$ We now show that $P$ is a homeomorphism.

Showing that $P$ is continuous. Since the topology on $\mathcal{Y}$ is
inherited from the product topology on $\prod_{n=0}^{\infty}\mathcal{Y}_{j}$,
it suffices to show that the map
$P:\mathcal{N}\to\prod_{n=0}^{\infty}\mathcal{Y}_{n}$ is continuous, i.e.
$P_{n}:\mathcal{N}\to\mathcal{Y}_{n}$ is continuous for each $n$. But this is
clear: if two measures in $\mathcal{N}$ are close in the Prokhorov topology,
then so is the time-average of these measures on a time interval.

Showing that $P$ is a bijection. Fix $f\in C(S^{1}\times[0,1])$. We claim that
for any $\varepsilon>0$, there exists $n_{0}=n_{0}(f,\varepsilon)$ such that
for all $m,n\geq n_{0}$ and $y=(y^{0},y^{1},\dots)\in\mathcal{Y}$ we have

$\left|(F_{m}(y^{m}))(f)-(F_{n}(y^{n}))(f)\right|<\varepsilon.$ (3.3)

To that end we note that $f$ is uniformly continuous; we can choose $\delta>0$
so that $|f(\zeta,t)-f(\zeta,t^{\prime})|<\varepsilon$ whenever
$|t-t^{\prime}|<\delta$. Choosing $n_{0}$ such that $2^{-n_{0}}<\delta$, we
obtain (3.3).

Now we show that $P$ is a bijection. By (3.3), for each
$y=(y^{0},y^{1},\dots)\in\mathcal{Y}$ we can define a bounded linear
functional $T_{y}:C(S^{1}\times[0,1])\to\mathbb{R}$ via

$T_{y}(f)=\lim_{n\to\infty}(F_{n}(y^{n}))(f)\quad\text{ for
}y=(y^{0},y^{1},\dots).$

Clearly $T_{y}$ maps nonnegative functions to nonnegative reals, so the Riesz-
Markov-Kakutani representation theorem tells us there is a unique111The
representation theorem yields a unique _regular_ Borel measure, but since
$S^{1}\times[0,1]$ is compact, _all_ Borel measures on $S^{1}\times[0,1]$ are
regular. Borel measure $\rho$ on $S^{1}\times[0,1]$ such that
$\rho(f)=T_{y}(f)$ for all $f\in C^{1}(S^{1}\times[0,1])$; it is easy to check
that $\rho\in\mathcal{N}$. Thus, for each $y\in\mathcal{Y}$ the equation
$P(\rho)=y$ has a unique solution in $\mathcal{N}$, so $P$ is a bijection.

Concluding that $P$ is a homeomorphism. We note that $\mathcal{N}$ is compact
(since $S^{1}\times[0,1]$ is compact) and $\mathcal{Y}$ is compact (since each
$\mathcal{Y}_{n}$ is compact and Hausdorff). Since $P$ is a continuous
bijection of compact sets, it is a homeomorphism. ∎

###  3.2 Almost sure limit of SLE driving measures

Consider a Brownian motion $B^{\kappa}_{t}$ on the unit circle
$S^{1}=\\{\zeta\in\mathbb{C}:\left\lvert\zeta\right\rvert=1\\}$ started at $1$
with variance $\kappa$ as (1.1). Define the _occupation measure_ of
$B^{\kappa}_{t}$:

$L^{\kappa}_{t}(A)=\int_{0}^{t}\mathbf{1}\\{B^{\kappa}_{s}\in
A\\}\,ds\quad\text{ for Borel sets }A\subset S^{1}.$

Let $\overline{L}^{\kappa}_{t}=t^{-1}L^{\kappa}_{t}$ be the _average
occupation measure_ of $B^{\kappa}$ at time $t$ (its normalization gives
$\overline{L}^{\kappa}_{t}\in\mathcal{M}_{1}(S^{1})$). An easy consequence of
the ergodic theorem is the following almost sure $t\to\infty$ limit of
$\overline{L}^{1}_{t}$; we include the proof for completeness.

######  Lemma 3.2.

Almost surely, as $t\to\infty$ we have
$\overline{L}^{1}_{t}\to(2\pi)^{-1}\operatorname{Leb}_{S^{1}}$ in
$\mathcal{M}(S^{1})$.

###### Proof.

It suffices to show that for any continuous function $f:S^{1}\to\mathbb{R}$ we
have almost surely

$\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}f(B^{1}_{s})\
ds=\frac{1}{2\pi}\int_{S^{1}}f(\zeta)\ d\zeta.$ (3.4)

Given this, by choosing a suitable countable collection of functions, we
obtain the lemma.

Let $(\Omega,\mathbb{P})$ be a Wiener space so that $\mathbb{P}$ is the law of
$B^{1}_{t}$. Consider the expanded probability space given by $(\Omega\times
S^{1},\mathbb{P}\otimes(2\pi)^{-1}\operatorname{Leb}_{S^{1}})$, and let
$(\omega,e^{i\theta})$ correspond to the random path
$e^{i\theta}B^{1}_{t}(\omega)$. That is, after sampling an instance of
Brownian motion $B^{1}_{t}(\omega)$ started at $1$, we apply an independent
uniform rotation to the circle so the Brownian motion starts at $e^{i\theta}$
instead. A consequence of Birkhoff’s ergodic theorem is that for a.e.
$(\omega,e^{i\theta})\in\Omega\times S^{1}$, we have

$\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}f(e^{i\theta}B^{1}_{s}(\omega))\
ds=\frac{1}{2\pi}\int_{S^{1}}f(\zeta)\ d\zeta.$ (3.5)

Equivalently, for a.e. $e^{i\theta}\in S^{1}$, we have (3.5) for a.e.
$\omega$. Taking a sequence of $e^{i\theta}$ converging to $1$ and using the
uniform continuity of $f$, we obtain (3.4). This concludes the proof of Lemma
3.2. ∎

Now we justify the heuristic argument in the introduction, which said that as
$\kappa\to\infty$, the Brownian motion $B^{\kappa}_{t}$ moves so quickly that
the driving measure converges to
$(2\pi)^{-1}\operatorname{Leb}_{S^{1}}\otimes\operatorname{Leb}_{[0,1]}$.

######  Lemma 3.3.

As $\kappa\to\infty$, $\\{\delta_{B^{\kappa}_{t}}\\}$ converges almost surely
to $(2\pi)^{-1}\operatorname{Leb}_{S^{1}}\otimes\operatorname{Leb}_{[0,1]}$ in
$\mathcal{N}$.

###### Proof.

Lemma 3.1 states that $\mathcal{N}$ is the projective limit of the spaces
$\mathcal{Y}_{n}$ defined in Section 3.1, with projection map from
$\mathcal{N}$ to $\mathcal{Y}_{n}$ given by
$(P^{i}_{n})_{i\in\mathcal{I}_{n}}$. It thus suffices to show that as
$\kappa\to\infty$, the random measure
$P^{i}_{n}(\\{\delta_{B^{\kappa}_{t}}\\})$ converges almost surely to
$P^{i}_{n}((2\pi)^{-1}\operatorname{Leb}_{S^{1}}\otimes\operatorname{Leb}_{[0,1]})=(2\pi)^{-1}\operatorname{Leb}_{S^{1}}$
in the Prokhorov topology, namely,

$\lim_{\kappa\to\infty}2^{n}\int_{i/2^{n}}^{(i+1)/2^{n}}\delta_{B^{\kappa}_{t}}\
dt=\frac{1}{2\pi}\operatorname{Leb}_{S^{1}}.$

This is true since Lemma 3.2 tells us that almost surely

$\lim_{\kappa\to\infty}\frac{2^{n}}{i}\int_{0}^{i/2^{n}}\delta_{B^{\kappa}_{t}}\
dt=\lim_{\kappa\to\infty}\frac{2^{n}}{i+1}\int_{0}^{(i+1)/2^{n}}\delta_{B^{\kappa}_{t}}\
dt=\frac{1}{2\pi}\operatorname{Leb}_{S^{1}}.$

Hence Lemma 3.3 holds. ∎

###### Proof of Proposition 1.1.

It follows immediately from Theorem 2.2 and Lemma 3.3. ∎

###  3.3 Large deviation principle of occupation measures

In this section, we discuss the large deviation principle of Brownian motion
occupation measures on $S^{1}$ as $\kappa\to\infty$.

Recall that $\overline{L}^{\kappa}_{t}=t^{-1}L^{\kappa}_{t}$ is the average
occupation measure of $B^{\kappa}$ at time $t$. By Brownian scaling we have
that (recall that the upper index is diffusivity and the lower index is time)
$\overline{L}^{1}_{\kappa t}$ and $\overline{L}^{\kappa}_{t}$ equal in law, so
it suffices to understand the large deviation principle for
$\overline{L}^{1}_{t}$ as $t\to\infty$. This follows from a more general
result of Donsker and Varadhan; we state the result for Brownian motion on
$S^{1}$.

######  Theorem 3.4 ([DV75, Theorem 3]).

Define $\tilde{I}:\mathcal{M}_{1}(S^{1})\to\mathbb{R}_{\geq 0}$ by

$\tilde{I}(\mu):=-\operatorname*{\vphantom{p}inf}_{u>0,\,u\in
C^{2}(S^{1})}\int_{S^{1}}\frac{u^{\prime\prime}}{2u}(\zeta)\mu(d\zeta)=-\operatorname*{\vphantom{p}inf}_{u>0,\,u\in
C^{2}(S^{1})}\int_{S^{1}}\frac{L(u)}{u}(\zeta)\mu(d\zeta),$ (3.6)

where $L(u)=u^{\prime\prime}/2$ is the infinitesimal generator of the Brownian
motion on $S^{1}$. The average occupation measure $\overline{L}^{\kappa}_{1}$
admits a large deviation principle as $\kappa\to\infty$, with rate function
$\tilde{I}$. That is, for any closed set $C\subset\mathcal{M}_{1}(S^{1})$,

$\limsup_{\kappa\to\infty}\frac{1}{\kappa}\log\mathbb{P}[\overline{L}^{\kappa}_{1}\in
C]\leq-\operatorname*{\vphantom{p}inf}_{\mu\in C}\tilde{I}(\mu),$ (3.7)

and for any open set $G\in\mathcal{M}_{1}(S^{1})$,

$\operatorname*{\vphantom{p}lim\,inf}_{\kappa\to\infty}\frac{1}{\kappa}\log\mathbb{P}[\overline{L}^{\kappa}_{1}\in
G]\geq-\operatorname*{\vphantom{p}inf}_{\mu\in G}\tilde{I}(\mu).$ (3.8)

Moreover, $\tilde{I}$ is good, lower-semicontinuous, and convex.

Note that the lower-semicontinuity (hence the goodness, since
$\mathcal{M}_{1}(S^{1})$ is compact) and convexity follow directly from the
expression of $\tilde{I}$. For the convenience of those readers who may not be
so familiar with the statement of Theorem 3.4, let us provide an outline of
the proof of the upper bound (3.7) in order to explain where this rate
function comes from.

Let $P_{\zeta}$ denote the law of a Brownian motion $B^{1}$ on $S^{1}$
starting from $\zeta\in S^{1}$ and $Q_{\zeta,t}$ the law of the average
occupation measure $\overline{L}^{1}_{t}$ under $P_{\zeta}$. Fix a small
number $h>0$, and let $\pi_{h}(\zeta,d\xi)$ be the law of $B_{h}$ under
$P_{\zeta}$. We consider the Markov chain $X_{n}:=B_{nh}$, so that $\pi_{h}$
is the transition kernel of $X$. We write $\mathbb{E}$ for the expectation
with respect to $P_{1}$.

Now let $u\in C^{2}(S^{1})$ such that $u>0$. From the Markov property, we
inductively get

$\displaystyle\mathbb{E}\left[\frac{u(X_{0})\cdots
u(X_{n-1})}{\pi_{h}u(X_{0})\cdots\pi_{h}u(X_{n-1})}u(X_{n})\right]$
$\displaystyle=\mathbb{E}[u(X_{0})]=u(1).$

Since the Brownian motion is a Feller process with infinitesimal generator
$L$, we have

$\log\frac{u(\zeta)}{\pi_{h}u(\zeta)}=\log\left(1-h\frac{Lu(\zeta)}{u(\zeta)}+o(h)\right)=-h\frac{Lu(\zeta)}{u(\zeta)}+o(h).$

Therefore,

$\displaystyle u(1)$
$\displaystyle=\mathbb{E}\left[\exp\left(-\sum_{i=0}^{n}h\frac{Lu(X_{i})}{u(X_{i})}+o(h)\right)\pi_{h}u(X_{n})\right]$
$\displaystyle=\mathbb{E}\left[\exp\left(-\int_{0}^{t}\frac{Lu(B_{s})}{u(B_{s})}ds\right)\pi_{h}u(B_{t})+o(1)\right],$

where $n$ is chosen to be the integer part of $t/h$. Hence,

$\displaystyle\mathbb{E}^{Q_{1,t}}\left[\exp\left(-t\int_{S^{1}}\frac{Lu(\zeta)}{u(\zeta)}\overline{L}^{1}_{t}(d\zeta)\right)\right]$
$\displaystyle\leq\frac{u(1)}{\operatorname*{\vphantom{p}inf}_{\xi\in
S^{1}}\pi_{h}u(\xi)}\leq\frac{u(1)}{\operatorname*{\vphantom{p}inf}_{\xi\in
S^{1}}u(\xi)}\leq M(u)$

for some $M(u)<\infty$ depending only on the function $u>0$. For any
measurable set $C\subset\mathcal{M}_{1}(S^{1})$, since

$\displaystyle M(u)$
$\displaystyle\geq\mathbb{E}^{Q_{1,t}}\left[\exp\left(-t\int_{S^{1}}\frac{Lu(\zeta)}{u(\zeta)}\overline{L}^{1}_{t}(d\zeta)\right)\right]$
$\displaystyle\geq Q_{1,t}(C)\exp\left(-t\sup_{\mu\in
C}\int_{S^{1}}\frac{Lu}{u}(\zeta)\,\mu(d\zeta)\right)$

for arbitrary $u$, we have

$\limsup_{t\to\infty}\frac{1}{t}\log
Q_{1,t}(C)\leq\operatorname*{\vphantom{p}inf}_{u>0,u\in C^{2}}\sup_{\mu\in
C}\int_{S^{1}}\frac{Lu}{u}(\zeta)\,\mu(d\zeta).$

When $C$ is closed (hence compact), some topological considerations allow us
to swap the $\operatorname*{\vphantom{p}inf}$ and $\sup$ in the above
expression, and we obtain

$\displaystyle\adjustlimits{\operatorname*{\vphantom{p}inf}}_{u>0,u\in
C^{2}}{\sup}_{\mu\in C}\int_{S^{1}}\frac{Lu}{u}(\zeta)\mu(d\zeta)$
$\displaystyle\leq\sup_{\mu\in C}\operatorname*{\vphantom{p}inf}_{u>0,u\in
C^{2}}\int_{S^{1}}\frac{Lu}{u}(\zeta)\mu(d\zeta)=-\operatorname*{\vphantom{p}inf}_{\mu\in
C}\tilde{I}(\mu),$

which is the upper bound (3.7). As it is often the case in the derivation of
large deviation principles, the lower bound turns out to be trickier, and uses
approximation by discrete time Markov chains and a change of measure argument.
We refer to the original paper [DV75] for more details.

The rate function $\tilde{I}$ of Theorem 3.4 is somewhat unwieldy but can be
simplified for Brownian motion as noted in [DV75]. We provide here an
alternative elementary proof.

######  Theorem 3.5 ([DV75, Theorem 5]).

For $\mu\in\mathcal{M}_{1}(S^{1})$, the rate function $\tilde{I}(\mu)$ is
finite if and only if $\mu=\phi^{2}(\zeta)d\zeta$ for some function $\phi\in
W^{1,2}$. In this case, we have $\tilde{I}(\mu)=I(\mu)$, where

$I(\mu)=\frac{1}{2}\int_{S^{1}}|\phi^{\prime}(\zeta)|^{2}\,d\zeta.$

###### Proof.

First assume that $\mu=\phi^{2}d\zeta$ for some $\phi\in W^{1,2}$ and that
$I(\mu)$ is finite, we will show that $\tilde{I}(\mu)=I(\mu)$. For this, take
a sequence $\phi_{n}\in C^{2}$ with $\phi_{n}>0$ converging to $\phi$ almost
everywhere such that
$\int_{S^{1}}(\phi_{n}^{\prime})^{2}\,d\zeta\rightarrow\int_{S^{1}}(\phi^{\prime})^{2}\,d\zeta$.
For any $u\in C^{2}$ and any $\varepsilon>0$, we have for sufficiently large
$n$ that

$\int_{S^{1}}\frac{u^{\prime\prime}}{2u}\phi^{2}\,d\zeta+\varepsilon\geq\int_{S^{1}}\frac{(v\phi_{n})^{\prime\prime}}{2(v\phi_{n})}\phi_{n}^{2}\,d\zeta=\int_{S^{1}}\frac{\phi_{n}^{\prime\prime}\phi_{n}}{2}\,d\zeta+\int_{S^{1}}\frac{(\phi_{n}^{2}v^{\prime})^{\prime}}{2v}\,d\zeta,$

where $v:=u/\phi_{n}\in C^{2}$. From integration by parts, this latter
expression equals to

$-\frac{1}{2}\int_{S^{1}}|\phi_{n}^{\prime}|^{2}\,d\zeta+\frac{1}{2}\int_{S^{1}}\frac{\phi_{n}^{2}v^{\prime
2}}{v^{2}}\,d\zeta\geq-\frac{1}{2}\int_{S^{1}}|\phi_{n}^{\prime}|^{2}\,d\zeta\geq-I(\mu)-\varepsilon$

by taking $n$ larger if necessary. Since $\varepsilon$ is arbitrary, we obtain
$-\int_{S^{1}}\frac{u^{\prime\prime}}{2u}\phi^{2}\,d\zeta\leq I(\mu)$, and
thus $\tilde{I}(\mu)\leq I(\mu)$ by taking supremum over $u$. The opposite
inequality can be shown by taking $u=\phi_{n}$ (i.e. $v=1$) and sending
$n\to\infty$. Therefore $\tilde{I}(\mu)=I(\mu)$ when $I(\mu)<\infty$.

It remains to prove that if $\tilde{I}(\mu)<\infty$ then $I(\mu)<\infty$, so
consider $\mu$ such that $\tilde{I}(\mu)$ is finite. Let
$\\{\eta_{\varepsilon}\\}_{\varepsilon>0}$ be a family of nonnegative smooth
functions on $S^{1}$ with $\int_{S^{1}}\eta_{\varepsilon}\,d\zeta=1$ and
converging weakly to the Dirac delta function at $1$ as $\varepsilon\to 0$.
Writing $\mu^{\xi}$ for $\mu$ rotated by $\xi\in S^{1}$, we define
$\mu_{\varepsilon}=\int_{S^{1}}\eta_{\varepsilon}(\xi)\mu^{\xi}d\xi$ as a
weighted average of probability measures so that $\mu_{\varepsilon}$ converges
weakly to $\mu$. Observe that $\tilde{I}$ is rotation invariant and convex.
Therefore by Jensen’s inequality,

$\tilde{I}(\mu_{\varepsilon})=\tilde{I}\left(\int_{S^{1}}\eta_{\varepsilon}(\xi)\mu^{\xi}\,d\xi\right)\leq\int_{S^{1}}\eta_{\varepsilon}(\xi)\tilde{I}(\mu^{\xi})\,d\xi=\tilde{I}(\mu).$

Write $\phi_{\varepsilon}:=\sqrt{\eta_{\varepsilon}*\mu}$, so that
$\mu_{\varepsilon}=\phi_{\varepsilon}^{2}(\zeta)d\zeta$. We will show that

$\int_{S^{1}}(\phi_{\varepsilon}^{\prime})^{2}\,d\zeta\leq
2\tilde{I}(\mu_{\varepsilon}).$ (3.9)

Given (3.9), we see that
$\int_{S^{1}}(\phi_{\varepsilon}^{\prime})^{2}\,d\zeta$ is uniformly bounded
above (by $2\tilde{I}(\mu)$); letting $\varepsilon\to 0$ implies $\mu$ is an
absolutely continuous measure, and furthermore $\sqrt{\mu(d\zeta)/d\zeta}\in
W^{1,2}$ so $I(\mu)<\infty$, concluding the proof of the theorem.

We turn to the proof of (3.9), which follows the argument of [DV80, Lemma
2.4]. In the definition (3.6), take $u=e^{\lambda h}$ where $h$ is smooth and
$\lambda$ is a real number. This gives

$\lambda^{2}\int_{S^{1}}h^{\prime
2}\phi_{\varepsilon}^{2}\,d\zeta+\lambda\int_{S^{1}}h^{\prime\prime}\phi_{\varepsilon}^{2}\,d\zeta=\lambda^{2}\int_{S^{1}}h^{\prime
2}\phi_{\varepsilon}^{2}\,d\zeta-2\lambda\int_{S^{1}}h^{\prime}\phi^{\prime}_{\varepsilon}\phi_{\varepsilon}\,d\zeta\geq-2\tilde{I}(\mu_{\varepsilon})$

which holds for any real number $\lambda$. By choosing $\lambda$ so that the
quadratic function takes the minimum, we have

$\left(\int_{S^{1}}h^{\prime}\phi_{\varepsilon}^{\prime}\phi_{\varepsilon}\,d\zeta\right)^{2}\leq
2\tilde{I}(\mu_{\varepsilon})\int_{S^{1}}h^{\prime
2}\phi_{\varepsilon}^{2}\,d\zeta.$ (3.10)

For $n\in\mathbb{N}$, consider an auxiliary function $\nu_{n}$ on positive
real numbers defined as

$\nu_{n}(x):=\begin{cases}0&\text{if }0<x\leq 1/2n\\\ 1/x&\text{if }x\geq
1/n\end{cases}$

and extended on $[1/2n,1/n]$ so that $0\leq\nu_{n}(x)\leq 1/x$ and $\nu_{n}$
is smooth for all $x$. And define $V_{n}(x)=\int_{0}^{x}\nu_{n}(y)\,dy$.
Plugging $h=V_{n}(\phi_{\varepsilon})$ to (3.10) gives

$\left(\int_{S^{1}}\nu_{n}(\phi_{\varepsilon})\phi_{\varepsilon}^{\prime
2}\phi_{\varepsilon}\,d\zeta\right)^{2}\leq
2\tilde{I}(\mu_{\varepsilon})\int_{S^{1}}\nu_{n}^{2}(\phi_{\varepsilon})\phi_{\varepsilon}^{\prime
2}\phi_{\varepsilon}^{2}\,d\zeta\leq
2\tilde{I}(\mu_{\varepsilon})\int_{S^{1}}\nu_{n}(\phi_{\varepsilon})\phi_{\varepsilon}^{\prime
2}\phi_{\varepsilon}\,d\zeta$

where $\nu_{n}(\phi_{\varepsilon})\leq 1/\phi_{\varepsilon}$ was used and the
common terms on both sides cancel out. As $n\to\infty$, Fatou’s lemma implies
(3.9) as desired. ∎

###  3.4 Large deviations for $\\{\delta_{B^{\kappa}_{t}}\\}$

In this section, we prove Theorem 1.2. That is, we establish the large
deviation principle for the Brownian trajectory measure
$\\{\delta_{B^{\kappa}_{t}}\\}$. We use the notation of Section 3.1.

The first step is the large deviation principle for
$P_{n}(\\{\delta_{B^{\kappa}_{t}}\\})$, which follows easily from Theorem 3.4.
Recall that $P_{n}(\\{\delta_{B^{\kappa}_{t}}\\})$ is a $2^{n}$-tuple of
elements of $\mathcal{M}_{1}(S^{1})$, the $i$th element being the average of
$\\{\delta_{B^{\kappa}_{t}}\\}$ on the time interval $[i/2^{n},(i+1)/2^{n}]$.

######  Lemma 3.6.

Fix $n\geq 1$. The random variable
$P_{n}(\\{\delta_{B^{\kappa}_{t}}\\})\in\mathcal{Y}_{n}$ satisfies the large
deviation principle as $\kappa\to\infty$, with good rate function
$I_{n}:\mathcal{Y}_{n}\to\mathbb{R}$ defined by

$I_{n}((\mu^{i})_{i\in\mathcal{I}_{n}}):=\frac{1}{2^{n}}\sum_{i\in\mathcal{I}_{n}}I(\mu_{i}),$
(3.11)

where $I:\mathcal{M}_{1}(S^{1})\to\mathbb{R}$ is the good rate function
defined in (1.5).

###### Proof.

Since the large deviation rate function $I$ is rotation invariant, the same
rate function is applicable to the setting of the occupation measure of
Brownian motion started at any $\zeta\in S^{1}$. Furthermore, the Markov
property of Brownian motion tells us that conditioned on the value
$B^{\kappa}_{j/2^{n}}$, the process
$\left(B^{\kappa}_{t}\right)_{[j/2^{n},(j+1)/2^{n}]}$ is independent of
$\left(B^{\kappa}_{t}\right)_{[0,j/2^{n}]}$. These observations, together with
Theorem 3.4, yield the lemma. ∎

Since $\mathcal{N}=\varprojlim\mathcal{Y}_{n}$, we can deduce the large
deviation principle for $\\{\delta_{B^{\kappa}_{t}}\\}$.

######  Proposition 3.7.

The random measure $\\{\delta_{B^{\kappa}_{t}}\\}\in\mathcal{N}$ has the large
deviation principle with good rate function

$\sup\nolimits_{n\geq 0}I_{n}(P_{n}(\rho))\quad\text{ for
}\rho\in\mathcal{N},$

where $I_{n}:\mathcal{Y}_{n}\to\mathbb{R}$ is defined in (3.11).

###### Proof.

This follows from the Dawson-Gärtner theorem [DG87] (or [DZ10, Thm 4.6.1]),
the fact that $\mathcal{N}=\varprojlim\mathcal{Y}_{n}$ by Lemma 3.1, and the
large deviation principle for $P_{n}(\\{\delta_{B^{\kappa}_{t}}\\})$ (Lemma
3.6). ∎

Finally, we can simplify the rate function $\sup_{n\geq 0}I_{n}(P_{n}(\rho))$.

######  Lemma 3.8.

Define $\mathcal{E}:\mathcal{N}\to\mathbb{R}$ by
$\mathcal{E}(\rho):=\int_{0}^{1}I(\rho_{t})\,dt,$ where $\\{\rho_{t}\\}$ is
any disintegration of $\rho$ with respect to $t$ (see (2.1)). Then, with
$I_{n}:\mathcal{Y}_{n}\to\mathbb{R}$ defined as in (3.11), we have

$\mathcal{E}(\rho)=\sup\nolimits_{n\geq 0}I_{n}(P_{n}(\rho)).$

###### Proof.

By definition we have
$P_{n}^{i}(\rho)=\frac{1}{2}(P_{n+1}^{2i}(\rho)+P_{n+1}^{2i+1}(\rho))$, so
Jensen’s inequality applied to the convex function $I$ yields
$I_{n}(P_{n}(\rho))\leq I_{n+1}(P_{n+1}(\rho))$, and hence

$\sup\nolimits_{n\geq
0}I_{n}(P_{n}(\rho))=\lim_{n\to\infty}I_{n}(P_{n}(\rho)).$

Next, we check that
$\mathcal{E}(\rho)\geq\lim_{n\to\infty}I_{n}(P_{n}(\rho))$. This again follows
from Jensen’s inequality:

$I_{n}(\rho)=\frac{1}{2^{n}}\sum_{i\in\mathcal{I}_{n}}I\left(2^{n}\int_{i/2^{n}}^{(i+1)/2^{n}}\rho_{t}\
dt\right)\leq\sum_{i\in\mathcal{I}_{n}}\int_{i/2^{n}}^{(i+1)/2^{n}}I\left(\rho_{t}\right)\
dt=\mathcal{E}(\rho).$

Thus, we are done once we prove the reverse inequality
$\mathcal{E}(\rho)\leq\lim_{n\to\infty}I_{n}(P_{n}(\rho))$.

Consider the probability space given by $[0,1]$ endowed with its Borel
$\sigma$-algebra $\mathcal{F}_{\infty}$, and let $\mu$ be the
$\mathcal{M}_{1}(S^{1})$-valued random variable defined by sampling
$t\sim\operatorname{Leb}_{[0,1]}$ then setting $\mu:=\rho_{t}$. Let
$\mathcal{F}_{n}$ be the $\sigma$-algebra generated by sets of the form
$[i/2^{n},(i+1)/2^{n}]$ for $i\in\mathcal{I}_{n}$; note that
$\mathcal{F}_{\infty}=\sigma(\cup_{n}\mathcal{F}_{n})$. Define
$\mu_{n}:=\mathbb{E}\left[\mu\big{|}\mathcal{F}_{n}\right].$ For any
continuous function $f\in C(S^{1})$, the bounded real-valued Doob martingale
$\mu_{n}(f)$ converges a.s. to $\mu(f)$. Taking a suitable countable
collection of $f$, we conclude that a.s. $\mu_{n}$ converges to $\mu$ in the
Prokhorov topology. By Fatou’s lemma and the lower-semicontinuity of $I$, we
have

$\operatorname*{\vphantom{p}lim\,inf}_{n\to\infty}\mathbb{E}[I(\mu_{n})]\geq\mathbb{E}[\operatorname*{\vphantom{p}lim\,inf}_{n\to\infty}I(\mu_{n})]\geq\mathbb{E}[I(\mu)].$
(3.12)

We can write $\mu_{n}$ explicitly as
$\mu_{n}=2^{n}\int_{i/2^{n}}^{(i+1)/2^{n}}\rho_{t}\ dt$ where
$i\in\mathcal{I}_{n}$ is the index for which $t\in[i/2^{n},(i+1)/2^{n}]$, so
$\mathbb{E}[I(\mu_{n})]=I_{n}(P_{n}(\rho)).$ We also have
$\mathbb{E}[I(\mu)]=\int_{0}^{1}I(\rho_{t})\ dt=\mathcal{E}(\rho).$ Combining
these with (3.12), we conclude that
$\lim_{n\to\infty}I_{n}(P_{n}(\rho))\geq\mathcal{E}(\rho)$. ∎

###### Proof of Theorem 1.2.

Proposition 3.7 says that $\\{\delta_{B^{\kappa}_{t}}\\}\in\mathcal{N}$ has a
large deviation principle with good rate function given by $\sup_{n\geq
0}I_{n}(P_{n}(\cdot))$, and Lemma 3.8 shows that this good rate function can
alternatively be expressed as $\mathcal{E}$. ∎

##  4 Comments

Let us make further comments and list a few questions in addition to those in
the introduction.

  1. 1.

As we have discussed in the introduction, one may wonder what the limit and
large deviations of chordal $\operatorname{SLE}_{\infty}$ are. Figure 4.1
shows two chordal $\operatorname{SLE}_{\kappa}$ curves on $[-1,1]^{2}$ from a
boundary point $-i$ to another boundary point $i$, for several large values of
$\kappa$. We see that the interfaces stretch out to the target point and are
close to horizontal lines after we map the square to $\mathbb{H}$ and the
target point $i$ to $\infty$.

Figure 4.1: An instance of chordal $\operatorname{SLE}_{128}$ and
$\operatorname{SLE}_{1000}$ on $[-1,1]^{2}$ from $-i$ to $i$. The simulation
of these _counterflow lines_ is done by imaginary geometry as described in
[MS16a], and are approximated via linear interpolation of an $800\times 800$
discrete Gaussian free field with suitable boundary conditions. The color
represents the time (capacity) parametrization of the $\operatorname{SLE}$
curve.

  2. 2.

Corollary 1.3 shows that $\operatorname{SLE}_{\infty}$ concentrates around the
family of Loewner chains driven by an absolutely continuous measure $\rho$
with $\mathcal{E}(\rho)<\infty$. In [VW] we geometrically characterize the
Loewner chains driven by such measures. Note that the answer to the same
question for the large deviation rate function of $\operatorname{SLE}_{0+}$,
namely the family of Jordan curves of finite Loewner energy, is well-
understood. That family has been shown to be exactly the family of Weil-
Petersson quasicircles [Wan19b], which has far-reaching connections to
geometric function theory and Teichmüller theory.

  3. 3.

The rate function (1.5) for the Brownian occupation measure coincides with the
rate function of the _square_ of the Brownian bridge (or Gaussian free field)
on $S^{1}$. Is there a profound reason or is this merely a coincidence? One
could attempt to relate the large deviations of the Brownian occupation
measure to the large deviations of the occupation measure of a _Brownian loop
soup_ on $S^{1}$.

  4. 4.

The fluctuations of the circular Brownian occupation measure $L^{\kappa}_{t}$
were studied by Bolthausen. We express this result in terms of the _local
time_ $\ell_{t}:S^{1}\to[0,\infty)$, defined via
$L^{1}_{t}=\ell_{t}(\zeta)d\zeta$. Note that $\ell_{t}$ is a.s. a random
continuous function.

######  Theorem 4.1 ([Bol79]).

Identify each $\zeta=e^{i\theta}\in S^{1}$ with $\theta\in[0,2\pi)$. As
$t\to\infty$, the stochastic process
$\sqrt{t}(\frac{\ell_{t}(\theta)}{t}-\frac{1}{2\pi})_{\theta\in[0,2\pi)}$
converges in distribution to
$(2b_{\theta}-\frac{1}{\pi}\int_{0}^{2\pi}b_{\tau}\
d\tau)_{\theta\in[0,2\pi)}$, where $b$ is a Brownian bridge on the interval
$[0,2\pi]$ with endpoints pinned at $b_{0}=b_{2\pi}=0$.

We wonder whether there are interesting consequences to the fluctuations of
$\operatorname{SLE}_{\infty}$.

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