# Surface states and related quantum interference in ab initio electron
transport

Dongzhe Li<EMAIL_ADDRESS>Department of Physics, Technical University of
Denmark, DK-2800 Kongens Lyngby, Denmark  CEMES, Université de Toulouse, CNRS,
29 rue Jeanne Marvig, F-31055 Toulouse, France  Jonas L. Bertelsen  Department
of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
Center for Nanostructured Graphene, Technical University of Denmark, DK-2800
Kongens Lyngby, Denmark  Nick Papior  DTU Computing Center, Department of
Applied Mathematics and Computer Science, Technical University of Denmark,
DK-2800 Kongens Lyngby, Denmark  Alexander Smogunov  SPEC, CEA, CNRS,
Université Paris-Saclay, CEA Saclay, F-91191 Gif-sur-Yvette, France  Mads
Brandbyge<EMAIL_ADDRESS>Department of Physics, Technical University of Denmark,
DK-2800 Kongens Lyngby, Denmark  Center for Nanostructured Graphene, Technical
University of Denmark, DK-2800 Kongens Lyngby, Denmark

###### Abstract

Shockley surface states (SS) have attracted much attention due to their role
in various physical phenomena occurring at surfaces. It is also clear from
experiments that they can play an important role in electron transport.
However, accurate incorporation of surface states in ab initio quantum
transport simulations remains still an unresolved problem. Here we go beyond
the state-of-the-art non-equilibrium Green’s function formalism through the
evaluation of the self-energy in real-space, enabling electron transport
without using artificial periodic in-plane conditions. We demonstrate the
method on three representative examples based on Au(111): a clean surface, a
metallic nanocontact, and a single-molecule junction. We show that SS can
contribute more than 30% of the electron transport near the Fermi energy. A
significant and robust transmission drop is observed at the SS band edge due
to quantum interference in both metallic and molecular junctions, in good
agreement with experimental measurements. The origin of this interference
phenomenon is attributed to the coupling between bulk and SS transport
channels and it is reproduced and understood by tight-binding model.
Furthermore, our method predicts much better quantized conductance for
metallic nanocontacts.

##  I INTRODUCTION

At the surfaces of noble metals lacking inversion symmetry, the Shockley
surface states (SS) appear in the projected band-gap of bulk states Davison
and Steslicka (1996). These SS are nearly free-electron model systems due to
their confinement to the surface, and thus have attracted significant
attention for the investigation of fundamental many-body effects in solids, in
particular by angle-resolved photoemission spectroscopy Kevan and Gaylord
(1987); Eiguren et al. (2002); Sirotti et al. (2014), and scanning tunneling
microscopy (STM) Berndt et al. (1999); Li et al. (1998); Kliewer et al.
(2000); Reinert et al. (2001). Crommie et al. Crommie et al. (1993) formed
quantum corrals on a metal surface where the SS create the wave phenomena.
Many transport measurements show that SS are actively involved in adsorption,
and chemical processes on surfaces Donath and Bertel (1995); Ren et al.
(2019). The manipulation of SS can influence the molecule-metal interaction,
altering the Kondo temperature Iancu et al. (2006); Granet et al. (2020),
while adsorption of single adatoms can induce further localization of the 2D
Shockley SS at the metal surfaces Madhavan et al. (2001); Olsson et al.
(2004); Merino and Gunnarsson (2004); Wahl et al. (2005); Limot et al. (2005);
Vitali et al. (2008). Moreover, tunneling between molecular orbitals and the
SS is shown to produce negative differential conductance in metal-organic
junctions Heinrich et al. (2011). These experimental observations invoke
several fundamental questions to be addressed: Can we describe SS accurately
in transport calculations? What is the role of SS in electron transport and
how can one use them to tune electrical conductance?

During the last two decades, the combination of density functional theory
(DFT) and non-equilibrium Green’s functions (NEGF) theory has been recognized
as the most powerful computational tool with predictive power. Currently,
there are mainly two different DFT+NEGF methodologies being applied. The first
one is based on supercell approach as implemented in TranSiesta Brandbyge et
al. (2002); Papior et al. (2017), Nanodcal Taylor et al. (2001), QuantumATK
Smidstrup et al. (2019), and many other codes Joon Choi and Ihm (1999);
Nardelli (1999); Wortmann et al. (2002); Smogunov et al. (2004); Wang (2005);
Rocha et al. (2006); Pecchia et al. (2008); Saha et al. (2009); Ozaki et al.
(2010); Verzijl and Thijssen (2012); Chen et al. (2012); Ferrer et al. (2014);
Refaely-Abramson et al. (2019); Takaki et al. (2020); Ye et al. (2021). They
typically remove periodic boundary conditions (PBCs) in the transport
direction using the recursive surface self-energy method Sancho et al. (1985)
while for the directions transverse to the transport, the PBCs are retained,
and corresponding $\mathbf{k}$-points are used. This means that the atomic-
scale junction under investigation, such as an atomic contact, an adatom, or a
molecular junction, as well as the connecting electrodes, such as a tip, are
repeated in the transverse direction, as shown in Fig. 1a. In this setup, the
only current-carrying asymptotic states are the bulk Bloch states propagating
deep into the electrodes. Conduction channels related to SS, propagating along
the surface (red arrows in Fig. 1b), are missing. The second approach is a
cluster-based method Jacob and Palacios (2011); Bagrets (2013); Frank and
Jacob (2015) which does not capture the surface state properly due to the
finite size of the cluster. Another particular cluster-based approach
developed by F. Pauly et al. Pauly et al. (2008), a kind of mixed cluster-
supercell method, which in principle could include the SS. However, the SS and
their contribution to the total self-energy have never been studied with their
approach. Moreover, due to its inherent need for large clusters, the SS is
approximated, to our knowledge.

Here, we demonstrate the proper incorporation of surface states’ contribution
in ab initio electron transport, going beyond the current state-of-the-art
DFT+NEGF framework by removing all PBCs. This is accomplished by calculating
the real-space self-energy (RSSE) via the $\mathbf{k}$-averaged Green’s
function. We show that the SS contributes more than 30% of the electron
transport near the Fermi energy ($E_{F}$) on noble metal surfaces, resulting
in a step-like increase of transmission function. Furthermore, a pronounced
transmission drop is observed just below $E_{F}$ due to quantum interference
(QI) when the surface potential is perturbed (by e.g., an STM tip, an adatom,
or a molecule). These results are in a good agreement with experimental
reports Madhavan et al. (2001); Hollen et al. (2015); Olsson et al. (2004);
Limot et al. (2005). The step- or drop-like features observed in transmission
at the SS band edge and their interpretation is the main goal of this
manuscript.

Figure 1:  (a) Supercell approach: Atomic-scale junctions with in-plane
periodic boundary conditions (PBCs), (b) RSSE method: Single atomic-scale
junctions without PBCs, (c) Clean Au(111) surface: Band structure projected on
the surface atom (left) and corresponding PDOS along the $\mathbf{k}$-path
MKGM (right).

##  II METHOD

The DFT+NEGF transport calculations within the local density approximation
(LDA) Perdew and Zunger (1981) were carried out using TranSiesta Brandbyge et
al. (2002); Papior et al. (2017), and the post-processing codes TBtrans, and
sisl Papior (2020). The transmission function is calculated using the NEGF
approach as

$T(E)=\Tr[\mathbf{\Gamma_{\text{L}}}\mathbf{G}\mathbf{\Gamma_{\text{R}}}\mathbf{G}^{\dagger}]$
(1)

where $\mathbf{G}$ is the retarded Green’s function, and
$\mathbf{\Gamma_{\text{L}}}$/$\mathbf{\Gamma_{\text{R}}}$ are matrices
describing the coupling to the left/right semi-infinite electrodes including
surface states. The ballistic conductance is evaluated from the corresponding
electron transmission function at $E_{F}$ using the Landauer formula Blanter
and Büttiker (2000), $G=G_{0}T(E_{F})$, where $G_{0}=2e^{2}/h$ is the
conductance quantum. We compare our RSSE method with the supercell approach,
where the open boundary condition is applied in the out-of-surface transport
direction $z$ while PBCs are used in the surface plane $xy$. The RSSE removes
all PBCs by evaluating the “surrounding/embedding” self-energy corresponding
to the surface supercell in real-space, which can be outlined as

$\mathbf{\Sigma}^{\mathrm{RSSE}}(E)=\mathbf{S}^{\mathrm{RS}}(E+i\eta)-\mathbf{H}^{\mathrm{RS}}-\bigg{[}\sum_{\mathbf{k}}\mathbf{G_{k}}(E)\bigg{]}^{-1}\,$
(2)

Here $\mathbf{G}_{\mathbf{k}}(E)$ is the Green’s function for a given
$\mathbf{k}$-point, $\mathbf{H}^{\mathrm{RS}}$ and $\mathbf{S}^{\mathrm{RS}}$
are the real-space representation of Hamiltonian and overlap matrices for an
infinitely large system projected on the real-space supercell. The self-energy
$\mathbf{\Sigma}^{\text{RSSE}}$ is only nonzero on the boundary of the
supercell, where orbitals have matrix elements with the states outside the
supercell. Since the self-energy describes a perfect, semi-infinite bulk
structure with the single supercell missing, we may reuse this self-energy for
any perturbed supercell of the same size containing for example an adatom,
vacancy, adsorbate, and STM-tip. All states in the supercell which originate
from bulk scattering states are represented in $\mathbf{G_{k}}(E)$. However,
the surface states are orthogonal to these and thus appear as poles in
$\mathbf{G_{k}}(E)$. In order to include them, we used a small imaginary
part/broadening $\eta=2\,\mathrm{meV}$, and a correspondingly dense
$\mathbf{k}$-points of 400 ${\times}$ 400 $\mathbf{k}$-mesh for the evaluation
of real-space quantities in Eq. (2). The implementation details are described
in our recent work Papior et al. (2019) where the method was applied only to
two-dimensional (2D) systems, here we show its importance in three-dimensional
(3D) systems of atomic-scale junctions with a particular focus on SS. We refer
to Appendix A for details on the computational parameters and $\mathbf{k}$
points convergence test is presented in Appendix B.

##  III RESULTS

We start our discussion with the band structure of the clean Au(111) surface.
Instead of the slab model Forster et al. (2007); Marathe et al. (2012);
Requist et al. (2015), we use a single surface connected to infinitely many
layers. This is done by calculating the $\mathbf{G}(E)$ for six Au(111) layers
of which the bottom three layers connect via the self-energy
$\mathbf{\Sigma}(E)$ to an infinite substrate, thus removing the PBCs in this
direction, but maintaining transverse $\mathbf{k}$-dependence:

$\mathbf{G_{k}}(E)=\left[(E+i\eta)\mathbf{S_{k}}-\mathbf{H_{k}}-\mathbf{\Sigma_{k}}(E)\right]^{-1}$
(3)

where $\mathbf{H_{k}}$ is the Hamiltonian and $\mathbf{S_{k}}$ the overlap
matrix of the surface layers in $\mathbf{k}$-space. We plot in Fig. 1c the
band structure of Au(111) surface projected on the surface atom, where the
well-known band gap and SS at the $\Gamma$ point are evident. The binding
energy at $\Gamma$ is about $0.4\,\mathrm{eV}$, in good agreement with
previous _ab initio_ calculations Marathe et al. (2012); Sirotti et al. (2014)
and experimental measurements Andreev et al. (2004); Forster et al. (2007);
Sławińska et al. (2012). From the $\mathbf{k}$-path PDOS, we further identify
that the SS are mainly from $p$ orbitals.

Having reproduced the electronic structure of SS of a clean Au(111) we pass
now to their effect on the transport properties of a point contact as shown in
Fig. 1b. Standard DFT+NEGF simulations based on the supercell approach are
problematic in two ways: i) the simulations have a periodic array of junctions
including the repetition of the STM probe tip; such an array of junctions will
have unphysical interactions and requires extra care with the convergence of
$\mathbf{k}$ points and supercell size Thygesen and Jacobsen (2005) and ii)
more importantly, the simulations are unable to capture the surface state
contribution to the transport. Our RSSE method solves both problems. In the
following, we compare RSSE with the supercell approach for three
representative examples based on Au(111): a clean surface, a metallic
nanocontact, and a single-molecule junction.

We plot in Fig. 2a the transmission function from the tip into the clean
Au(111) surface with a tip-surface distance ($d$) of 4 Å. Most importantly, we
find a crucial difference for energy higher than $-0.4\,\mathrm{eV}$ where the
surface band appears and opens an additional transport channel. However, this
surface channel is completely missing in the supercell approach. Therefore,
this demonstrates the critical role of SS contributing more than 30% to
electron transmission around $E_{F}$. This is in very good agreement with the
step-like feature observed in experiments Madhavan et al. (2001); Hollen et
al. (2015). We further investigate the $d$ dependence on transmission, as
shown in Fig. 2b. Instead of an increase of the transmission at the SS band
edge, we find a drop with decreasing $d$, and it more pronounced for smaller
$d$. The transport through bulk and surface bands is independent in the
tunneling regime while the two channels get coupled in the contact regime,
which as we will argue later is at the origin of the transmission drop.

Figure 2:  Clean Au(111): (a) Transmission functions from the tip to surface
by supercell (black) and RSSE (red) approaches with $d=4$ Å. A strong increase
of the transmission is seen with the RSSE approach at about
$E_{F}-0.4\,\mathrm{eV}$ where the surface band appears. The supercell
approach completely overlooks this contribution, resulting in over 30%
underestimation of the transmission coefficient near $E_{F}$. (b) RSSE
transmissions at different $d$. The transmission shows a jump or a drop in the
tunneling ($d=4$ Å) or contact ($d=3$ Å or $d=2.5$ Å) regimes, respectively,
at the SS band edge.  Figure 3:  Adatom and molecule on Au(111): The same as
in Fig. 2(a) but for adatom (a) and carbon monoxide molecule (b) deposited on
surface. In the case of RSSE, a pronounced drop is found for both junctions at
about $E_{F}-0.4\,\mathrm{eV}$. The distances between the tip and adatom or
molecule are 4.5 Å and 3.6 Å, respectively.

As another system, we consider a junction made of a single Au atom adsorbed in
a hollow site on Au(111) with $d=4.5\,\textup{\AA}$, as shown in Fig. 3a. Very
interestingly, at about $E_{F}-0.4\,\mathrm{eV}$, we see again a transmission
drop similar to the case of a clean surface. Due to the stronger interaction
between adatom and surface, the drop becomes more pronounced. Our _ab initio_
results reproduce well the experimental measurements Olsson et al. (2004);
Limot et al. (2005) in terms of energy level alignment and width of the peak.
A similar transmission dip has also been recently reported in the case of Co
adsorbed on Cu(111) by ab inito theory Tacca et al. (2021). Moreover, the
energy derivative of the transmission changes a sign in two methods leading to
the opposite signs of the Seebeck coefficient which could be tested
experimentally. We further note that this transmission drop is robust with
respect to $d$ (see Appendix C). On the other hand, the supercell approach
fails to reproduce the highly energy-dependent SS feature even with a very
large lateral periodicity (see Fig. 8a in Appendix D). Finally, the Fano-
factor (e.g., shot noise experiment) is very different when SS are included
(see Fig. 8b in Appendix D). We argue that the transmission drop is a general
effect resulting from the mixing of surface and bulk channels controlled by
the interaction with, for example, an STM tip or an adsorbate. To verify this
point, we investigated a carbon monoxide molecule adsorbed on a hollow site
(energetically most favorable Piccolo et al. (2004)) of the Au(111) surface as
presented in Fig. 3b. Although the main conducting LUMO is located at about
$E_{F}+1.8$ eV, very different from the gold adatom discussed before, a
similar drop-like feature is again reproduced at the same energy, pointing out
the general SS-related character.

Figure 4:  Tight-binding model with two bands (left side), simulating the bulk
and the surface channels of the Au(111) surface, and one band to the right,
simulating the Au tip. (a) Model for clean Au(111) surface. The inclusion of
interband coupling, $V$, allows to reproduce the transmission drop at the SS
band edge; (b) Model for impurity on Au(111). The transmission drop is
observed for any energy of impurity level but disappears if impurity/SS
coupling is switched off (dashed red), confirming its QI interpretation.

To understand the origin of the transmission drop at $-0.4\,\mathrm{eV}$
observed with the RSSE method, we construct a tight-binding (TB) model,
depicted in Fig. 4, with two bands for the electrode (left), simulating bulk
(1) and surface (2) channels, and one band (3) for the STM tip (right)
simulated by an atomic tight-binding chain with on-site and hopping energies
being 0 and 5 eV, respectively. The surface (left) is characterized by
unperturbed Green’s functions at two contact orbitals, $g_{1}$ and $g_{2}$,
constructed on bulk and surface channels, respectively. They are chosen for
simplicity to be purely imaginary (a wide-band approximation) as
$g_{i}(E)=2\pi i/W_{i}$ for $|E-\varepsilon_{i}|<W_{i}$ and 0 otherwise. The
band centers, $\varepsilon_{i}$, and half band-widths, $W_{i}$, are set to
{$\varepsilon_{1}=0,W_{1}=10~{}{\rm eV}$} and {$\varepsilon_{2}=3.5~{}{\rm
eV},W_{2}=4~{}{\rm eV}$} for bulk and surface bands, respectively, which gives
in particular a SS band edge at $E=-0.5$ eV as required.

We first consider the clean Au(111) contacted by the STM tip (Fig. 4a). The
tip apex atom (3) is considered as a central region of Eq. 1, the left side
coupling $\Gamma_{L}$ is constructed on the $2{\times}2$ Green’s function $G$
(from sites 1 and 2) of the surface while the right side $\Gamma_{R}$ is given
by the tip TB chain expression. Necessary tip/surface coupling constants were
set to $t_{13}=1$ eV and $t_{23}=0.35$ eV. If two channels of the surface are
independent ($V=0$, so $G_{ij}=\delta_{ij}g_{i}$) the regular-shaped
transmission is obtained with superposed contributions from bulk and surface
bands (red). Interestingly, when the interband mixing $V$ is switched on,
which corresponds to the increased bulk and SS mixing due to approaching STM
tip, the transmission drop at the SS band edge is developed (blue). This drop
can be attributed to destructive QI between the two electron pathways: i) the
direct tunneling to the principal bulk channel, tip $\rightarrow$ bulk band
(1) and ii) the pathway mediated by the SS, the tip $\rightarrow$ SS (2)
$\rightarrow$ bulk band (1). Since both pathways involve the same terminal
bulk band they can interfere destructively, producing the transmission drop.
Overall the TB reproduces well the DFT results.

For the impurity case, an additional orbital, $\epsilon_{0}$, should be
introduced (Fig. 4b). The $\Gamma_{L}$ for the tip apex orbital (3) is now
constructed on the $3{\times}3$ Green’s function $G$ (from sites 1, 2, and 0)
whereas $\Gamma_{R}$ remains the same. The hopping parameters $t_{01},t_{02}$,
and $t_{03}$ were set to 5.0 eV, 3.0 eV, and 1.0 eV, respectively. It is
assumed that the tunneling from the tip goes through the impurity orbital only
without explicit inclusion of the interband mixing such that the two bands are
coupled via the impurity level. The transmission drop appears again,
independent of $\epsilon_{0}$. It is well reproduced, in particular, for
$\epsilon_{0}=0$ (red) and $\epsilon_{0}=2$ eV (blue), which should correspond
to Au adatom and the molecule, respectively, discussed before in Fig. 3.
Similar to the previous case, we can interpret this transmission drop to
destructive QI due to two pathways: i) tip $\rightarrow$ impurity
$\rightarrow$ bulk band and ii) tip $\rightarrow$ impurity $\rightarrow$ SS
$\rightarrow$ impurity $\rightarrow$ bulk band. Note that if the impurity/SS
coupling is turned off, $t_{20}=0$, the transmission recovers its normal shape
(red dashed), which confirms our QI interpretation.

Figure 5:  Conductance .vs. tip displacement for Au atomic contacts calculated
by supercell (black) and RSSE (red) approaches. A conductance plateau of about
1 $G_{0}$ is found by the RSSE method, whereas the supercell approach yields
only 0.8 $G_{0}$.

As mentioned before, the drop in transmission at the SS band edge is a robust
feature showing up for different tip-adatom distances. As the tip is
approaching the adatom, the SS are expected to give a more important
contribution to the transmission function. We show, therefore, in Fig. 5 the
conductance as a function of $d$. Clearly, close to the contact regime ($d<8$
Å), the difference between the RSSE and conventional supercell approach starts
to increase, reflecting the enhanced contribution of SS into the transport.
Due to SS contribution, missing in the supercell approach, the RSSE method
provides much better quantized electron transport with the conductance of
$1\,G_{0}$ improving strongly an agreement with experimental data Brandbyge et
al. (1995); Vardimon et al. (2013); Chen et al. (2014). Being generic, the
conductance jump at the edge of the SS should be robust in STM setup but less
pronounced in break junction setup where the SS are strongly perturbed by a
pyramid-like cluster formed on the surface.

##  IV CONCLUSIONS

In summary, using the newly developed RSSE method, we have demonstrated how
surface states can be properly incorporated in ab initio electron transport
going beyond current state-of-the-art DFT+NEGF. We show that the SS can
contribute more than 30% in electron transport near $E_{F}$ for noble metal
surfaces. Moreover, we find a robust QI induced by SS in both atomic and
molecular junctions ranging from tunneling to contact regime, in good
agreement with experimental measurements. We also find that the RSSE gives a
considerably better quantized conductance for metallic nanocontacts. Our
method should be important for a variety of systems where surface propagating
channels can play an important role, such as topologically derived SS Seo et
al. (2010); Yan et al. (2015), graphene edge states Yao et al. (2009), and 2D
materials/metal interfaces Dahal and Batzill (2014).

Acknowledgment We acknowledge funding from the EU Horizon 2020 programme
(766726), Villum foundation (13340), and the Danish Research Foundation
(DNRF103). Part of the calculations was done using HPC resources from CALMIP
(P21023).

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## Appendix A: Computational details

Figure 6:  The convergence of the self-energy with respect to planar 2D
$\mathbf{k}$ points in $6{\times}6$ unit cell. The surface states’
transmission, at $E>-0.4$ eV, converges much slower compared to the bulk one.

Ab initio quantum transport calculations were done using TranSIESTA Brandbyge
et al. (2002); Papior et al. (2017) in the LDA with the parametrization of
Perdew and Zunger Perdew and Zunger (1981) for exchange-correlation
functional. The valence–electron wavefunctions were expanded in a basis set of
local orbitals in SIESTA Soler et al. (2002); García et al. (2020). Concerning
the basis set, the cutoff radius ($r_{c}$ in Bohr units) and orbitals are
$r_{c}$ = 6.083, 5.715 (6$s$, polarized) and $r_{c}$ = 4.287, 2.838 (5$d$) for
the Au. The pseudopotentials were taken from Ref. Rivero et al., 2015 where
the parameters have been carefully checked, producing a good agreement with
plane-wave ab initio code. The mesh cutoff and filter cutoff were set to 400
Ry and 300 Ry, respectively. The lattice constants of the Au were set
according to their bulk experimental lattice constants of 4.07 Å and the
interlayer spacing within the slab was held fixed. The spin-orbit coupling was
not considered. For the conductance evaluation with respect to displacement
distance (see Fig. 5), we allow only STM apex and adatom atoms to relax until
atomic forces became lower than 0.01 eV/Å, and repeat it until forming atomic
contacts. Six atomic layers were used to describe the surface with bulk self-
energy on the lower three. All the results presented were calculated within a
large supercell of $6{\times}6$ Au(111) unit cells for both supercell and RSSE
methods.

In the supercell approach, the density matrix was converged using $6{\times}6$
$\mathbf{k}$-points while the transmission functions were evaluated using
denser $24{\times}24$ $\mathbf{k}$-points. For the RSSE method, the self-
energy was constructed using a very dense $\mathbf{k}$-mesh of
$400{\times}400$ and the transmission spectrum was calculated within a single
$\mathbf{k}$-points. Physical quantities like transmission, the PDOS, and
Fano-factor were extracted using TBtrans and SISL Papior (2020).

## Appendix B: $\mathbf{k}$ points convergence

For the pre-calculation of RSSE, we need a dense integration grid in
reciprocal space to obtain well-converged surface states’ transmission as
shown in Fig. 6, at $E>-0.4$ eV. The bulk contribution, $E<-0.4$ eV, converges
very fast (even with $30{\times}30$) while much denser $\mathbf{k}$ mesh is
needed for the surface states. Throughout the paper, the real-space self-
energy was calculated from $400{\times}400$ $\mathbf{k}$ points.

## Appendix C: From tunneling to contact regimes

The destructive quantum interference is found to be rather robust, appearing
for different tip-atom distances ranging from tunneling to contact regime as
Fig. 7 shows. Note that the smaller the tip-adatom distance, the larger
contribution of surface states around the Fermi energy. In particular, at the
contact regime ($d=2.9$ Å), the transmission calculated with our RSSE approach
starts to increase at the surface band edge ($E>-0.4$ eV) deviating
considerably from the one obtained with the conventional supercell method. As
a consequence, with RSSE we recover the conductance very close to the
quantized value of 1 $G_{0}$, often reported in single-atom contact
measurements.

Figure 7:  Adatom on Au(111): Transmission functions calculated by two
different approaches with different tip-adatom distances ranging from
tunneling to contact regime: (a) $d=2.9$ Å, (b) $d=3.7$ Å, (c) $d=4.5$ Å, and
(d) $d=5.3$ Å.

## Appendix D: Periodicity size effect for the supercell approach

To investigate how the in-plane periodicity influences the transport behavior,
we plot in Fig. 8a the transmission functions calculated by the supercell
approach with different sizes of the periodic array. Even for the relatively
wide $8{\times}8$ superlattice (434 atoms in total), a large difference is
found compared to the RSSE result. Interestingly, for the PBCs, we find
standing wave-like patterns in the SS energy window which become denser when
the in-plane periodicity is increased, similar to confined states in a quantum
well of increasing width. Even if the size of the supercell could be made
sufficiently large, while likely computationally unfeasible, we would not
necessarily end up with the RSSE result, drawn with the red line, which lacks
these oscillations. This is due to the fact that for PBCs it is only the
asymptotic bulk scattering states which are represented in the coupling
matrices, $\mathbf{\Gamma_{\text{L/R}}}$, entering the transmission. On the
other hand, the adsorbate atom potential may provide a coupling between
surface and bulk states, decreasing the importance of the asymptotic SS
channel.

Figure 8:  Adatom on Au(111): Transmission functions (a) and Fano-factor (b)
calculated by RSSE (red) and the supercell approach with different in-plane
periodicity, $4{\times}4$ (blue), $6{\times}6$ (black) and $8{\times}8$
(brown). For supercell approaches, standing-wave patterns are found and they
become denser when the periodicity is increased. The $\mathbf{k}$-dependence
on transport is also reflected in $\mathbf{k}_{\parallel}$-resolved
transmission coefficients at the Fermi level. The tip atom to adatom distance
is 4.5 Å.

On the other hand, fig. 8b shows the corresponding Fano-factor calculated from
$\mathbf{k}$-dependent $i$-th transmission eigenchannels,
$\tau_{i,\mathbf{k}}$.

$F=\frac{\sum_{i,\mathbf{k}}\tau_{i,\mathbf{k}}(1-\tau_{i,\mathbf{k}})\omega_{\mathbf{k}}}{\sum_{i,\mathbf{k}}\tau_{i,\mathbf{k}}\omega_{\mathbf{k}}}$
(4)

where $\omega_{\mathbf{k}}$ is $\mathbf{k}$-points weight-factors. Due to the
PBCs and $\mathbf{k}$-point sampling, the Fano-factor is sensitive to the
periodic repetition. This is also reflected in
$\mathbf{k}_{\parallel}$-resolved transmission coefficients at the Fermi level
in first-Brillouin. The use of RSSE removes the artificial interactions
between the periodic images and $\mathbf{k}$-dependence.


# THE RANDOM CLUSTER MODEL ON THE COMPLETE GRAPH VIA LARGE DEVIATIONS

Darion Mayes

###### Abstract

We study the emergence of the giant component in the random cluster model on
the complete graph, which was first studied by Bollobás, Grimmett, and Janson
in [4]. We give an alternative analysis using a thermodynamic/large deviations
approach introduced by Biskup, Chayes, and Smith in [3] for the case of
percolation. In particular, we compute the rate function for large deviations
of the size of the largest connected component of the random graph for $q\geq
1$.

##  1 Introduction

The random cluster model was introduced by Fortuin and Kasteleyn in [7] as a
generalisation of several existing models in statistical physics. For a
finite, simple graph $G=(V,E)$, the random cluster model with edge weight
$p\in[0,1]$ and cluster weight $q>0$ is the probability measure $\phi_{G,p,q}$
on the state space $\Omega_{G}=\\{0,1\\}^{E}$ defined by

$\phi_{G,p,q}[\omega]:=\dfrac{\bigl{\\{}\prod_{e\in
E}p^{\omega_{e}}(1-p)^{1-\omega_{e}}\bigr{\\}}q^{k(\omega)}}{Z_{G,p,q}},\
\omega=(\omega_{e})_{e\in E}\in\Omega_{G},$ (1)

where

$Z_{G,p,q}:=\sum_{\omega\in\Omega_{G}}\bigl{\\{}\prod_{e\in
E}p^{\omega_{e}}(1-p)^{1-\omega_{e}}\bigr{\\}}q^{k(\omega)}$ (2)

is the normalising partition function, and $k(\omega)$ is the number of
connected components in the graph $G(\omega)=(V,E(\omega))$ with
$E(\omega)=\\{e\in E:\ \omega_{e}=1\\}$. For an element $\omega\in\Omega_{G}$,
we say that the edge $e$ is open if $e\in E(\omega)$, otherwise, it is closed.
The measure $\phi_{G,p,q}$ may be equivalently interpreted as a probability
distribution on the set of random edge subgraphs $G(\omega)=(V,E(\omega))$ of
$G$. For $q=1$, we recover the percolation model, for which we use the
standard shorthand $\phi_{G,p}$.

We will be interested in the particular case of the complete graph
$K_{n}=(V_{n},E_{n})$ when we choose $\lambda\in[0,\infty)$ and rescale the
edge weight as $p=\lambda/n$. For the corresponding random cluster measure, we
use the shorthand $\phi_{n,\lambda,q}:=\phi_{K_{n},\lambda/n,q}$. Similarly,
we denote the partition function as $Z_{n,\lambda,q}$, and use the shorthand
$\phi_{n,\lambda}$ in the case of percolation. For fixed $q\geq 1$, the random
cluster model is stochastically ordered in $\lambda$, and we study how the
random graph obtained under the measure $\phi_{n,\lambda,q}$ varies with
respect to this parameter.

One particular quantity of interest is the size of the largest connected
component in the random graph. In their groundbreaking paper [6], Erdős and
Rényi studied the measure $\phi_{n,\lambda}$ and showed the existence of a
critical parameter $\lambda_{c}=1$ marking the emergence of a giant component
containing a positive proportion of the total number of vertices
asymptotically almost surely. More specifically, they showed that, with
probability tending to one as $n\to\infty$ under the percolation measure
$\phi_{n,\lambda}$, the largest component of the graph
$K_{n}(\omega)=(V_{n},E_{n}(\omega))$ is of order $\log n$ for $\lambda<1$ and
of order $n$ for $\lambda>1$. This abrupt change in the size of the largest
component (in the limit as $n\to\infty$) is known as an asymptotic phase
transition.

The main idea in the proof of the asymptotic phase transition of [6] is an
exploration process whereby one chooses a vertex and sequentially inspects
which vertices are connected to it in the random graph. Provided one has not
yet explored a large fraction of the vertices, this exploration can be
approximated by a Poisson branching process with mean $\lambda$ in the limit
as $n\to\infty$. The emergence of a giant component then corresponds to the
survival of the corresponding Poisson branching process. This method is not
crucially dependent on the structure of the complete graph, and has been
successfully applied to establish similar asymptotic phase transitions on a
variety of families of graphs, such as the hypercube in [1], and expander
graphs in [2].

For $q\neq 1$, the presence of the factor $q^{k(\omega)}$ in the random
cluster measure $\phi_{n,\lambda,q}$ creates an additional complexity—the
states of the edges in the random graph are no longer independent and it is no
longer clear that the exploration process may be used. Nevertheless, it has
been established by Bollobás, Grimmett, and Janson in [4] that the largest
component of the complete graph undergoes an asymptotic phase transition at a
particular value $\lambda_{c}(q)$. The claim is proven by randomly colouring
the vertices using two colours (say red and green) in a particular way so that
the distribution of edges on the red vertices is given by a percolation
measure, to which we may apply the exploration process used in [6]. Note that
the colouring argument depends crucially on the fact that for a fixed set $R$
of red vertices, the conditional distribution of edges yields a random
percolation on the complete graph induced on the set $R$. This is a particular
fact that does not generalise to more structured families of graphs.

The purpose of the present paper is to provide a new analysis of the random
cluster model $\phi_{n,\lambda,q}$ on the complete graph, with the hope that
it may be extended to wider families of graphs. Our analysis employs the
methods introduced by Biskup, Chayes and Smith in [3], who analysed the
largest component for percolation on the complete graph using a large
deviations approach instead of branching processes. We extend this approach to
the random cluster model on the complete graph for $q\geq 1$. In particular,
we compute the rate function for large deviations of the size of the largest
connected component, thereby recovering the asymptotic phase transition proven
in [4]. In addition, we obtain a limit for the free energy of the random
cluster model. This was also computed in [4] via the colouring argument, but
used to study the large deviations of the number of connected components,
rather than their size. As a byproduct of our analysis, we also obtain the
exponential decay rate for the events that the random graph is connected and
acyclic, respectively.

##  2 Statement of Results

In this section, we state the main results of the paper. Fix $q>0$ and
$\lambda>0$. We use $\phi_{n,\lambda,q}$ to denote the random cluster
probability measure on the complete graph $K_{n}$ with edge weight
$p=\lambda/n$ and cluster weight $q$. Similarly, we denote the corresponding
partition function by $Z_{n,\lambda,q}$. In some of our results, we will
analyse the measure of an event $A\subset\Omega_{K_{n}}$ before normalisation,
in which case we write
$Z_{n,\lambda,q}[A]:=Z_{n,\lambda,q}\phi_{n,\lambda,q}[A]$. When $q=1$, we
drop the subscript $q$ entirely and use the standard percolation notation
$\phi_{n,\lambda}$.

Let $\mathcal{N}_{r}$ be the number of connected components in $K_{n}(\omega)$
of size larger than $r$, and let $\mathcal{V}_{r}$ be the set of vertices in
$K_{n}(\omega)$ belonging to components of size larger than $r$. In addition,
recall the entropy function, defined for $\theta\in(0,1)$ by

$S(\theta)=\theta\log\theta+(1-\theta)\log(1-\theta).$ (3)

Following [3], we define two functions on $[0,\infty)$ by

$\pi_{1}(x)=1-e^{-x},\ \Psi(x)=\bigg{(}\log
x-\frac{1}{2}\bigg{[}x-\frac{1}{x}\bigg{]}\bigg{)}\wedge 0.$ (4)

Note that $\Psi(x)<0$ if and only if $x>1$. Finally, we define the function

$\displaystyle\Phi(\theta,\lambda,q)=-S(\theta)$
$\displaystyle+(1-\theta)\log[1-\pi_{1}(\lambda\theta)]+\theta\log\pi_{1}(\lambda\theta)$
(5)
$\displaystyle+(1-\theta)\\{\Psi(\tfrac{\lambda(1-\theta)}{q})-(\tfrac{q-1}{2q})\lambda(1-\theta)+\log
q\\}.$

Our main result may then be stated as follows:

######  Theorem 2.1.

Fix $q>0$ and $\lambda>0$. Then, for every $\theta\in[0,1]$,

$\lim_{\epsilon\downarrow
0}\lim_{n\to\infty}\frac{1}{n}\log\phi_{n,\lambda,q}\big{[}\lvert\mathcal{V}_{\epsilon
n}\rvert=\lfloor\theta
n\rfloor\big{]}=\Phi(\theta,\lambda,q)-\sup_{\theta\in[0,1]}\Phi(\theta,\lambda,q).$
(6)

In the language of large deviations, Theorem 2.1 says that in the limit
$\epsilon\downarrow 0$, the sequence of random variables
$\lvert\mathcal{V}_{\epsilon n}\rvert/n$ satisfies the large deviations
principle with rate function $\Phi(\theta,\lambda,q)$. Moreover, if
$\theta^{*}\in[0,1]$ maximises $\Phi(\theta,\lambda,q)$, then Theorem 2.1
implies that approximately $\theta^{*}n$ vertices of $K_{n}(\omega)$ belong to
components of order $n$. The following lemma tells us that in fact, these
vertices all belong to a single component of size $\theta^{*}n$:

######  Lemma 2.2.

Fix $q>0$ and $\lambda>0$. Then, for every $\lambda>0$ and $\epsilon>0$ there
exists $c=c(\lambda,\epsilon)>0$ such that for every $\theta>\epsilon>0$,

$\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor,\mathcal{N}_{\epsilon
n}=1]\geq(1-e^{-cn})\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon
n}\rvert=\lfloor\theta n\rfloor].$ (7)

It remains to specify the maximiser $\theta^{*}$. We will see in Section $5$
that the value $\theta^{*}$ maximising $\Phi(\theta,\lambda,q)$ is given by

$\theta(\lambda,q)=\begin{cases}0&\text{ if }\lambda<\lambda_{c}(q)\\\
\theta_{\text{max}}&\text{ if }\lambda\geq\lambda_{c}(q)\end{cases}$ (8)

where the critical value $\lambda_{c}$ is defined as

$\lambda_{c}(q)=\begin{cases}q&\text{ if }q\leq 2\\\
2\bigg{(}\frac{q-1}{q-2}\bigg{)}\log(q-1)&\text{ if }q>2\end{cases}$ (9)

and $\theta_{\text{max}}$ is the largest solution of the mean field equation

$e^{-\lambda\theta}=\frac{1-\theta}{1+(q-1)\theta}.$ (10)

The solutions to (10) are discussed in detail in ([4], Lemma $2.5$). In
particular, it is shown there that $\theta_{\max}>0$ for
$\lambda>\lambda_{c}$. Consequently, Theorem 2.1 implies that the largest
component of the graph $K_{n}(\omega)$ is of order $n$ asymptotically almost
surely when $\lambda>\lambda_{c}$. In this way, we recover the asymptotic
phase transition for the size of the largest connected component established
in [4]. The behaviour for $\lambda=\lambda_{c}$ is more complicated, and will
not be discussed here.

Theorem 2.1 was proven in [3] for the special case of percolation by
conditioning on the set $A$ of vertices contained in the largest component and
computing the exponential rates of the events that $A$ is connected and that
$A^{c}$ does not contain any large components. We will extend this approach to
the random cluster model. To this end, let $K$ be the event that
$K_{n}(\omega)$ is connected, let $B_{r}$ be the event that $K_{n}(\omega)$
contains no components of size larger than $r$ and let $L$ be the event that
$K_{n}(\omega)$ is acyclic. The following three theorems, which generalise
([3], Theorems $2.3$, $2.4$ and $2.5$), yield the exponential rates of these
three events, beginning with the event $K$:

######  Theorem 2.3.

Fix $q>0$ and $\lambda>0$. Then

$\lim_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}K\big{]}=\log\pi_{1}(\lambda).$ (11)

Moreover, convergence is uniform for $\lambda$ belonging to compact subsets of
$[0,\infty)$.

Observe that the limit obtained in (11) is independent of $q$. This is no
coincidence; on the event $K$, $K_{n}(\omega)$ is connected and so
$k(\omega)=1$. It follows that $Z_{n,\lambda,q}[K]=q\phi_{n,\lambda}[K]$, and
so the proof given in ([3], Theorem $2.3$) for percolation trivially
generalises to the random cluster model.

Next, we compute the exponential rate of the event $L$. On this event, we have
the correspondence $k(\omega)=n-\lvert E_{n}(\omega)\rvert$ between the number
of components and edges of the graph $K_{n}(\omega)$. In particular, it is
possible to absorb the cluster weight $q$ into the edge weight and extend
([3], Theorem $2.4$) in the following form:

######  Theorem 2.4.

Fix $q>0$ and $\lambda>0$. Then

$\lim_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}L\big{]}=\lim_{r\to\infty}\lim_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}B_{r}\cap
L\big{]}=\Psi(\tfrac{\lambda}{q})-(\tfrac{q-1}{2q})\lambda+\log q.$ (12)

Moreover, convergence is uniform for $\lambda$ belonging to compact subsets of
$(0,\infty)\setminus\\{q\\}$.

As in [3], one may prove that the exponential rate of the event that
$K_{n}(\omega)$ contains only small components coincides with the exponential
rate of the event that $K_{n}(\omega)$ is acyclic. This is not surprising, as
in [4] it was shown that almost all vertices outside of the giant component
belong to trees. This argument is summarised in the following analogue of
([3], Theorem $2.5$):

######  Theorem 2.5.

Fix $q>0$ and $\lambda>0$. Then

$\lim_{r\to\infty}\liminf_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}B_{r}\big{]}=\lim_{r\to\infty}\limsup_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}B_{r}\big{]}=\lim_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}L\big{]}$ (13)

and

$\lim_{\epsilon\downarrow 0}\liminf_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}B_{\epsilon n}\big{]}=\lim_{\epsilon\downarrow
0}\limsup_{n\to\infty}\frac{1}{n}\log Z_{n,\lambda,q}\big{[}B_{\epsilon
n}\big{]}=\lim_{n\to\infty}\frac{1}{n}\log Z_{n,\lambda,q}\big{[}L\big{]}.$
(14)

Moreover, convergence is uniform for $\lambda$ belonging to compact subsets of
$(0,\infty)\setminus\\{q\\}$.

Observe that the preceding three theorems compute the exponential rates of the
events $K$, $L$ and $B_{r}$ for the random cluster measure before
normalisation. Indeed, we will compute the rate function for the size of the
largest connected component in the following form:

######  Theorem 2.6.

Fix $q>0$ and $\lambda>0$. Then, for every $\theta\in[0,1]$,

$\lim_{\epsilon\downarrow 0}\lim_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}\big{[}\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor\big{]}=\Phi(\theta,\lambda,q).$ (15)

Moreover, convergence is uniform for $\lambda$ belonging to compact subsets of
$(0,\infty)\setminus\\{q\\}$.

In order to turn these theorems into statements about probabilities in the
random cluster model, we will reintroduce the partition function using the
following result:

######  Theorem 2.7.

Fix $q>0$ and $\lambda>0$. Then

$\lim_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}=\sup_{\theta\in[0,1]}\Phi(\theta,\lambda,q).$ (16)

The limit of equation (16) is known as the free energy of the random cluster
model, and is not a new quantity of interest; it was computed in ([4], Theorem
$2.6$). In Lemma 5.2, we show that our computation agrees with theirs.

The structure of the free energy provides some hint as to its derivation.
Indeed, for fixed $\epsilon>0$, one may decompose the partition function
$Z_{n,\lambda,q}$ according to the number of vertices in components of size at
least $\epsilon n$ to obtain

$Z_{n,\lambda,q}=\sum_{k=0}^{n}Z_{n,\lambda,q}\big{[}\lvert\mathcal{V}_{\epsilon
n}\rvert=(\tfrac{k}{n})n\big{]}.$ (17)

We seek to apply Theorem 2.6 to each of the terms on the right hand side of
(17). We may then dominate the sum in (17) in terms of its largest summand,
and apply the Laplace Principle when taking the appropriate limit. However,
these steps involve some technicalities, which will be discussed in detail in
Section $5$. In particular, uniform convergence of the rate function is
required in order to pass to a supremum in the limit. This is the reason why
it is specified in our theorems, while it was not required for the random
graphs in [3].

##  3 Acyclic graphs

In this section, we prove Theorems 2.4 and 2.5 by analysing the quantity
$Z_{n,\lambda,q}[L\cap B_{r}]$, as was done for percolation in [3]. Let
$m_{l}$ be the number of connected components of size $l$ in $K_{n}(\omega)$.
Then, the weight of a configuration $\omega\in L\cap B_{r}$ is given by

$Z_{n,\lambda,q}[\omega]=q^{\sum m_{l}}(1-\tfrac{\lambda}{n})^{{n\choose
2}-n+\sum m_{l}}\prod_{l\geq 1}(\tfrac{\lambda}{n})^{(l-1)m_{l}}.$ (18)

Moreover, conditionally on the numbers $m_{1},\cdots,m_{n}$, there are
$\frac{n!}{\prod_{l}m_{l}!(l!)^{m_{l}}}$ ways of choosing appropriately sized
components in $K_{n}(\omega)$. In addition, each component of size $l$ has
$a_{l}=l^{l-2}$ possible spanning trees. Thus

$Z_{n,\lambda,q}[L\cap B_{r}]=\displaystyle\sum_{\begin{subarray}{c}\sum
lm_{l}=n\\\ m_{l}=0\forall
l>r\end{subarray}}\frac{n!}{\prod_{l}[m_{l}!(l!)^{m_{l}}]}q^{\sum
m_{l}}(1-\tfrac{\lambda}{n})^{{n\choose 2}-n+\sum m_{l}}\prod_{l\geq
1}\bigg{[}a_{l}(\tfrac{\lambda}{n})^{l-1}\bigg{]}^{m_{l}}.$ (19)

Introducing the sum

$Q_{n,k,r}=\displaystyle\sum_{\begin{subarray}{c}\sum lm_{l}=n\\\ \sum
m_{l}=k\\\ m_{l}=0\forall l>r\end{subarray}}\prod_{l\geq
1}\bigg{(}\frac{a_{l}}{l!}\bigg{)}^{m_{l}}\frac{1}{m_{l}!}$ (20)

allows us to rewrite (19) as

$Z_{n,\lambda,q}[L\cap
B_{r}]=n!(\tfrac{\lambda}{n})^{n}(1-\tfrac{\lambda}{n})^{{n\choose
2}-n}\sum_{k=1}^{n}(\tfrac{\lambda}{n})^{-k}[q(1-\tfrac{\lambda}{n})]^{k}Q_{n,k,r}.$
(21)

The equation (21) employs the same re-arrangement as that done for percolation
in [3], with the random cluster model introducing an extra factor of $q^{k}$
in the summand. We cite the following proposition regarding the asymptotic
behaviour of the quantity $Q_{n,k,r}$:

######  Proposition 3.1 ([3], Proposition $4.1$).

Consider the polynomial

$F_{r}(s)=\sum_{l=1}^{r}\frac{s^{l}a_{l}}{l!}.$ (22)

Then for all $n,k,r\geq 1$

$Q_{n,k,r}\leq\frac{1}{k!}\inf_{s>0}\frac{F_{r}(s)^{k}}{s^{n}}.$ (23)

Moreover, for each $\eta>0$ there is an $n_{0}<\infty$ and a sequence
$(c_{r})_{r\geq 1}$ of positive numbers such that for all $n\geq n_{0}$,
$k\geq 1$ and $r\geq 2$ such that $k<(1-\eta)n$ and $rk>n(1+\eta)$, we have

$Q_{n,k,r}\geq\frac{c_{r}}{\sqrt{n}}\frac{1}{k!}\inf_{s>0}\frac{F_{r}(s)^{k}}{s^{n}}.$
(24)

Applying Stirling’s Formula to the factorial term, we may use Proposition 3.1
to obtain an upper bound on the summand in (21) of

$(\tfrac{\lambda}{n})^{-k}[q(1-\tfrac{\lambda}{n})]^{k}Q_{n,k,r}\leq
e^{o(n)}\inf_{s>0}\exp\\{n\Theta_{r}(s,k/n,\lambda/q)\\},$ (25)

where the error term is bounded uniformly for $\lambda$ belonging to compact
subsets of $(0,\infty)$, and the function $\Theta_{r}(s,\theta,\alpha)$ is
given by

$\Theta_{r}(s,\theta,\alpha)=-\theta\log\alpha-\theta\log\theta+\theta+\theta\log
F_{r}(s)-\log s.$ (26)

The notation $\alpha=\lambda/q$ is chosen to agree with [3]. Notice that a sum
of $n$ terms is sandwiched between its maximal term and $n$ times its maximal
term. This implies an upper bound on the equation (21) of

$Z_{n,\lambda,q}[L\cap B_{r}]\leq
n!n(\tfrac{\lambda}{n})^{n}(1-\tfrac{\lambda}{n})^{{n\choose
2}-n}e^{o(n)}\sup_{k}\inf_{s>0}\exp\\{n\Theta_{r}(s,k/n,\lambda/q)\\}$ (27)

and a lower bound (for fixed $\eta>0$) on the equation (21) of

$Z_{n,\lambda,q}[L\cap B_{r}]\geq
n!(\tfrac{\lambda}{n})^{n}(1-\tfrac{\lambda}{n})^{{n\choose
2}-n}e^{o(n)}\sup_{k}\inf_{s>0}\exp\\{n\Theta_{r}(s,k/n,\lambda/q)\\},$ (28)

where the supremum is now taken over $k$ satisfying
$\tfrac{1}{r}(1+\eta)n<k<(1-\eta)n$. If we can show that the supremum of
$\Theta$ is contained in this interval (for sufficiently small $\eta$ and
sufficiently large $r$) then the bounds (27) and (28) will coincide. This
supremum is evaluated in ([3], Lemma $4.2$). We reproduce their lemma here, as
it will be important to check that any convergence is uniform:

######  Lemma 3.2 ([3], Lemma $4.2$).

Let $\alpha>0$ and $r\geq 2$. Then there is a unique
$(s_{r},\theta_{r})\in[0,\infty]\times[1/r,1]$ for which

$\Theta_{r}(s_{r},\theta_{r},\alpha)=\sup_{1/r\leq\theta\leq
1}\inf_{s>0}\Theta_{r}(s,\theta,\alpha).$ (29)

Moreover, $s_{r}\in(0,\infty)$ satisfies the limit

$\lim_{r\to\infty}s_{r}=\begin{cases}\alpha e^{-\alpha}&\text{ if }\alpha\leq
1\\\ \tfrac{1}{e}&\text{ if }\alpha>1\end{cases}$ (30)

and $\theta_{r}\in(1/r,1)$ satisfies the limit

$\lim_{r\to\infty}\theta_{r}=\begin{cases}1-\tfrac{\alpha}{2}&\text{ if
}\alpha\leq 1\\\ \tfrac{1}{2\alpha}&\text{ if }\alpha>1\end{cases}$ (31)

Combining (30) and (31) yields the limit

$\lim_{r\to\infty}\Theta_{r}(s_{r},\theta_{r},\alpha)=\begin{cases}1+\tfrac{\alpha}{2}-\log\alpha&\text{
if }\alpha\leq 1\\\ 1+\tfrac{1}{2\alpha}&\text{ if }\alpha>1\end{cases}$ (32)

Moreover, convergence is uniform for $\alpha$ belonging to compact subsets of
$(0,\infty)\setminus\\{1\\}$.

Note that we may rewrite the limit (32) as

$\lim_{r\to\infty}\Theta_{r}(s_{r},\theta_{r},\alpha)=1+\tfrac{\alpha}{2}-\log\alpha+\Psi(\alpha).$
(33)

As $\alpha=\lambda/q$, the convergence (in $r$) will be uniform for $\lambda$
belonging to compact subsets of $(0,\infty)\setminus\\{q\\}$, which is
precisely the claim of Theorem 2.4.

###### Proof.

We cite the fact that $(s_{r},\theta_{r})\in(0,\infty)\times(1/r,1)$ from
([3], Lemma $4.2$). By setting the partial derivatives of $\Theta_{r}$ equal
to $0$, we see the maximising pair $(s_{r},\theta_{r})$ is a solution of the
equations

$sF_{r}^{\prime}(s)=\alpha,\ F_{r}(s)=\alpha\theta.$ (34)

We analyse each of these equations in turn, beginning with the equation
$sF_{r}^{\prime}(s)=\alpha$. By definition, $sF_{r}^{\prime}(s)$ is equal to
the sum

$\sum_{l=1}^{r}a_{l}\frac{s^{l}}{(l-1)!}=\sum_{l=1}^{r}\frac{1}{l}\frac{l^{l}}{l!}s^{l}.$
(35)

Applying Stirling’s Formula to the term $l^{l}/l!$, we see that for $s>1/e$,
the sum in (35) diverges. In the case $s\leq 1/e$, (35) converges to the
function $W(s)$ satisfying $We^{-W}=s$ (a fact which we again cite from [3]).

We now split the argument into two cases, supposing first that $\alpha$
belongs to a bounded interval $[\alpha_{0},\alpha_{1}]$ with $\alpha_{1}<1$.
Define

$r_{1}=\inf\\{r:\ (1/e)F_{r}^{\prime}(1/e)>\alpha_{1}\\},$ (36)

and observe that for every $r\geq r_{1}$ and $\alpha\leq\alpha_{1}$, we must
have $s_{r}<1/e$. In particular, the error

$\Delta_{r}(s):=W(s)-sF_{r}^{\prime}(s)$ (37)

may be uniformly bounded by $\Delta_{r}:=\Delta_{r}(1/e)$, which converges to
$0$ as $r\to\infty$. As $s_{r}F_{r}^{\prime}(s_{r})=\alpha$, we see that
$\lvert W(s_{r})-\alpha\rvert\leq\Delta_{r}$. Moreover, by the Mean Value
Theorem, we have

$\lvert s_{r}-\alpha e^{-\alpha}\rvert=\lvert W(s_{r})e^{-W(s_{r})}-\alpha
e^{-\alpha}\rvert\leq ce^{-c}\lvert W(s_{r})-\alpha\rvert$ (38)

for some $c\in(\alpha-\Delta_{r},\alpha+\Delta_{r})$. As $ce^{-c}\leq 1$, we
deduce that $\lvert s_{r}-\alpha e^{-\alpha}\rvert\leq\Delta_{r}$, which
converges to $0$ uniformly.

Next, suppose that $\alpha\in[\alpha_{0},\alpha_{1}]$ with $\alpha_{0}>1$. In
this case, $s_{r}$ is bounded below by $1/e$, and we may discard lower order
terms in the sum (39) to obtain the bound

$sF_{r}^{\prime}(s)\geq r^{-3/2}(es)^{r}$ (39)

If we write $s=\gamma/e$, then (39) exceeds $\alpha_{1}$ if

$\gamma\geq\exp(\tfrac{1}{r}\log\alpha_{1}+\tfrac{3}{2r}\log r):=\gamma_{r}.$
(40)

Thus $\lvert s_{r}-1/3\rvert\leq(\gamma_{r}-1)/e$, which converges to $0$
uniformly.

It remains to analyse the equation $F_{r}(s)=\alpha\theta$, separating the
argument into the same two cases as before. Suppose first that
$\alpha\in[\alpha_{0},\alpha_{1}]$ with $\alpha_{1}<1$. To find a limit for
$\theta_{r}$, we find a limit for $F_{r}(s_{r})$ by integrating the equation

$sF_{r}^{\prime}(s)=W(s)-\Delta_{r}(s).$ (41)

Using the identity $We^{-W}=s$ and its derivative $W^{\prime}(1-W)e^{-W}=1$
with respect to $s$, we obtain the differential equation

$F_{r}^{\prime}(s)=W^{\prime}(1-W)-\frac{\Delta_{r}(s)}{s},$ (42)

which has the solution

$F_{r}(s)=W-\tfrac{1}{2}W^{2}-\int_{0}^{s}\tfrac{\Delta_{r}(t)}{t}dt.$ (43)

Plugging in the equations $F_{r}(s_{r})=\alpha\theta_{r}$ and
$W(s_{r})=\alpha+\Delta_{r}(s_{r})$ yields

$\theta_{r}=1-\tfrac{1}{2}\alpha+\tfrac{1}{\alpha}\bigg{(}\Delta_{r}(s_{r})-\alpha\Delta_{r}(s_{r})-\tfrac{1}{2}\Delta_{r}(s_{r})^{2}-\int_{0}^{s_{r}}\tfrac{\Delta_{r}(t)}{t}dt\bigg{)}.$
(44)

As $\alpha$ is bounded away from $0$, $s_{r}$ is bounded below $1/e$ for $r$
sufficiently large, and $\Delta_{r}(t)/t$ converges to $0$ as $r\to\infty$,
the error term in (44) converges to $0$ uniformly.

Finally, suppose $\alpha\in[\alpha_{0},\alpha_{1}]$ with $\alpha_{0}>1$, and
recall that in this case, $s_{r}$ converges to $e^{-1}$ uniformly. By the
triangle inequality, we have

$\lvert F_{r}(s_{r})-\tfrac{1}{2}\rvert\leq\lvert
F_{r}(s_{r})-F_{r}(e^{-1})\rvert+\lvert F_{r}(e^{-1})-\tfrac{1}{2}\rvert.$
(45)

The second term converges to $0$ independently of $\alpha$. For the first, we
apply the Mean Value Theorem to obtain

$\lvert F_{r}(s_{r})-F_{r}(e^{-1})\rvert\leq F_{r}^{\prime}(c)\lvert
s_{r}-e^{-1}\rvert,$ (46)

for some $c\in(e^{-1},s_{r})$. Noting that $F_{r}^{\prime}(s)$ is increasing
and $F_{r}^{\prime}(s_{r})=\alpha/s_{r}$ is bounded, we deduce that the first
term must also converge uniformly. The result follows after substituting
$F_{r}(s_{r})=\alpha\theta_{r}$. ∎

Note that the supremum in Lemma 3.2 is taken over the interval
$\theta\in[1/r,1]$, whereas the sums in the bounds of equations (27) and (28)
are over discrete subsets of the interval $[1/r,1]$. We claim that the
supremums over these two sets coincide in the limit as $n\to\infty$. Let
$\theta_{r,n}=\tfrac{1}{n}\lfloor\theta_{r}n\rfloor$ and define $s_{r,n}$ to
be the number $s$ satisfying

$\Theta_{r}(s_{r,n},\theta_{r,n},\alpha)=\inf_{s>0}\Theta_{r}(s,\theta_{r,n},\alpha).$
(47)

It is sufficient to prove that the pair $(s_{r,n},\theta_{r,n})$ converges to
the pair $(\theta_{r},s_{r})$ as $n\to\infty$:

######  Lemma 3.3.

Fix $\alpha>0$ and $r\geq 2$. Then

$\lim_{n\to\infty}\Theta_{r}(s_{r,n},\theta_{r,n},\alpha)=\Theta_{r}(s_{r},\theta_{r},\alpha).$
(48)

Moreover, convergence is uniform for $\alpha$ in compact subsets of
$(0,\infty)\setminus\\{1\\}$.

To prove Lemma 3.3, we will need the following proposition:

######  Proposition 3.4.

Let $P(x)=\sum_{r=0}^{n}a_{r}x^{r}$ be a polynomial with non-negative co-
efficients, and consider the function

$Q(x)=\frac{P(x)}{xP^{\prime}(x)}.$ (49)

Then for $x>0$, $Q(x)$ is decreasing.

###### Proof.

Differentiation yields

$Q^{\prime}(x)=\frac{1}{x^{2}P^{\prime}(x)^{2}}\bigg{(}xP^{\prime}(x)^{2}-P(x)P^{\prime}(x)-xP(x)P^{\prime\prime}(x)\bigg{)}.$
(50)

It will be sufficient to show that

$x^{3}P^{\prime}(x)^{2}Q^{\prime}(x)=\sum_{s=0}^{n}\sum_{t=0}^{n}[sta_{s}a_{t}-ta_{s}a_{t}-t(t-1)a_{s}a_{t}]x^{s+t},$
(51)

is negative. Writing $m=s+t$ (and noting that the sum is the same when
interchanging the order of summation of $s$ and $t$) this may be rewritten as

$\displaystyle x^{3}P^{\prime}(x)^{2}Q(x)$
$\displaystyle=\tfrac{1}{2}\sum_{m=0}^{2n}\sum_{r=0}^{n}[2r(m-r)-m-r(r-1)-(m-r)(m-r-1)]a_{r}a_{m-r}x^{m}$
$\displaystyle=-\tfrac{1}{2}\sum_{m=0}^{2n}\sum_{r=0}^{n}(m-2r)^{2}a_{r}a_{m-r}x^{m}$
(52)

where we have set $a_{l}=0$ for any $l<0$. This is negative, as required. ∎

###### Proof of Lemma 3.3.

Recall that

$\Theta_{r}(s,\theta,\alpha)=-\theta\log\alpha-\theta\log\theta+\theta+\theta\log
F_{r}(s)-\log s.$ (53)

Using the triangle inequality, we may bound the difference between
$\Theta_{r}(s_{r,n},\theta_{r,n},\alpha)$ and
$\Theta_{r}(s_{r},\theta_{r},\alpha)$ by the sum of the differences of the
terms in (53). If we can show that each of these differences converges to $0$,
we will be done:

  1. 1.

First, consider the term $\lvert
1-\log\alpha\rvert\lvert\theta_{r}-\theta_{r,n}\rvert$, and note that
$\lvert\theta_{r}-\theta_{r,n}\rvert\leq 1/n$ by definition. As the function
$1-\log\alpha$ is uniformly bounded on compact subsets of
$\alpha\in(0,\infty)\setminus\\{1\\}$, this term converges uniformly to $0$ as
$n\to\infty$.

  2. 2.

Next, consider the term
$\lvert\theta_{r}\log\theta_{r}-\theta_{r,n}\log\theta_{r,n}\rvert$. By the
triangle inequality, we have

$\lvert\theta_{r}\log\theta_{r}-\theta_{r,n}\log\theta_{r,n}\rvert\leq\lvert\theta_{r}(\log\theta_{r}-\log\theta_{r,n})\rvert+\lvert(\theta_{r}-\theta_{r,n})\log\theta_{r,n}\rvert,$
(54)

where both of the terms in the right hand side converge uniformly to $0$ by
uniform continuity of the logarithm away from $0$.

  3. 3.

Next, consider the term $\lvert\log s_{r}-\log s_{r,n}\rvert$. Recall that, at
a stationary point, $\theta$ is given by

$\theta=\frac{F_{r}(s)}{sF_{r}^{\prime}(s)}.$ (55)

By Proposition 3.4, $\theta$ is decreasing when viewed as a function of $s$.
Fix $a<b$ such that, on our chosen compact subset of
$\alpha\in(0,\infty)\setminus\\{1\\}$,

$\theta(b)<\theta_{r,n}<\theta_{r}<\theta(a).$ (56)

On $[a,b]$, $\theta(s)$ is continuous and injective. In particular, it has a
uniformly continuous inverse $s=h(\theta)$ on the compact set
$[\theta(b),\theta(a)]$, and so $s_{r}-s_{r,n}$ converges uniformly to $0$. As
the logarithm is uniformly continuous on intervals bounded away from $0$, the
same holds for $\log s_{r}-\log s_{r,n}$.

  4. 4.

Finally, consider the term $\lvert\theta_{r}\log F_{r}(s_{r})-\theta_{r,n}\log
F_{r}(s_{r,n})\rvert$. By the triangle inequality, we have

$\displaystyle\lvert\theta_{r}\log F_{r}(s_{r})-\theta_{r,n}\log
F_{r}(s_{r,n})\rvert\leq$ $\displaystyle\lvert\theta_{r}\log
F_{r}(s_{r})-\theta_{r}\log F_{r}(s_{r,n})\rvert$
$\displaystyle+\lvert\theta_{r}\log F_{r}(s_{r,n})-\theta_{r,n}\log
F_{r}(s_{r,n})\rvert.$ (57)

The second term converges to $0$ as $n\to\infty$ by continuity of $F_{r}$. For
the first term, we observe that

$\log F_{r}(s_{r,n})-\log
F_{r}(s_{r})=\int_{s_{r}}^{s_{r,n}}\frac{F_{r}^{\prime}(t)}{F_{r}(t)}dt\leq
r\int_{s_{r}}^{s_{r,n}}t\ dt\leq rs_{r,n}(s_{r,n}-s_{r}).$ (58)

For $n$ sufficiently large, $s_{r,n}$ is uniformly bounded and so this
converges to uniformly.

∎

We have established that $\theta_{r}$ converges to a limit uniformly for
$\lambda$ belonging to compact subsets of $(0,\infty)\setminus\\{q\\}$.
Moreover, this limit belongs to a compact subset of $(0,1)$. In particular,
one may choose $\eta>0$ sufficiently small and $r>0$, $n>0$ sufficiently large
to ensure that $\theta_{r,n}n$ belongs to the interval
$[\tfrac{1}{r}(1+\eta)n,(1-\eta)n]$. Then, we may apply Proposition 3.1 to the
bounds of equations (27) and (28) to obtain

$Z_{n,\lambda,q}[L\cap
B_{r}]=e^{o(n)}\exp\\{n(-1-\tfrac{1}{2}\lambda+\log\lambda+\Theta_{r}(s_{r,n},\theta_{r,n},\lambda/q))\\},$
(59)

where the $o(n)$ term is bounded uniformly for $\lambda$ belonging to compact
subsets of $(0,\infty)\setminus\\{q\\}$.

###### Proof of Theorem 2.4.

From (59), we have that

$\displaystyle\frac{1}{n}\log Z_{n,\lambda,q}[L\cap B_{r}]$
$\displaystyle=\Theta_{r}(s_{r,n},\theta_{r,n},\lambda/q)-\Theta_{r}(s_{r},\theta_{r},\lambda/q)$
$\displaystyle+\Theta_{r}(s_{r},\theta_{r},\lambda/q)-1-\tfrac{1}{2}\lambda+\log\lambda$
$\displaystyle+\tfrac{o(n)}{n}.$ (60)

Take the limit of (3) as $n\to\infty$ and $r\to\infty$ in that order. The
first line converges to $0$ by Lemma 3.3. The second line converges to the
required rate by Lemma 3.2. ∎

Next, we prove Theorem 2.5. We need one preliminary lemma, corresponding to
([3], Lemma $5.1$):

######  Lemma 3.5.

Fix $q>0$ and $\lambda>0$. Then

$Z_{n,\lambda,q}[B_{r}]\leq
Z_{n,\lambda,q}[L](1-\tfrac{\lambda}{n})^{-\tfrac{1}{2}rn}.$ (61)

###### Proof.

We follow the proof of ([3], Lemma $5.1$), showing that

$\phi_{n,\lambda,q}[B_{r}]\leq\phi_{n,\lambda,q}[L](1-\tfrac{\lambda}{n})^{-\tfrac{1}{2}rn}.$
(62)

Given a set of vertices $S\subset\\{1,\cdots,n\\}$, let $C_{S}$ denote the
restriction of the graph $K_{n}(\omega)$ to $S$, and let $T$ be a tree on $S$.
Conditionally on the event $C_{S}\supset T$, all vertices of $S$ belong to the
same component of $K_{n}(\omega)$. In particular, any edge in $E(S)\setminus
T$ is open independently with probability $\lambda/n$, and so

$\frac{\phi_{n,\lambda,q}[C_{S}=T]}{\phi_{n,\lambda,q}[C_{S}\supset
T]}=(1-\tfrac{\lambda}{n})^{{|S|\choose
2}-|S|+1}\geq(1-\tfrac{\lambda}{n})^{\tfrac{1}{2}|S|^{2}}.$ (63)

Let $K_{S}$ be the event that $C_{S}$ is connected. Then

$\phi_{n,\lambda,q}[K_{S}]\leq\sum_{T}\phi_{n,\lambda,q}[C_{S}\supset
T]\leq(1-\tfrac{\lambda}{n})^{-\tfrac{1}{2}|S|^{2}}\phi_{n,\lambda,q}[C_{S}\text{
is a tree}].$ (64)

Now, let $L_{r}$ denote the event that each component of $K_{n}(\omega)$ is
either acyclic or has size at most $r$, and note that $B_{r}\subset L_{r}$.
Let $\\{S_{j}\\}$ be a partition of $\\{1,\cdots,n\\}$ and let
$\phi_{n,\lambda,q}[\\{S_{j}\\}]$ denote the probability that $\\{S_{j}\\}$
are the connected components of $K_{n}(\omega)$. Conditioning the event
$L_{r}$ on the partition $\\{S_{j}\\}$ of connected components, we have

$\phi_{n,\lambda,q}[L_{r}]=\sum_{\\{S_{j}\\}}\phi_{n,\lambda,q}[\\{S_{j}\\}]\phi_{n,\lambda,q}[L_{r}\mid\\{S_{j}\\}].$
(65)

Moreover, conditionally on the partition $\\{S_{j}\\}$ of connected
components, the states of edges in different components are independent. In
particular, we may write

$\displaystyle\phi_{n,\lambda,q}[L_{r}|\\{S_{j}\\}]$
$\displaystyle=\prod_{j:\lvert
S_{j}\rvert>r}\phi_{n,\lambda,q}[C_{S_{j}}\text{ is a tree}\mid K_{S_{j}}]$
$\displaystyle=\prod_{j}\phi_{n,\lambda,q}[C_{S_{j}}\text{ is a tree}\mid
K_{S_{j}}]\prod_{j:\lvert S_{j}\rvert\leq r}\phi_{n,\lambda,q}[C_{S_{j}}\text{
is a tree}\mid K_{S_{j}}]^{-1}$
$\displaystyle\leq\prod_{j}\phi_{n,\lambda,q}[C_{S_{j}}\text{ is a tree}\mid
K_{S_{j}}]\prod_{j:|S_{j}|\leq r}(1-\tfrac{\lambda}{n})^{-\tfrac{1}{2}\lvert
S_{j}\rvert^{2}}$

where the inequality in the third line is a consequence of (63). As $\lvert
S_{j}\rvert<r$ for every term in the second product and the sum over $\lvert
S_{j}\rvert$ is at most $n$, we obtain

$\phi_{n,\lambda,q}[L_{r}\mid\\{S_{j}\\}]\leq\phi_{n,\lambda,q}[L\mid\\{S_{j}\\}](1-\tfrac{\lambda}{n})^{-\tfrac{1}{2}rn}.$
(66)

Finally, we see that

$\displaystyle\phi_{n,\lambda,q}[B_{r}]$
$\displaystyle\leq\phi_{n,\lambda,q}[L_{r}]$
$\displaystyle=\sum_{\\{S_{j}\\}}\phi_{n,\lambda,q}[\\{S_{j}\\}]\phi_{n,\lambda,q}[L_{r}\mid\\{S_{j}\\}]$
$\displaystyle\leq\sum_{\\{S_{j}\\}}\phi_{n,\lambda,q}[\\{S_{j}\\}]\phi_{n,\lambda,q}[L\mid\\{S_{j}\\}](1-\tfrac{\lambda}{n})^{-\tfrac{1}{2}rn}$
$\displaystyle=\phi_{n,\lambda,q}[L](1-\tfrac{\lambda}{n})^{-\tfrac{1}{2}rn}$

which establishes (62). We obtain (61) by multiplying both sides of (62) by
the partition function. ∎

###### Proof of Theorem 2.5.

The theorem is a consequence of the following inequalities:

$\displaystyle\lim_{n\to\infty}\frac{1}{n}\log Z_{n,\lambda,q}[L]$
$\displaystyle\leq\lim_{r\to\infty}\liminf_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[B_{r}],$ (67)
$\displaystyle\lim_{r\to\infty}\liminf_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[B_{r}]$ $\displaystyle\leq\lim_{\epsilon\downarrow
0}\liminf_{n\to\infty}\frac{1}{n}\log Z_{n,\lambda,q}[B_{\epsilon n}],$
$\displaystyle\lim_{r\to\infty}\limsup_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[B_{r}]$ $\displaystyle\leq\lim_{\epsilon\downarrow
0}\limsup_{n\to\infty}\frac{1}{n}\log Z_{n,\lambda,q}[B_{\epsilon n}],$
$\displaystyle\lim_{\epsilon\downarrow 0}\limsup_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[B_{\epsilon n}]$
$\displaystyle\leq\lim_{n\to\infty}\frac{1}{n}\log Z_{n,\lambda,q}[L].$

To prove the first inequality, we apply the inclusion $B_{r}\supset B_{r}\cap
L$ and Theorem 2.4 to see that

$\liminf_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[B_{r}]\geq\liminf_{n\to\infty}\frac{1}{n}Z_{n,\lambda,q}[B_{r}\cap
L]\to\lim_{n\to\infty}\frac{1}{n}\log Z_{n,\lambda,q}[L]$ (68)

as $r\to\infty$. To prove the second inequality, fix $r\geq 2,\ \epsilon>0$,
and let $N=\lceil r/\epsilon\rceil$. Then, for every $n\geq N$, we have
$\epsilon n\geq\epsilon\lceil r/\epsilon\rceil\geq r.$ As a result,
$B_{\epsilon n}\supset B_{r}$, and

$\frac{1}{n}\log Z_{n,\lambda,q}[B_{r}]\leq\frac{1}{n}\log
Z_{n,\lambda,q}[B_{\epsilon n}].$ (69)

One may take the infimum over $m\geq n$, followed by the limit as
$n\to\infty$, on both sides to obtain

$\liminf_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[B_{r}]\leq\liminf_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[B_{\epsilon n}].$ (70)

As $r$ and $\epsilon$ were arbitrary, taking the limits $r\to\infty$ and
$\epsilon\downarrow 0$ yields the second inequality. The proof of the third
inequality is similar. To prove the fourth inequality, we apply Lemma 3.5 to
obtain

$\frac{1}{n}\log Z_{n,\lambda,q}[B_{\epsilon n}]\leq\frac{1}{n}\log
Z_{n,\lambda,q}[L]+\tfrac{1}{2}\lambda\epsilon,$ (71)

from which the inequality follows after taking the limit superior as
$n\to\infty$ and the limit as $\epsilon\downarrow 0$. ∎

##  4 Uniqueness and the largest component

In this section, we prove Theorem 2.6 and Lemma 2.2. The proof of the latter
is identical to that of ([3], Lemma $6.2$), and requires only the following
analogue of ([3], Lemma $6.1$), which estimates the probability of the event
$K_{\epsilon,2}$ that $K_{n}(\omega)$ is connected or has exactly two
connected components each of size at least $\epsilon n$:

######  Lemma 4.1 ([3], Lemma $6.1$).

Fix $q>0$. Then for all $\lambda_{0}>0$ and $\epsilon_{0}>0$ there exists
$c_{1}=c_{1}(\lambda_{0},\epsilon_{0})>0$ such that for all
$\epsilon\geq\epsilon_{0}$ and $\lambda\leq\lambda_{0}$, we have

$\limsup_{n\to\infty}\frac{1}{n}\log\phi_{n,\lambda,q}[K^{c}|K_{\epsilon,2}]<-c_{1}.$
(72)

###### Proof.

Observe that

$\phi_{n,\lambda,q}[K^{c}|K_{\epsilon,2}]=\frac{\phi_{n,\lambda,q}[K_{\epsilon,2}\setminus
K]}{\phi_{n,\lambda,q}[K_{\epsilon,2}\setminus
K]+\phi_{n,\lambda,q}[K]}\leq\frac{\phi_{n,\lambda,q}[K_{\epsilon,2}\setminus
K]}{\phi_{n,\lambda,q}[K]}.$ (73)

It suffices to show that the ratio on the right hand side of (73) decays to
zero exponentially in $n$, with a rate that is uniformly bounded in
$\epsilon\geq\epsilon_{0}$ and $\lambda\leq\lambda_{0}$. Observe that
$\omega\in K_{\epsilon,2}\setminus K$ if and only if we may find a set
$A\subset K_{n}$ (where $A$ depends on $\omega$) of vertices of size between
$\epsilon n$ and $n-\epsilon n$ such that $A$, $A^{c}$ are connected
components of $K_{n}(\omega)$ and there are no open edges between them. We
count the configurations satisfying these conditions. Let $E(A,A^{c})$ be the
set of edges between $A$ and its complement in $K_{n}(\omega)$, and suppose
that $|A\rvert=k$. It can be seen that $A$ is disconnected from $A^{c}$ in
$K_{n}(\omega)$ with probability

$\phi_{n,\lambda,q}[E(A,A^{c})=\emptyset]=\frac{Z_{k,\lambda
k/n,q}Z_{n-k,\lambda(1-k/n),q}}{Z_{n,\lambda,q}}(1-\lambda/n)^{k(n-k)}.$ (74)

Conditionally on the above event, $A$ is connected in $K_{n}(\omega)$ with
probability

$\phi_{k,\lambda k/n,q}[K]=Z_{k,\lambda k/n,q}[K]/Z_{k,\lambda k/n,q},$ (75)

and $A^{c}$ is connected in $K_{n}(\omega)$ with probability

$\phi_{n-k,\lambda(1-k/n),q}[K]=Z_{n-k,\lambda(1-k/n),q}[K]/Z_{n-k,\lambda(1-k/n),q}.$
(76)

Note that there are ${n\choose k}$ choices for $A$, and that we have counted
any pair $(A,A^{c})$ twice. Thus, we have the equation

$Z_{n,\lambda,q}[K_{\epsilon,2}\setminus K]=\frac{1}{2}\sum_{\epsilon n\leq
k\leq n-\epsilon n}{n\choose
k}\bigg{(}1-\frac{\lambda}{n}\bigg{)}^{k(n-k)}Z_{k,\lambda
k/n,q}[K]Z_{n-k,\lambda(1-k/n),q}[K].$

Applying Theorem 2.3, we obtain

$\frac{\phi_{n,\lambda,q}[K_{\epsilon,2}\setminus
K]}{\phi_{n,\lambda,q}[K]}=e^{o(n)}\sum_{\epsilon n\leq k\leq n-\epsilon
n}{n\choose
k}\frac{\pi_{1}(\lambda\tfrac{k}{n})^{k}\pi_{1}(\lambda(1-\tfrac{k}{n}))^{n-k}}{\pi_{1}(\lambda)^{n}}\bigg{(}1-\frac{\lambda}{n}\bigg{)}^{k(n-k)},$
(77)

where $\pi_{1}(\lambda)=1-e^{-\lambda}$, and the $e^{o(n)}$ term is the error
term found in Theorem 2.3. The lemma now concludes identically to ([3], Lemma
$6.1$), rewriting (77) as

$\frac{\phi_{n,\lambda,q}[K_{\epsilon,2}\setminus
K]}{\phi_{n,\lambda,q}[K]}=e^{o(n)}\sum_{\epsilon n\leq k\leq n-\epsilon
n}e^{n[\Xi(k/n)-\Xi(0)]},$ (78)

where the function $\Xi$ is defined by

$\Xi(\theta)=-S(\theta)+\theta\log\pi_{1}(\lambda\theta)+(1-\theta)\log\pi_{1}(\lambda(1-\theta))-\lambda\theta(1-\theta).$
(79)

We now bound the sum in (78) by $n$ times its maximal summand. As $\Xi$ is
convex and symmetric around the point $1/2$, the summand is maximised at the
endpoints. In particular, we have the bound

$\frac{\phi_{n,\lambda,q}[K_{\epsilon,2}\setminus
K]}{\phi_{n,\lambda,q}[K]}\leq e^{o(n)}e^{n[\Xi(\epsilon)-\Xi(0)]}.$ (80)

More explicitly, we may take any value $c_{1}<\Xi(0)-\Xi(\epsilon)$ provided
that $n$ is sufficiently large. ∎

###### Proof of Lemma 2.2.

The proof is identical to that of ([3], Lemma $6.2$). In particular, it will
suffice to prove that

$\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor,\ \mathcal{N}_{\epsilon n}>1]\leq
e^{-cn}\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor].$ (81)

For a given vertex $x$, let $\mathcal{C}_{x}$ denote the component of
$K_{n}(\omega)$ containing $x$. On the event $\\{\lvert\mathcal{V}_{\epsilon
n}\rvert=\lfloor\theta n\rfloor,\mathcal{N}_{\epsilon n}>1\\}$, we may find a
pair of vertices $x,y\in[n]$ in $K_{n}(\omega)$ such that
$\lvert\mathcal{C}_{x}\rvert\geq\epsilon n$,
$\lvert\mathcal{C}_{y}\rvert\geq\epsilon n$ and $x\nleftrightarrow y$. Define
the following two events for the random graph:

$\displaystyle A_{1}$
$\displaystyle=\\{\lvert\mathcal{C}_{x}\rvert\geq\epsilon
n\\}\cap\\{\lvert\mathcal{C}_{y}\rvert\geq\epsilon
n\\}\cap\\{x\nleftrightarrow y\\},$ $\displaystyle A_{2}$
$\displaystyle=\\{\lvert\mathcal{C}_{x}\rvert\geq\epsilon
n\\}\cap\\{\lvert\mathcal{C}_{y}\rvert\geq\epsilon n\\}.$

As the complete graph is transitive, the probabilities of the events $A_{1}$
and $A_{2}$ do not depend on the particular choices of $x$ and $y$. By the
union bound, it follows that

$\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor,\mathcal{N}_{\epsilon n}>1]\leq
n^{2}\phi_{n,\lambda,q}[\\{\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor\\}\cap A_{1}].$ (82)

We now condition further on the set $\mathcal{C}_{x}\cup\mathcal{C}_{y}$. For
a given set $\mathcal{C}\subset[n]$, let $D$ be the event that $\mathcal{C}$
is disjoint from $\mathcal{C}^{c}$ in $K_{n}(\omega)$ and that its complement
contains $\lfloor\theta n\rfloor-\lvert\mathcal{C}\rvert$ vertices in
components of size at least $\epsilon n$. Then

$\displaystyle\phi_{n,\lambda,q}[\\{\lvert\mathcal{V}_{\epsilon
n}\rvert=\lfloor\theta n\rfloor\\}\cap A_{1}]$
$\displaystyle=\sum_{\mathcal{C}\subset[n]}\phi_{n,\lambda,q}[A_{1}\cap\\{\mathcal{C}_{x}\cup\mathcal{C}_{y}=\mathcal{C}\\}\cap
D],$
$\displaystyle=\sum_{\mathcal{C}\subset[n]}\phi_{n,\lambda,q}[A_{1}\cap\\{\mathcal{C}_{x}\cup\mathcal{C}_{y}=\mathcal{C}\\}\mid
D]\phi_{n,\lambda,q}[D].$

Write $m=\lvert\mathcal{C}\rvert$, $\tilde{\lambda}=\lambda\theta$, and
$\tilde{\epsilon}=\epsilon/\theta$. On the event $D$, the random cluster
measure restricts to the random cluster measure $\phi_{\theta
n,\tilde{\lambda},q}$ on $\mathcal{C}$. Moreover, we have the following
correspondences between events:

$\displaystyle A_{1}\cap\\{\mathcal{C}_{x}\cup\mathcal{C}_{y}=\mathcal{C}\\}$
$\displaystyle=K^{c}\cap K_{\tilde{\epsilon},2},$ $\displaystyle
A_{2}\cap\\{\mathcal{C}_{x}\cup\mathcal{C}_{y}=\mathcal{C}\\}$
$\displaystyle=K_{\tilde{\epsilon},2}.$

As $\tilde{\epsilon}\geq\epsilon$ for every $\theta>0$, we may apply Lemma 4.1
to deduce that

$\phi_{n,\lambda,q}[A_{1}\cap\\{\mathcal{C}_{x}\cup\mathcal{C}_{y}=\mathcal{C}\\}\mid
D]\leq
e^{-c_{1}m}\phi_{n,\lambda,q}[A_{2}\cap\\{\mathcal{C}_{x}\cup\mathcal{C}_{y}=\mathcal{C}\\}\mid
D],$ (83)

which allows us to rewrite (82) as

$\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor,\mathcal{N}_{\epsilon n}>1]\leq
n^{2}e^{-c_{1}m}\phi_{n,\lambda,q}[\\{\lvert\mathcal{V}_{\epsilon
n}\rvert=\lfloor\theta n\rfloor\\}\cap A_{2}].$ (84)

The result follows as $\\{\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor\\}\cap A_{2}\subset\\{\lvert\mathcal{V}_{\epsilon
n}\rvert=\lfloor\theta n\rfloor\\}$ and $m\geq\epsilon n$. ∎

###### Proof of Theorem 2.6.

By Lemma 2.2, it is sufficient to prove that

$\lim_{\epsilon\downarrow 0}\lim_{n\to\infty}\frac{1}{n}\log
Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=\lfloor\theta
n\rfloor,\mathcal{N}_{\epsilon n}=1]=\Phi(\theta,\lambda,q).$ (85)

The case $\theta=1$ reduces to Theorem 2.3 and the case $\theta=0$ reduces to
Theorems 2.4 and 2.5. Let $\theta\in(0,1)$, $\epsilon\in(0,\theta)$ and assume
that $\theta n$ is an integer. Given a configuration $\omega$, observe that
$\omega\in\\{\lvert\mathcal{V}_{\epsilon n}\rvert=\theta n,N_{\epsilon
n}=1\\}$ if and only if we may find a subset $A\subset K_{n}$ of vertices
(where $A$ depends on $\omega$) of size $\theta n$ such that $A$ is a
connected component of $K_{n}(\omega)$, $A^{c}$ contains no connected
components of $K_{n}(\omega)$ of size exceeding $\epsilon n$, and
$E(A,A^{c})=0$. We count the possible configurations which satisfy these
conditions. Note that there are ${n\choose\theta n}$ possible choices for $A$,
and $A$ is disconnected from $A^{c}$ in $K_{n}(\omega)$ with probability

$\phi_{n,\lambda,q}[E(A,A^{c})=0]=\frac{Z_{\theta
n,\lambda\theta,q}Z_{(1-\theta)n,\lambda(1-\theta),q}}{Z_{K_{n},p,q}}(1-\lambda/n)^{\theta(1-\theta)n^{2}}.$
(86)

Conditionally on this event, $A$ is connected in $K_{n}(\omega)$ with
probability

$\phi_{\theta n,\lambda\theta,q}[K]=Z_{\theta n,\lambda\theta,q}[K]/Z_{\theta
n,\lambda\theta,q},$ (87)

and $A^{c}$ does not contain any components of size exceeding $\epsilon n$ in
$K_{n}(\omega)$ with probability

$\phi_{(1-\theta)n,\lambda(1-\theta),q}[B_{\epsilon
n}]=Z_{(1-\theta)n,\lambda(1-\theta),q}[B_{\epsilon
n}]/Z_{(1-\theta)n,\lambda(1-\theta),q}.$ (88)

Thus, we have the equation

$Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=\theta n,N_{\epsilon
n}=1]={n\choose\theta n}(1-\tfrac{\lambda}{n})^{\theta n(1-\theta)n}Z_{\theta
n,\lambda\theta,q}[K]Z_{(1-\theta)n,\lambda(1-\theta),q}[B_{\epsilon n}].$
(89)

We now take logarithms, divide by $n$ and take the limit as $n\to\infty$. Note
first that

$\lim_{n\to\infty}\frac{1}{n}\log\bigg{(}{n\choose\theta
n}(1-\lambda/n)^{\theta
n(1-\theta)n}\bigg{)}=-S(\theta)+(1-\theta)\log[1-\pi_{1}(\lambda\theta)].$
(90)

Applying Theorem 2.3, we also have

$\lim_{n\to\infty}\frac{1}{n}\log Z_{\theta
n,\lambda\theta,q}[K]=\theta\log\pi_{1}(\lambda\theta).$ (91)

Finally, we apply Theorems 2.4 and 2.5 to obtain

$\displaystyle\lim_{\epsilon\downarrow 0}\lim_{n\to\infty}\frac{1}{n}\log
Z_{(1-\theta)n,\lambda(1-\theta),q}[B_{\epsilon n}]=$
$\displaystyle(1-\theta)\times$ (92)
$\displaystyle\bigg{\\{}\Psi(\tfrac{\lambda(1-\theta)}{q})-(\tfrac{q-1}{2q})\lambda(1-\theta)+\log
q\bigg{\\}}.$

Summing these limits yields the result. Convergence is uniform as each of the
individual limits converge uniformly. ∎

##  5 Free energy

In this section, we prove Theorem 2.7. To begin, we let $\epsilon>0$. Then, we
may decompose the partition function as

$Z_{n,\lambda,q}=Z_{n,\lambda,q}[B_{\epsilon n}]+\sum_{k>\epsilon
n}Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=k].$ (93)

By Lemma 2.2, we may write

$Z_{n,\lambda,q}=Z_{n,\lambda,q}[B_{\epsilon n}]+(1-o(1))\sum_{k>\epsilon
n}Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=k,\
\mathcal{N}_{\epsilon n}=1].$ (94)

We aim to apply Theorem 2.6 to each summand in (94). Recall that

$Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert=k,N_{\epsilon
n}=1]={n\choose k}(1-\tfrac{\lambda}{n})^{k(n-k)}Z_{k,\lambda
k/n,q}[K]Z_{n-k,\lambda(1-k/n),q}[B_{\epsilon n}].$ (95)

In order to apply Theorem 2.6 to all of the summands in (94) simultaneously,
we require that the term $\lambda(1-k/n)$ belongs to a compact subset of
$(0,\infty)\setminus\\{q\\}$. It will suffice to prove that the terms for
which $k/n$ is close to $1$ or $1-q/\lambda$ have negligible probability,
which we do via the following two tail inequalities for sufficiently small
$\epsilon$:

$\displaystyle Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon
n}\rvert\geq(1-\epsilon)n,\ \mathcal{N}_{\epsilon n}=1]$ $\displaystyle\leq
o(1)Z_{n,\lambda,q},$ (96) $\displaystyle
Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon
n}\rvert\leq(1-q/\lambda+\epsilon)n,\ \mathcal{N}_{\epsilon n}=1]$
$\displaystyle\leq o(1)Z_{n,\lambda,q}.$

Equivalently, we show that

$\displaystyle\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon
n}\rvert\geq(1-\epsilon)n,\ \mathcal{N}_{\epsilon n}=1]$ $\displaystyle\leq
o(1),$ (97) $\displaystyle\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon
n}\rvert\leq(1-q/\lambda+\epsilon)n,\ \mathcal{N}_{\epsilon n}=1]$
$\displaystyle\leq o(1).$

Both inequalities in (97) may be proven via direct comparisons with
percolation, using the following law of large numbers for the size of the
largest component of $K_{n}(\omega)$ under the percolation measure
$\phi_{n,\lambda}$, cited from [9]:

######  Theorem 5.1 ([9], Theorem $4.8$).

Fix $\lambda>1$ and let $p=\lambda/n$. Then, for every
$\nu\in(\tfrac{1}{2},1)$ there exists $\delta=\delta(\lambda,\nu)>0$ such that

$\phi_{n,\lambda}[\lvert\lvert\mathcal{C}_{1}\rvert-\theta(\lambda,1)n\rvert\geq
n^{\nu}]=O(n^{-\delta}).$ (98)

We begin with the first inequality of (97). Note that the random cluster
measure $\phi_{n,\lambda,q}$ is stochastically dominated by the percolation
measure $\phi_{n,\lambda}$ for $q\geq 1$ (see e.g. ([8], Theorem $3.21$)),
yielding the upper bound

$\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon n}\rvert\geq(1-\epsilon)n,\
\mathcal{N}_{\epsilon n}=1]\leq\phi_{n,\lambda}[\lvert\mathcal{V}_{\epsilon
n}\rvert\geq(1-\epsilon)n,\ \mathcal{N}_{\epsilon n}=1].$ (99)

As the percolation measure is stochastically ordered in the edge weight $p$,
we may assume that $\lambda>1$, in which case we are done by Theorem 5.1
provided we take $\epsilon<1-\theta(\lambda,1)$.

We now turn to the second inequality of (97), noting first that it is only
relevant if $\lambda>q$. If this is the case, then the random cluster measure
$\phi_{n,\lambda,q}$ stochastically dominates the supercritical percolation
measure $\phi_{n,\lambda/q}$, again by ([8], Theorem $3.21$). As a result,
Theorem 5.1 may be applied to show that

$\phi_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon
n}\rvert\leq(\theta(\lambda/q,1)-\epsilon)n,\ \mathcal{N}_{\epsilon
n}=1]=O(n^{-\delta}).$ (100)

Write $\alpha=\lambda/q$, as before. We claim that
$\theta(\alpha,1)>1-\alpha^{-1}$. As $\theta(\alpha,1)$ solves the equation

$\alpha=-\frac{1}{\theta(\alpha,1)}\log(1-\theta(\alpha,1))$ (101)

it will be sufficient to prove that

$-\frac{1}{\theta(\alpha,1)}\log(1-\theta(\alpha,1))<(1-\theta(\alpha,1))^{-1},$
(102)

which is a consequence of the inequality $(1-x)\log(1-x)+x>0$ for $x\in(0,1)$.
In particular, we are done provided we take
$\epsilon<\theta(\lambda/q,1)-1+q/\lambda$.

Applying both inequalities of (96) to (94), we see that

$(1-o(1))Z_{n,\lambda,q}=Z_{n,\lambda,q}[B_{\epsilon
n}]+\sum_{k>\max\\{\epsilon,\theta(\lambda/q,1)-\epsilon\\}}^{(1-\epsilon)n}Z_{n,\lambda,q}[\lvert\mathcal{V}_{\epsilon
n}\rvert=k,\ \mathcal{N}_{\epsilon n}=1].$ (103)

We are finally in a position to prove Theorem 2.7:

###### Proof of Theorem 2.7.

Let $s_{n}$ be the maximal summand of Equation (103). As the summands converge
uniformly, so too does the maximal summand, with limit

$\lim_{\epsilon\downarrow 0}\lim_{n\to\infty}\frac{1}{n}\log
s_{n}=\sup_{\theta>\theta(\lambda/q,1)}\Phi(\theta,\lambda,q).$ (104)

Moreover, as the sum is bounded between $s_{n}$ and $ns_{n}$, we have

$\frac{1}{n}\log
s_{n}\leq\frac{1}{n}\log((1-o(1))Z_{n,\lambda,q})\leq\frac{1}{n}\log
s_{n}+\frac{1}{n}\log n.$ (105)

Taking the limits as $n\to\infty$ and $\epsilon\downarrow 0$ in that order
yields the result. ∎

We have proven that the free energy of the random cluster model converges to
the supremum of the function $\Phi(\theta,\lambda,q)$. Finally, we evaluate
this supremum. In particular, the following lemma shows that our computation
of the free energy agrees with the one found in ([4], Theorem $2.6$):

######  Lemma 5.2.

Let $q>0$ and $\lambda>0$. Then

$\sup_{\theta\in[0,1]}\Phi(\theta,\lambda,q)=\sup_{\theta>\theta(\lambda/q,1)}\Phi(\theta,\lambda,q)=\frac{g(\theta(\lambda,q))}{2q}-\bigg{(}\frac{q-1}{2q}\bigg{)}\lambda+\log
q$ (106)

where the function $g:(0,1)\to\mathbb{R}$ is defined by

$g(\theta)=-(q-1)(2-\theta)\log(1-\theta)-[2+(q-1)\theta]\log[1+(q-1)\theta].$
(107)

###### Proof.

We separate the argument into the cases $\lambda(1-\theta)>q$ and
$\lambda(1-\theta)<q$, corresponding to the regions in which the $\Psi$
function is defined. We use the shorthand notation

$a=\frac{1-\theta}{q\theta},\
b=\frac{e^{-\lambda\theta}}{1-e^{-\lambda\theta}},\ k=\lambda\theta.$ (108)

When $\lambda(1-\theta)>q$, the derivative of $\Phi$ with respect to $\theta$
is given by

$\frac{\partial}{\partial\theta}\Phi(\theta,\lambda,q)=(kb)-\log(kb)-1.$ (109)

As $x-\log x-1\geq 0$ with equality if and only if $x=1$, the derivative in
(109) is equal to zero only if $kb=1$. This is equivalent to the equation
$1+\lambda\theta=e^{\lambda\theta}$, for which the only solution is
$\theta=0$. When $\lambda(1-\theta)<q$, we obtain the derivative

$\frac{\partial}{\partial\theta}\Phi(\theta,\lambda,q)=(\log a-ka)-(\log
b-kb).$ (110)

The function $\log x-kx$ is convex, with a maximum at $x=\tfrac{1}{k}$. We
know that $a\leq\tfrac{1}{k}$ by assumption, and $b\leq\tfrac{1}{k}$ is a
consequence of the inequality $1+\lambda\theta\leq e^{\lambda\theta}$. As a
result, the derivative in (110) is equal to zero only if $a=b$, which may be
rearranged to see that the maximising value $\theta^{*}$ satisfies the mean
field equation (10). Conversely, any solution $\theta$ to the mean-field
equation satisfies the assumption (and so is a stationary point), as

$\frac{\lambda(1-\theta)}{q}=ka=kb\leq 1.$ (111)

We may now assume $\theta^{*}$ satisfies the mean-field equation. Under this
assumption, one may rewrite $\Phi(\theta^{*},\lambda,q)$ in the form

$\Phi(\theta^{*},\lambda,q)=\frac{1}{2q}g(\theta^{*})-\frac{q-1}{2q}\lambda+\log
q.$ (112)

This is the form taken in ([4], Theorem $2.6$). It remains to show that (112)
is maximised when we take the solution $\theta(\lambda,q)$ to the mean field
equation. We quote the following properties of the function $g$ from [4]:

$\displaystyle g(0)$ $\displaystyle=g^{\prime}(0)=0,$ (113) $\displaystyle
g^{\prime\prime}(\theta)$
$\displaystyle=-\frac{q(q-1)[q-2-2(q-1)\theta]\theta}{(1-\theta)^{2}[1+(q-1)\theta]^{2}}.$

For $q\leq 2$, $g(\theta)$ is a convex, increasing function and the result is
clear. For $q>2$, $g(\theta)$ is initially decreasing. Moreover,
$g^{\prime\prime}(\theta)$ has a zero at $\theta=\tfrac{q-2}{2(q-1)}$, and is
increasing thereafter. In particular, $g(\theta)$ is convex for
$\theta>\tfrac{q-2}{2(q-1)}$ and has only one zero in this region, which we
may compute as $\theta_{c}=\tfrac{q-2}{q-1}$. Note that $\theta_{c}$ is the
largest solution to the mean-field equation for $\lambda=\lambda_{c}$.

We claim that $\theta_{\text{max}}$ is increasing as a function of $\lambda$.
If $\theta_{\text{max}}(\lambda)=0$ then this is obvious, so we may assume
that $\theta_{\text{max}}(\lambda)>0$. Let $\epsilon>0$, and define the
function

$h(\theta):=e^{-(\lambda+\epsilon)\theta}-\frac{1-\theta}{1+(q-1)\theta}.$
(114)

Noting that $h(\theta_{\text{max}}(\lambda))<0$ and $h(1)>0$, it follows that
$h$ has a zero in the interval $(\theta_{\text{max}}(\lambda),1)$ i.e.
$\theta_{\text{max}}(\lambda+\epsilon)>\theta_{\text{max}}(\lambda)$.

We may now conclude. If $\lambda<\lambda_{c}$, then
$\theta_{\text{max}}(\lambda)<\theta_{\text{max}}(\lambda_{c})$ and so
$g(\theta_{\text{max}}(\lambda))<0$. In particular, $\theta^{*}=0$ maximises
the free energy. Conversely, if $\lambda>\lambda_{c}$ then it follows that
$g(\theta_{\text{max}}(\lambda))>0$. As $g(\theta)$ is convex for
$\theta>\tfrac{1}{2}\theta_{c}$, it follows that $\theta_{\text{max}}$ is the
solution maximising the function $g(\theta)$, and so
$\theta^{*}=\theta_{\text{max}}$. ∎

## Acknowledgements

This research was supported by the EPSRC grant EP/N509796/1/1935605. The
author would also like to thank Roman Kotecký for his advice and encouragement
and acknowledge the support by the GAČR grant 20-08468S during his visit to
Prague in January 2020.

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