$|S_{k,\ell}|>n/2$ and consequently,

$\mathcal{S}_{\ell}^{c}\subset\Big{\\{}\sum_{k=1}^{q}|S_{k,\ell}|\leq(1-\varepsilon)n\Big{\\}}\bigcup_{k=1}^{q}\Big{\\{}|S_{k,\ell}|\in[\varepsilon
n,n/2]\Big{\\}}\,.$

Now, fix $\varepsilon\in(0,\frac{\eta}{4q})$, let
$\lambda=\lambda_{\varepsilon}\wedge 2$ for $\lambda_{\varepsilon}>0$ of
Theorem 1.18(ii). The event $|S_{k,\ell}|\in[\varepsilon n,n/2]$ implies by
the edge expansion property of ${\sf G}_{n}$ that $|\partial
S_{k,\ell}|\geq\lambda\varepsilon n:=2\delta n$. Further, noting that

$\sum_{i=1}^{n}\Delta_{i}{\bf 1}\big{(}{\sf N}_{\ell}^{(1)}(i)-{\sf
N}_{\ell}^{(2)}(i)\leq\eta\big{)}+\left|\left\\{(i,j)\in
E_{n}:\mathscr{K}_{\ell}(i)\neq\mathscr{K}_{\ell}(j)\right\\}\right|\geq\max_{k}\\{|\partial
S_{k,\ell}|\\}\,,$

it follows from the preceding that under $\mathcal{S}_{\ell}^{c}$ either the
event $\big{\\{}\sum_{i=1}^{n}\mathcal{B}_{\ell,n}^{\eta}(i)\geq
n\delta\big{\\}}$ or $\mathcal{A}^{\delta}_{\ell,n}$ holds. Thus, from Lemma
4.6 we deduce that, as claimed earlier,

$\lim_{\ell\to\infty}\limsup_{n\to\infty}\mu_{n}^{\beta,0}(\mathcal{S}^{c}_{\ell})=0.$
(4.25)

Having to contend only with the (disjoint) events $\mathcal{S}_{k,\ell}$ of
(4.24), we arrive at (4.23) upon showing that for any $k\neq k^{\prime}\in[q]$
and $\ell\in\mathbb{N}$,

$\mathbb{E}_{n}\left[\mu_{n}^{\beta,0}(\mathcal{S}_{k,\ell},\mathscr{K}_{\ell}(I_{n})=k^{\prime})\right]\leq
q\varepsilon\quad\text{ and
}\quad\limsup_{n\to\infty}\mu_{n}^{\beta,0}(\mathcal{S}_{k^{\prime},\ell},\mathscr{K}_{n}=k)=0.$
(4.26)

To this end, as $|S_{k,\ell}|\geq(1-q\varepsilon)n$ on $\mathcal{S}_{k,\ell}$,
it is immediate that for any $k^{\prime}\neq k$,

${\bf 1}(\mathcal{S}_{k,\ell})\frac{1}{n}\sum_{i=1}^{n}{\bf
1}(\mathscr{K}_{\ell}(i)=k^{\prime})\leq q\varepsilon\,,$

which upon taking the expectation gives the lhs of (4.26). Next, ${\sf
N}(i,k)\leq 1$, hence on $\mathcal{S}_{k^{\prime},\ell}$

$\widehat{\sf N}_{\ell}(k^{\prime},k):=\frac{1}{n}\sum_{i=1}^{n}[{\sf
N}_{\ell}(i,k^{\prime})-{\sf N}_{\ell}(i,k)]\geq\frac{1}{n}\sum_{i\in
S_{k^{\prime},\ell}}[{\sf N}^{(1)}_{\ell}(i)-{\sf
N}^{(2)}_{\ell}(i)]-\varepsilon q\geq(1-\varepsilon q)\eta-\varepsilon
q\geq\frac{\eta}{2}\,$

(due to our choice of $\varepsilon q<\eta/4<1/4$). Thus,

$\mu_{n}^{\beta,0}\big{(}\widehat{\sf N}_{\ell}(k^{\prime},k){\bf
1}\\{\mathcal{S}_{k^{\prime},\ell},\mathscr{K}_{n}=k\\}\big{)}\geq\frac{\eta}{2}\mu_{n}^{\beta,0}(\mathcal{S}_{k^{\prime},\ell},\mathscr{K}_{n}=k).$
(4.27)

Moreover, note that for any $k^{\prime}\neq k$,

$\widehat{\sf N}_{\ell}(k^{\prime},k)=\frac{1}{n}\sum_{v=1}^{n}{\bf
1}\big{(}{\sf B}_{v}(2\ell)\cong{\sf
T}_{d}(2\ell)\big{)}[\delta_{\sigma_{v},k^{\prime}}-\delta_{\sigma_{v},k}]\,.$

Hence, with ${\sf G}_{n}\stackrel{{\scriptstyle loc}}{{\longrightarrow}}{\sf
T}_{d}$,

$\lim_{n\to\infty}\sup_{{\underline{\sigma}}\in[q]^{n}}\Big{|}\widehat{\sf
N}_{\ell}(k^{\prime},k)-\widehat{\sf
N}(k^{\prime},k)\Big{|}=0,\qquad\text{where}\qquad\widehat{\sf
N}(k^{\prime},k):=\frac{1}{n}\sum_{v=1}^{n}(\delta_{\sigma_{v},k^{\prime}}-\delta_{\sigma_{v},k})\,.$
(4.28)

Clearly, $\widehat{\sf N}(k^{\prime},k){\bf 1}(\mathscr{K}_{n}=k)\leq 0$,
hence (4.28) entails that

$\limsup_{n\to\infty}\mu_{n}^{\beta,0}\big{(}\widehat{\sf
N}_{\ell}(k^{\prime},k){\bf
1}\\{\mathcal{S}_{k^{\prime},\ell},\mathscr{K}_{n}=k\\}\big{)}\leq 0.$ (4.29)

From (4.27) and (4.29) we get the rhs of (4.26), thus completing the proof of
Theorem 1.18(ii). ∎

##  5\. The critical line: proof of Theorem 1.13

Recall Lemma 2.9(a) on the existence of sub-sequential local weak limit points
for both $\mu_{n}^{\beta,B}$ and $\varphi_{n}^{\beta,B}$ and that as in Step I
of the proof of Theorem 1.12, fixing $d,q\geq 3$, $B>0$ and $(\beta,B)\in{\sf
R}_{c}$, it suffices to show that such limit point
$\mathfrak{m}_{\operatorname{b}}$ of $\varphi_{n}^{\beta,B}$ is supported on
$\mathcal{Q}_{\rm m}:=\mathcal{M}_{\operatorname{b}}^{\beta,B}$. Recalling
Definition 2.8 of $\mathcal{Q}_{\star}(t)$ and Lemma 2.9(d) that such
$\mathfrak{m}_{\operatorname{b}}$ must be supported on
$\mathcal{Q}_{\star}:=\\{\varphi\in\mathcal{P}(\mathbb{B}_{\infty}^{\star}):\varphi^{t}\in\mathcal{Q}_{\star}(t)\text{
for all }t\in\mathbb{N}\\}$, we show that
$\mathfrak{m}_{\operatorname{b}}(\mathcal{Q}_{\star}\backslash\mathcal{Q}_{\rm
m})>0$ contradicts [6, Theorem 1]. Indeed, Section 5.1 identifies
$\mathcal{Q}_{\rm m}$ in terms of the support of the rcm messages, whereby
Section 5.2 completes this argument by combining this characterization with
the symmetric form $\Psi^{\rm sym}(\cdot)$ of the free energy density limit of
the rcm in case $d\in 2\mathbb{N}$.

###  5.1. Properties of the rcm messages

We start with the definition of our rcm messages.

######  Definition 5.1 (rcm messages).

Let $\mathscr{T}^{\star}$ denote the tail $\sigma$-algebra of the bonds on
${\sf T}_{d}^{\star}$. That is, for the subsets
$\mathbb{B}_{t}^{\star}:=\\{0,1\\}^{E({\sf T}_{d}^{\star}(t))}$ of
$\mathbb{B}_{\infty}^{\star}:=\\{0,1\\}^{E({\sf T}_{d}^{\star})}$,

$\mathscr{T}^{\star}:=\bigcap_{t\geq
1}\mathscr{T}^{\star}_{t}\,,\quad\text{where}\quad\mathscr{T}^{\star}_{t}:=\sigma(\bigcup_{r>t}\mathscr{T}_{t}^{\star,r})\quad\text{and}\quad\mathscr{T}_{t}^{\star,r}:=\sigma\left(\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t}^{\star}\right)\,.$

For $(u,v)\in E({\sf T}_{d})$ and $\varphi\in\mathcal{Q}_{\star}$, the rcm
message from $u$ to $v$ is then

$s_{u\to v}(\varphi):=\varphi[u\leftrightarrow
v^{\star}|\eta_{(u,v)}=0,\mathscr{T}^{\star}].$ (5.1)

######  Remark 5.2.

While not needed here, one can define the infinite volume random cluster Gibbs
measures on ${\sf T}_{d}$, in presence of an external field, via the so called
Dobrushin-Lanford-Ruelle condition, as done in [3, Section III], and show that
the messages of (5.1) are indeed those that characterize such Gibbs measures.

The rcm messages are limits of the rcm pre-messages which we define next (and
later relate to the local functions $\mathfrak{s}^{t,r}_{u\to w}$ on the
finite graphs ${\sf G}_{n}$). To this end, for finite graph ${\sf G}$, $w\in
V({\sf G})$ and $t<r$, we denote hereafter
$\eta_{w,t}^{\star,r}:=\underline{\eta}_{{\sf B}_{w}^{\star}(r)\backslash{\sf
B}_{w}^{\star}(t)}$ and $\eta_{w,t}^{r}:=\underline{\eta}_{{\sf
B}_{w}(r)\backslash{\sf B}_{w}(t)}$ with
$\eta_{t}^{\star,r}:=\eta_{o,t}^{\star,r}$ and $\eta_{t}^{r}:=\eta_{o,t}^{r}$
the corresponding objects on the tree ${\sf T}_{d}$ rooted at $o$.

######  Definition 5.3 (rcm pre-messages).

For $(u,v)\in E({\sf T}_{d}(t))$ consider the (local) function

$s_{u\to v}^{t,r}(\underline{y}):=\varphi_{{\sf
T}^{\star}_{d}(r)}^{\beta,B}(u\leftrightarrow
v^{\star}|\eta_{(u,v)}=0,\eta_{t}^{\star,r}=\underline{y}),\quad\text{ of
}\quad\underline{y}\in\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t}^{\star}\,.$

Similarly, for finite graph ${\sf G}$, $w\in V({\sf G})$ and $(u,v)\in{\sf
B}_{w}(t)$ let

$\mathfrak{s}_{u\to v}^{t,r}(w,\underline{y}):=\varphi_{{\sf
B}_{w}(r)}^{\beta,B}(u\leftrightarrow
v^{\star}|\eta_{(u,v)}=0,\eta_{w,t}^{\star,r}=\underline{y}),\quad\underline{y}\in\\{0,1\\}^{{\sf
B}_{w}^{\star}(r)\backslash{\sf B}_{w}^{\star}(t)},$

using hereafter $s_{\partial v}(\varphi)$ for the vector $(s_{u\to
v}(\varphi))_{u\in\partial v}$ and similarly $s_{\partial v}^{t,r}$ and
$\mathfrak{s}_{\partial v}^{t,r}:=(\mathfrak{s}_{u\to
v}^{t,r}(v,\cdot))_{u\in\partial v}$.

Clearly $\mathscr{T}_{t}^{\star,r}\uparrow\mathscr{T}^{\star}_{t}$ and
therefore the Doob’s martingale $\\{{s}_{u\to
v}^{t,r}(\eta_{t}^{\star,r})\\}_{r}$ converges $\varphi$-a.e. to
$\varphi[u\leftrightarrow v^{\star}|\eta_{(u,v)}=0,\mathscr{T}^{\star}_{t}]$.
Further, $\mathscr{T}^{\star}_{t}\downarrow\mathscr{T}^{\star}$ and hence the
latter backward martingale converges $\varphi$-a.e. to $s_{u\to v}(\varphi)$.
In conclusion, fixing any $\varphi\in\mathcal{Q}_{\star}$, for $\varphi$-a.e.
bond configuration,

$s_{\partial v}(\varphi)=\lim_{t\to\infty}\lim_{r\to\infty}{s}_{\partial
v}^{t,r}(\eta_{t}^{\star,r})\,,\qquad\forall v\in V({\sf T}_{d})\,.$ (5.2)

Now, for $\ddagger\in\\{\operatorname{f},1\\}$ and $\nu_{\ddagger}^{\beta,B}$
of Definition 1.3, let

$b_{\ddagger}=b_{\ddagger}^{\beta,B}:=\frac{q\nu_{\ddagger}^{\beta,B}(1)-1}{q-1}\geq
0\,,$ (5.3)

setting hereafter $b_{\operatorname{w}}:=b_{1}^{\beta,B}$,
${\gamma:=(e^{\beta}-1)/(e^{\beta}+q-1)}$, and

$\widehat{\sf
BP}(s_{1},\ldots,s_{d-1};x):=\frac{e^{x}\displaystyle{\prod_{i=1}^{d-1}}\big{(}1+(q-1)\gamma
s_{i}\big{)}-\prod_{i=1}^{d-1}\big{(}1-\gamma
s_{i}\big{)}}{e^{x}\displaystyle{\prod_{i=1}^{d-1}}\big{(}1+(q-1)\gamma
s_{i}\big{)}+(q-1)\prod_{i=1}^{d-1}\big{(}1-\gamma s_{i}\big{)}}.$ (5.4)

Utilizing these objects, we proceed to characterize $\mathcal{Q}_{\rm m}$ via
the support of the rcm messages.

######  Lemma 5.4.

Fix $B>0$, $\beta\geq 0$ and $\varphi\in\mathcal{Q}_{\star}$.

  1. (a)

For $\varphi$-a.e. bond configurations and any $(u,v)\in E({\sf T}_{d})$,

$\displaystyle s_{u\to v}(\varphi)=\widehat{\sf BP}(($ $\displaystyle s_{w\to
u}(\varphi))_{w\in\partial u\backslash\\{v\\}};B),$ (5.5) $\displaystyle
b_{\operatorname{f}}\leq$ $\displaystyle s_{u\to v}(\varphi)\leq
b_{\operatorname{w}}\,.$ (5.6)

  2. (b)

For $\ddagger\in\\{\operatorname{f},\operatorname{w}\\}$ and any $(u,v)\in
E({\sf T}_{d})$, we have $s_{u\to
v}(\varphi_{\ddagger}^{\beta,B})=b_{\ddagger}$, for
$\varphi_{\ddagger}^{\beta,B}$-a.e. configurations.

  3. (c)

If $\varphi$-a.e. the random vector $s_{\partial o}(\varphi)$ is supported on
$\\{(b_{\ddagger},b_{\ddagger},\ldots,b_{\ddagger})\\}_{\ddagger\in\\{\operatorname{f},\operatorname{w}\\}}$,
then $\varphi\in\mathcal{Q}_{{\rm m}}$.

###### Proof.

(a). For any $(i,j)\in E({\sf T}_{d})$ let ${\sf T}_{i\to j}$ be the connected
component of the sub-tree of ${\sf T}_{d}$ rooted at $i$ after deleting the
edge $(i,j)$. Set ${\sf T}_{i\to j}^{\star}$ to be the graph obtained from
${\sf T}_{i\to j}$ by adding the edges from $v^{\star}$ to $V({\sf T}_{i\to
j})\backslash\\{i\\}$ (so there is no edge between $i$ and $v^{\star}$ in
${\sf T}_{i\to j}^{\star}$). Denoting by $\Omega_{i\to j}$ the event that $i$
is connected to $v^{\star}$ using only the open bonds within ${\sf T}_{i\to
j}^{\star}$ and by $\mathfrak{p}(i)$ the (unique) parent of $i\neq o$ (the
root) in ${\sf T}_{d}$, let

$\widehat{\mathscr{T}}_{t}^{r}:=\sigma(\mathscr{T}_{t}^{\star,r},\left\\{\Omega_{i\to\mathfrak{p}(i)},i\in\partial{\sf
T}_{d}(r)\right\\}).$

By the one-to-one correspondence between the power set of
$\mathfrak{D}_{\star}(r)$ and
$\sigma(\\{\Omega_{i\to\mathfrak{p}(i)},i\in\partial{\sf T}_{d}(r)\\})$, the
$\sigma$-algebra $\widehat{\mathscr{T}}_{t}^{r}$ is generated by the finite
partition of
$(\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t}^{\star})\times\mathfrak{D}_{\star}(r)$
to pairs $(\eta_{t}^{\star,r},\mathscr{C}^{r})$. Thus, for any $B>0$,
$\beta\geq 0$, $\varphi\in\mathcal{Q}_{\star}$ and $r>t$

$\varphi(\cdot|\widehat{\mathscr{T}}_{t}^{r})(\eta_{t}^{\star,r}=\underline{y},\mathscr{C}^{r}=\mathscr{C})=\varphi_{\mathscr{C}}^{\beta,B,r}(\cdot|\eta_{t}^{\star,r}=\underline{y})\,,\qquad\forall\underline{y}\in\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t}^{\star}\,,\;\;\forall\mathscr{C}\in\mathfrak{D}_{\star}(r)\,.$
(5.7)

In particular,

$\varphi[u\leftrightarrow
v^{\star}|\eta_{(u,v)}=0,\widehat{\mathscr{T}}_{t}^{r}]=\hat{s}_{u\to
v}^{t,r}(\eta_{t}^{\star,r},\mathscr{C}^{r})\,,\qquad\forall(u,v)\in E({\sf
T}_{d}(t))\,,$

where

$\hat{s}_{u\to
v}^{t,r}(\underline{y},\mathscr{C}):=\varphi_{\mathscr{C}}^{\beta,B,r}(u\leftrightarrow
v^{\star}|\eta_{(u,v)}=0,\eta_{t}^{\star,r}=\underline{y})\,.$ (5.8)

As $\widehat{\mathscr{T}}_{t}^{r}\uparrow\mathscr{T}_{t}^{\star}$, fixing any
$\varphi\in\mathcal{Q}_{\star}$, we get similarly to (5.2), that
$\varphi$-a.e.

$s_{u\to v}(\varphi)=\lim_{t\to\infty}\lim_{r\to\infty}\hat{s}_{u\to
v}^{t,r}(\eta_{t}^{\star,r},\mathscr{C}^{r})\,,\qquad\forall(u,v)\in E({\sf
T}_{d})$ (5.9)

and by the continuity of the mapping $\widehat{\sf BP}(\cdot;B)$ it suffices
for (5.5) to show that for any $t<r$,

$\hat{s}^{t,r}_{u\to v}=\widehat{\sf BP}((\hat{s}^{t,r}_{w\to
u})_{w\in\partial u\backslash\\{v\\}};B)\,,\qquad\forall(u,v)\in E({\sf
T}_{d}(t)).$ (5.10)

By Lemma 2.2, using an argument similar to the proof of Corollary 2.3 we find
that

$\displaystyle\bar{s}^{t,r}_{u\to
v}(\underline{y},\mathscr{C}):=\varpi^{\beta,B,r}_{\mathscr{C}}(\sigma_{u}=1|\eta_{(u,v)}=0,\eta_{t}^{\star,r}=\underline{y})$
$\displaystyle=\big{(}1-\frac{1}{q}\big{)}\hat{s}^{t,r}_{u\to
v}(\underline{y},\mathscr{C})+\frac{1}{q}\,,$ (5.11)

for any
$\underline{y}\in\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t}^{\star}$,
$\mathscr{C}\in\mathfrak{D}_{\star}(r)$. Fixing such $(u,v)$, $\underline{y}$,
and $\mathscr{C}$, the identity (5.10) thus amounts to

$\bar{s}^{t,r}_{u\to v}=\overline{\sf BP}((\bar{s}^{t,r}_{w\to
u})_{w\in\partial u\backslash\\{v\\}}),$ (5.12)

where $\overline{\sf
BP}(\bar{s}_{1},\ldots,\bar{s}_{d-1}):=\big{(}1-\frac{1}{q}\big{)}\widehat{\sf
BP}\Big{(}\frac{q\bar{s}_{1}-1}{q-1},\ldots,\frac{q\bar{s}_{d-1}-1}{q-1};B\Big{)}+\frac{1}{q}$.
Alternatively, in view of (5.4),

$\overline{\sf
BP}(\bar{s}_{1},\ldots,\bar{s}_{d-1})=\frac{e^{B}\displaystyle{\prod_{i=1}^{d-1}}\left(1+(e^{\beta}-1)\bar{s}_{i}\right)}{e^{B}\displaystyle{\prod_{i=1}^{d-1}}\left(1+(e^{\beta}-1)\bar{s}_{i}\right)+(q-1)\displaystyle{\prod_{i=1}^{d-1}}\Big{(}1+\frac{e^{\beta}-1}{q-1}(1-\bar{s}_{i})\Big{)}}.$
(5.13)

The spin marginal of
$\varpi^{\beta,B,r}_{\mathscr{C}}(\cdot|\eta_{(u,v)}=0,\eta_{t}^{\star,r}=\underline{y})$
on the sub-tree ${\sf T}_{u\to v}\cap{\sf T}_{d}(t)$ is a Potts measure.
Further, with $\mathscr{C}\in\mathscr{D}_{\star}(r)$, the boundary condition
for the spins at ${\sf T}_{u\to v}\cap\partial{\sf T}_{d}(t)$, as determined
by $(\underline{y},\mathscr{C})$, must be a product measure, where each
marginal spin is either uniform over $[q]$ or supported on $1$. Thus,
restricting this Potts measure to $u$ and $\partial u\backslash\\{v\\}$,
yields a Potts measure on $q$ spins, whose boundary marginals on $\partial
u\backslash\\{v\\}$ are mutually independent and uniform over
$\\{2,\ldots,q\\}$. Thus, upon expressing the rhs of (5.12) in terms of
(5.13), the identity (5.12) follows by a direct computation.

Setting
$\widetilde{b}_{\ddagger,t^{\prime}}(u,v):=\varphi_{\ddagger,t^{\prime}}^{\beta,B}(u\leftrightarrow
v^{\star}|\eta_{(u,v)}=0)$ for
$\ddagger\in\\{\operatorname{f},\operatorname{w}\\}$, $(u,v)\in E({\sf
T}_{d}(t))$ and $\varphi_{\ddagger,t^{\prime}}^{\beta,B}$ as in (1.14), we
show in part (b) that $\widetilde{b}_{\ddagger,t^{\prime}}(u,v)\to
b_{\ddagger}$ as $t^{\prime}\to\infty$. Thus, in view of (5.8) and (5.9), it
suffices for (5.6) to fix such $(u,v)$ and show that for $r>t^{\prime}>t$, any
$\underline{y}\in\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t^{\prime}}^{\star}$
and $\mathscr{C}\in\mathfrak{D}_{\star}(r)$,

$\widetilde{b}_{\operatorname{f},t^{\prime}}(u,v)\leq\varphi_{\mathscr{C}}^{\beta,B,r}(u\leftrightarrow
v^{\star}|\eta_{(u,v)}=0,\eta_{t^{\prime}}^{\star,r}=\underline{y})\leq\widetilde{b}_{\operatorname{w},t^{\prime}}(u,v)\,.$
(5.14)

The conditional measure in (5.14) is merely the rcm on ${\sf T}_{u\to
v}\cap\widehat{\sf T}^{\star}_{d}(t^{\prime})$ with the induced boundary
condition
$\mathscr{C}^{\prime}(\underline{y},\mathscr{C})\in\mathfrak{D}_{\star}(t^{\prime})$
at ${\sf T}_{u\to v}\cap{\sf K}^{\star}(\partial{\sf T}_{d}(t^{\prime}))$. Any
such $\mathscr{C}^{\prime}$ stochastically dominates the free boundary
condition and is stochastically dominated by the corresponding wired boundary
condition (see Definition 1.9), with (5.14) thus a consequence of the
monotonicity of the rcm $\varphi^{\beta,B}_{{\sf T}_{u\to v}\cap\widehat{\sf
T}_{d}^{\star}(t^{\prime})}(\cdot)$.

(b). The rcm-s $\varphi_{\ddagger}^{\beta,B}$, for
$\ddagger\in\\{\operatorname{f},\operatorname{w}\\}$, are both tail trivial
(indeed, this follows as in the proof of [8, Theorem 4.19(d)]). Moreover, by
definition $\varphi^{\beta,B}_{\ddagger}\in\mathcal{Q}_{\star}$ are invariant
under automorphisms of ${\sf T}_{d}$. Hence,
$\varphi^{\beta,B}_{\ddagger}$-a.s. the values of $s_{u\to
v}(\varphi^{\beta,B}_{\ddagger})$ for $(u,v)\in E({\sf T}_{d})$, must equal
the same non-random constant
$\widetilde{b}_{\ddagger}:=\varphi^{\beta,B}_{\ddagger}(w\leftrightarrow
v^{\star}|\eta_{(w,o)}=0)$, where $w\in\partial o$. Recall (1.15) that
$\widetilde{b}_{\ddagger,t}(w,o)\to\widetilde{b}_{\ddagger}$ as $t\to\infty$.
Now, for any finite graph ${\sf G}$ and $e\in E({\sf G})$, the conditional
measure $\varphi_{{\sf G}}^{\beta,B}(\cdot|\eta_{e}=0)$ is merely the rcm
$\varphi_{\widetilde{\sf G}}^{\beta,B}(\cdot)$, for the subgraph
$\widetilde{\sf G}$ obtained upon deleting the edge $e$ from ${\sf G}$. Hence,
it follows from Lemma 2.2 that

$\widetilde{b}_{\ddagger}=\lim_{t\to\infty}\widetilde{b}_{\ddagger,t}=\lim_{t\to\infty}\frac{q\mu^{\beta,B}_{w\to
o,\ddagger,t}(\sigma_{w}=1)-1}{q-1},$ (5.15)

where $\mu^{\beta,B}_{w\to o,\ddagger,t}$ denotes the Potts measure on ${\sf
T}_{w\to o}\cap{\sf T}_{d}(t)$ with free and $1$-boundary conditions at the
spins of ${\sf T}_{w\to o}\cap\partial{\sf T}_{d}(t)$, for
$\ddagger=\operatorname{f}$ and $\ddagger=\operatorname{w}$, respectively.
Since ${\sf T}_{w\to o}$ is a $(d-1)$-ary tree (i.e. every vertex has $(d-1)$
children), it thus follows from Definition 1.3 of $\nu_{\ddagger}^{\beta,B}$,
that the rhs of (5.15) is precisely $b_{\ddagger}^{\beta,B}$ of (5.3), as
claimed.

(c). Setting $\mathscr{A}_{\ddagger}:=\\{(s_{\partial
o}(\varphi))=(b_{\ddagger},b_{\ddagger},\ldots,b_{\ddagger})\\}$ and fixing
$\varphi\in\mathcal{Q}_{\star}$ with
$\varphi(\mathscr{A}_{\operatorname{f}}\cup\mathscr{A}_{\operatorname{w}})=1$,
our claim that $\varphi\in\mathcal{Q}_{\rm m}$ amounts to having for any
$t\in\mathbb{N}$ that $\varphi$-a.e.

$\varphi(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\cdot\mid\mathscr{T}^{\star})=\varphi_{\operatorname{f}}^{\beta,B}(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\cdot){\bf
1}_{\mathscr{A}_{\operatorname{f}}}+\varphi_{\operatorname{w}}^{\beta,B}(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\cdot){\bf 1}_{\mathscr{A}_{\operatorname{w}}}\,.$ (5.16)

To this end, fixing $t<t^{\prime}<r$, recall that by (5.7)

$\varphi(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\underline{y}\mid\widehat{\mathscr{T}}_{t^{\prime}}^{r})=\sum_{\mathscr{C}\in\mathfrak{D}_{\star}(t)}\varphi_{\mathscr{C}}^{\beta,B,t}(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\underline{y})\rho_{\varphi}(\mathscr{C}|\widehat{\mathscr{T}}_{t^{\prime}}^{r}),\qquad\forall\underline{y}\in\mathbb{B}_{o}^{\star}(t),$

with $\rho_{\varphi}(\cdot\mid\widehat{\mathscr{T}}_{t^{\prime}}^{r})$ the
conditional distribution induced on $\mathfrak{D}_{\star}(t)$ by the finitely
generated $\widehat{\mathscr{T}}_{t^{\prime}}^{r}$. Recall the one-to-one
correspondence between $\mathfrak{D}_{\star}(t)$ and subgraphs
$C_{\star}\subset\\{(u,v^{\star}),u\in\partial{\sf T}_{d}(t)\\}$, with

$\rho_{\varphi}(\\{u\in
C_{\star}\\}\mid\widehat{\mathscr{T}}_{t^{\prime}}^{r})=\varphi\big{[}u\leftrightarrow
v^{\star}|\eta_{(u,\mathfrak{p}(u))}=0,\eta_{(u,v^{\star})}=0,\widehat{\mathscr{T}}_{t^{\prime}}^{r}\big{]}=:\tilde{s}_{u\to\mathfrak{p}(u)}^{t^{\prime},r}(\eta_{t^{\prime}}^{\star,r},\mathscr{C}^{r}),\quad\forall
u\in\partial{\sf T}_{d}(t)\,.$

Using standard and backward martingale convergence theorems, it then follows
that $\varphi$-a.e.

$\displaystyle\varphi(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\cdot\mid\mathscr{T}^{\star})$
$\displaystyle=\sum_{\mathscr{C}\in\mathfrak{D}_{\star}(t)}\varphi_{\mathscr{C}}^{\beta,B,t}(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\cdot)\rho_{\varphi}(\mathscr{C}|\mathscr{T}^{\star}),\,$
(5.17) $\displaystyle\rho_{\varphi}(\\{u\in
C_{\star}\\}\mid\mathscr{T}^{\star})$
$\displaystyle=\tilde{s}_{u\to\mathfrak{p}(u)}(\varphi):=\lim_{t^{\prime}\to\infty}\lim_{r\to\infty}\tilde{s}_{u\to\mathfrak{p}(u)}^{t^{\prime},r}\,.$
(5.18)

In view of (5.17), we get (5.16) and thereby complete the proof, upon showing
that $\varphi$-a.e.

$\rho_{\varphi}(\cdot|\mathscr{T}^{\star})=\rho_{\varphi_{\operatorname{f}}^{\beta,B}}(\cdot){\bf
1}_{\mathscr{A}_{\operatorname{f}}}+\rho_{\varphi_{\operatorname{w}}^{\beta,B}}(\cdot){\bf
1}_{\mathscr{A}_{\operatorname{w}}}\,.$ (5.19)

Turning to prove (5.19), recall that $\varphi\in\mathcal{Q}_{\star}$ and that
the rcm $\varphi_{\widehat{\sf T}_{d}^{\star}(t^{\prime})}^{\beta,B}$ is
monotonic. Hence,

$\rho_{\varphi_{\operatorname{f},t^{\prime}}^{\beta,B}}(\cdot)\preccurlyeq\rho_{\varphi}(\cdot|\widehat{\mathscr{T}}_{t^{\prime}}^{r})\preccurlyeq\rho_{\varphi_{\operatorname{w},t^{\prime}}^{\beta,B}}(\cdot)\,,$

which upon taking first $r\to\infty$ and then $t^{\prime}\to\infty$, yields
that $\varphi$-a.e.

$\rho_{\varphi_{\operatorname{f}}^{\beta,B}}(\cdot)\preccurlyeq\rho_{\varphi}(\cdot|\mathscr{T}^{\star})\preccurlyeq\rho_{\varphi_{\operatorname{w}}^{\beta,B}}(\cdot)\,.$

Given such stochastic ordering, (5.19) follows once all the corresponding one-
dimensional marginals match (see [8, Proposition 4.6]). That is, in view of
(5.18) and our assumption that
$\varphi(\mathscr{A}_{\operatorname{f}}\cup\mathscr{A}_{\operatorname{w}})=1$,
it remains only to show that $\varphi$-a.e. on the event
$\mathscr{A}_{\ddagger}$

$\tilde{s}_{u\to\mathfrak{p}(u)}(\varphi)=\tilde{s}_{u\to\mathfrak{p}(u)}(\varphi^{\beta,B}_{\ddagger})\,,\qquad\forall
u\in{\sf T}_{d}\,.$

To this end, note that as in the proof of (5.5), we also have that
$\varphi$-a.e.

$\tilde{s}_{u\to\mathfrak{p}(u)}(\varphi)=\widehat{\sf BP}((s_{w\to
u}(\varphi))_{w\in\partial u\backslash\\{\mathfrak{p}(u)\\}};0),\qquad\forall
u\in{\sf T}_{d}\,.$

Recalling from part (b) that $s_{w\to
u}(\varphi^{\beta,B}_{\ddagger})=b_{\ddagger}$ for all $(w,u)\in E({\sf
T}_{d})$, is thus suffices to verify that on the event
$\mathscr{A}_{\ddagger}$, also $s_{w\to u}(\varphi)=b_{\ddagger}$ (outside a
null set). Indeed, the latter is a direct consequence of (5.5) and the
identity

$b_{\ddagger}^{\beta,B}=\widehat{\sf
BP}(b^{\beta,B}_{\ddagger},\ldots,b^{\beta,B}_{\ddagger};B)\,,$

which in view of (5.3), is merely the fact that
$\nu_{\operatorname{f}}^{\beta,B}$ and $\nu_{1}^{\beta,B}$ are both fixed
points of (1.5). ∎

######  Remark 5.5.

With $\widetilde{b}_{\operatorname{w},\ell}(w,o)\downarrow
b_{\operatorname{w}}$ and $\widetilde{b}_{\operatorname{f},\ell}(w,o)\uparrow
b_{\operatorname{f}}$ (for $w\in\partial o$), we deduce from (5.14) that for
some finite $\kappa_{o}=\kappa_{o}(\varepsilon)$ and any $r>t+\kappa_{o}$,

$b_{\operatorname{f}}-\varepsilon\leq s_{u\to
v}^{t+\kappa_{o},r}(\underline{y})\leq
b_{\operatorname{w}}+\varepsilon,\qquad\forall(u,v)\in E({\sf
T}_{d}(t)),\;\underline{y}\in\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t+\kappa_{o}}^{\star}\,.$

Moreover, if $w\in V({\sf G})$ for a finite graph ${\sf G}$ such that ${\sf
B}_{w}(r)\cong{\sf T}_{d}(r)$ for $r>t+\kappa_{o}$, then

$b_{\operatorname{f}}-\varepsilon\leq\mathfrak{s}_{u\to
v}^{t+\kappa_{o},r}(w,\underline{y})\leq
b_{\operatorname{w}}+\varepsilon,\quad\forall(u,v)\in{\sf
B}_{w}(t),\;\underline{y}\in\\{0,1\\}^{{\sf B}_{w}^{\star}(r)\backslash{\sf
B}_{w}^{\star}(t+\kappa_{o})}\,.$ (5.20)

Denoting hereafter by ${\sf G}^{-}(w)$ the graph ${\sf G}$ without $w\in
V({\sf G})$ and all edges to it, we next show that for $B>0$, when $r\gg t$,
uniformly over ${\sf G}$ the law induced on ${\sf B}_{w}^{\star}(t)$ by the
rcm on ${\sf G}$ does not depend much on the boundary conditions outside ${\sf
B}_{w}^{\star}(r)$.

######  Lemma 5.6.

Fix $B>0$, $\beta\geq 0$, $1<t<r$ and $o\in V({\sf G})$ for a finite graph
${\sf G}$. Denoting by $\mathscr{S}_{t,r}^{c}$ the set of
$\eta_{o,t}^{\star,r}$ with an open cluster, not containing $v^{\star}$, that
intersects both $\partial{\sf B}_{o}(t)$ and $\partial{\sf B}_{o}(r)$,

${\varphi_{\sf G}^{\beta,B}(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\cdot\mid\underline{\eta}_{{\sf G}^{\star}\backslash{\sf
B}^{\star}_{o}(t)}=\underline{z})}={\varphi^{\beta,B}_{{\sf
B}_{o}(r)}(\underline{\eta}_{{\sf
B}_{o}^{\star}(t)}=\cdot\mid\eta_{o,t}^{\star,r}=\underline{z}_{{\sf
B}^{\star}_{o}(r)\backslash{\sf B}_{o}^{\star}(t)})},$ (5.21)

whenever $\underline{z}_{{\sf B}_{o}^{\star}(r)\backslash{\sf
B}_{o}^{\star}(t)}\in\mathscr{S}_{t,r}$. Further, for any $\varepsilon>0$,
some $r_{0}=r_{0}(\varepsilon,q,B,t,|{\sf B}_{o}(t)|)<\infty$,

$\displaystyle\sup_{r\geq r_{0}}\sup_{{\sf
G},\underline{y}}\big{\\{}\varphi^{\beta,B}_{\sf
G}(\mathscr{S}^{c}_{t,r}\mid\underline{\eta}_{{\sf G}\backslash{\sf
B}_{o}(t)}=\underline{y})\big{\\}}$ $\displaystyle\leq\varepsilon,$ (5.22)
$\displaystyle\sup_{r\geq r_{o}}\max_{{\sf
G},\underline{y}}\Big{|}\frac{\varphi_{{\sf
G}}^{\beta,B}(\underline{\eta}_{{\sf
B}_{o}(t)}=\cdot\mid\underline{\eta}_{{\sf G}\backslash{\sf
B}_{o}(t)}=\underline{y})}{\varphi^{\beta,B}_{{\sf
B}_{o}(r)}(\underline{\eta}_{{\sf
B}_{o}(t)}=\cdot\mid\eta_{o,t}^{r}=\underline{y}_{{\sf B}_{o}(r)\backslash{\sf
B}_{o}(t)})}-1\Big{|}$ $\displaystyle\leq\varepsilon\,.$ (5.23)

In addition, (5.22) holds for $\varphi^{\beta,B}_{{\sf G}^{-}(o)}(\cdot)$.

###### Proof.

Note that (5.21) amounts to having

$\mathscr{X}_{\star}(\underline{y},\underline{\widehat{y}}):=\frac{\varphi_{\sf
G}^{\beta,B}(\underline{y})}{\varphi_{\sf
G}^{\beta,B}(\underline{\widehat{y}})}\frac{\varphi_{{\sf
B}_{o}(r)}^{\beta,B}(\underline{\widehat{y}}_{{\sf
B}_{o}^{\star}(r)})}{\varphi_{{\sf B}_{o}(r)}^{\beta,B}(\underline{y}_{{\sf
B}_{o}^{\star}(r)})}=1,$ (5.24)

whenever $\underline{y}_{{\sf B}_{o}^{\star}(r)\backslash{\sf
B}_{o}^{\star}(t)}\in\mathscr{S}_{t,r}$ and $y_{e}=\widehat{y}_{e}$ for all
$e\notin{\sf B}^{\star}_{o}(t)$. Turning to prove (5.24), recall by Definition
1.7 of the rcm on a finite graph, that

$\mathscr{X}_{\star}(\underline{y},\underline{\widehat{y}})=q^{|{{\sf C}_{{\sf
G}^{\star}}(\underline{y})}|-|{\sf C}_{{\sf
G}^{\star}}(\underline{\widehat{y}})|+|{\sf C}_{{\sf
B}_{o}^{\star}(r)}(\underline{\widehat{y}}_{{\sf B}_{o}^{\star}(r)})|-|{\sf
C}_{{\sf B}_{o}^{\star}(r)}(\underline{y}_{{\sf B}_{o}^{\star}(r)})|}.$ (5.25)

The open cluster containing $v^{\star}$ appears once in each of these four
counts. Thus, with $y_{e}=\widehat{y}_{e}$ for all $e\notin{\sf
B}^{\star}_{o}(t)$, the rhs of (5.25) is one, unless some other open cluster
induced by $\underline{y}$ or by $\underline{\widehat{y}}$ intersects both
$\partial{\sf B}_{o}(t)$ and $\partial{\sf B}_{o}(r)$, a scenario which is
precluded for $\underline{\widehat{y}}_{{\sf B}_{o}^{\star}(r)\backslash{\sf
B}_{o}^{\star}(t)}=\underline{y}_{{\sf B}_{o}^{\star}(r)\backslash{\sf
B}_{o}^{\star}(t)}\in\mathscr{S}_{t,r}$.

As for (5.22), we can assume wlog that $\underline{y}_{{\sf G}\backslash{\sf
B}_{o}(t)}$ induces $L\geq 1$ disjoint open clusters $C_{i}$ that intersect
both $\partial{\sf B}_{o}(t)$ and $\partial{\sf B}_{o}(r)$. In particular,
$L\leq|{\sf B}_{o}(t)|$ and $|C_{i}|\geq r-t$. Since $\mathscr{S}_{t,r}^{c}$
implies that all edges from $v^{\star}$ to some $C_{i}$ must be closed, we
thus deduce by a union over $i\leq L$, that

$\varphi^{\beta,B}_{\sf G}(\mathscr{S}^{c}_{t,r}\mid\underline{\eta}_{{\sf
G}\backslash{\sf B}_{o}(t)}=\underline{y})\leq\sum_{i=1}^{L}qe^{-B|C_{i}|}\leq
q|{\sf B}_{o}(t)|e^{-B(r-t)},$

which since $B>0$, completes the proof for $\varphi_{\sf G}^{\beta,B}$. With
$t>1$, the same applies for $\varphi^{\beta,B}_{{\sf G}^{-}(o)}$.

Turning to prove (5.23), we move to the marginal rcm-s
$\widetilde{\varphi}^{\beta,B}_{{\sf G}}(\cdot)$ of (1.13) and note that as in
our proof of (5.21), it suffices to find
$\varepsilon_{r}=\varepsilon_{r}(q,B,t,|{\sf B}_{o}(t)|)\to 0$, such that if
$y_{e}=\widehat{y}_{e}$ for all $e\notin{\sf B}_{o}(t)$, then

$\mathscr{X}(\underline{y},\underline{\widehat{y}}):=\frac{\widetilde{\varphi}_{\sf
G}^{\beta,B}(\underline{y})}{\widetilde{\varphi}_{\sf
G}^{\beta,B}(\underline{\widehat{y}})}\frac{\widetilde{\varphi}_{{\sf
B}_{o}(r)}^{\beta,B}(\underline{\widehat{y}}_{{\sf
B}_{o}(r)})}{\widetilde{\varphi}_{{\sf
B}_{o}(r)}^{\beta,B}(\underline{y}_{{\sf B}_{o}(r)})}\leq
1+\varepsilon_{r}\,.$ (5.26)

To this end, since $\underline{y}$ and $\underline{\widehat{y}}$ agree outside
${\sf B}_{o}(r)$, the four products over edges in (5.26) match and perfectly
cancel each other, with $\mathscr{X}(\underline{y},\underline{\widehat{y}})$
thereby being the ratio of product of contributions from open clusters.
Further, as $\underline{y}$ coincides with $\underline{\widehat{y}}$ outside
${\sf B}_{o}(t)$, the collections ${\sf C}_{{\sf G}}(\underline{y})$ and ${\sf
C}_{{\sf G}}(\underline{\widehat{y}})$ contain exactly the same open clusters
$C$ which do not intersect ${\sf B}_{o}(t)$, and this applies also for ${\sf
C}_{{\sf B}_{o}(r)}(\underline{y}_{{\sf B}_{o}(r)})$ versus ${\sf C}_{{\sf
B}_{o}(r)}(\underline{\widehat{y}}_{{\sf B}_{o}(r)})$. Similarly, a cluster
$C\subset{\sf B}_{o}(r-1)$ appears in ${\sf C}_{{\sf G}}(\underline{y})$ if
and only if it appears in ${\sf C}_{{\sf B}_{o}(r)}(\underline{y}_{{\sf
B}_{o}(r)})$ (and the same holds for $\underline{\widehat{y}}$). Denoting by
${\sf C}_{+}$ those elements of ${\sf C}_{{\sf G}}(\underline{y})\cup{\sf
C}_{{\sf B}_{o}(r)}(\underline{\widehat{y}}_{{\sf B}_{o}(r)})$, counted twice
if needed, which intersect both $\partial{\sf B}_{o}(t)$ and $\partial{\sf
B}_{o}(r)$, and by ${\sf C}_{-}$ all such elements in ${\sf C}_{{\sf
G}}(\underline{\widehat{y}})\cup{\sf C}_{{\sf B}_{o}(r)}(\underline{y}_{{\sf
B}_{o}(r)})$, we thus deduce that

$\mathscr{X}(\underline{y},\underline{\widehat{y}})=\frac{\displaystyle{\prod_{C\in{\sf
C}_{+}}}\big{(}1+(q-1)e^{-B|C|}\big{)}}{\displaystyle{\prod_{C\in{\sf
C}_{-}}}\big{(}1+(q-1)e^{-B|C|}\big{)}}\leq(1+(q-1)e^{-B(r-t)})^{2|{\sf
B}_{o}(t)|}=:1+\varepsilon_{r}$

(at most $2|{\sf B}_{o}(t)|$ elements in ${\sf C}_{+}$, each of at least size
$r-t$, and product over ${\sf C}_{-}$ exceeding one). ∎

###  5.2. The rcm free energy density for $d\in 2\mathbb{N}$: proof of Theorem
1.13

Recall [6, Proposition 4.2] that for $(\beta,B)\in{\sf R}_{c}$ and $d\in
2\mathbb{N}$ the Potts free energy density limit

$\widehat{\Phi}=\widehat{\Phi}(\beta,B):=\max\\{\Phi(\nu_{\operatorname{f}}^{\beta,B}),\Phi(\nu_{1}^{\beta,B})\\}\,,$

has the symmetric rcm formulation,

$\displaystyle\widehat{\Phi}=\log(\Psi^{{\rm
sym}}(b_{\operatorname{f}},\ldots,b_{\operatorname{f}}))=\log(\Psi^{{\rm
sym}}(b_{\operatorname{w}},\ldots,b_{\operatorname{w}}))$
$\displaystyle=\sup_{\underline{b}\in[b_{\operatorname{f}},b_{\operatorname{w}}]^{d}}\\{\log(\Psi^{{\rm
sym}}(\underline{b}))\\},$ (5.27)

where for the symmetric group $\mathfrak{S}_{d}$ of size $d$,

$\Psi^{{\rm sym}}(\underline{b}):=\frac{\Psi^{{\rm
vx}}(\underline{b})}{\Psi^{{\rm e},{\rm sym}}(\underline{b})},\qquad\Psi^{{\rm
e},{\rm
sym}}(\underline{b}):=\frac{1}{d!}\sum_{\pi\in\mathfrak{S}_{d}}\Psi^{{\rm
e}}(b_{\pi(1)},\ldots,b_{\pi(d)}),$

while for any $\underline{b}=(b_{1},b_{2},\ldots,b_{d})\in[0,1]^{d}$ and $d\in
2\mathbb{N}$,

$\displaystyle\Psi^{{\rm vx}}(\underline{b})$
$\displaystyle:=(1-\gamma)^{-d}\Big{(}e^{B}\prod_{i=1}^{d}\big{(}1+(q-1)\gamma
b_{i}\big{)}+(q-1)\prod_{i=1}^{d}\big{(}1-\gamma
b_{i}\big{)}\Big{)},\quad\gamma:=\frac{e^{\beta}-1}{e^{\beta}+q-1},$
$\displaystyle\Psi^{{\rm e}}(\underline{b})$
$\displaystyle:=(1-\gamma)^{-d/2}\prod_{i=1}^{d/2}\Big{(}1+(q-1)\gamma
b_{2i-1}b_{2i}\Big{)}.$

We prove Theorem 1.13 by passing wlog to a locally weakly convergent sub-
sequence of $\varphi_{n}^{\beta,B}$ and procuring from
$\mathfrak{m}_{\operatorname{b}}(\mathcal{Q}_{\star}\backslash\mathcal{Q}_{\rm
m})>0$ some graphs ${\sf G}^{\prime}_{n}\stackrel{{\scriptstyle
loc}}{{\longrightarrow}}{\sf T}_{d}$ of free energy densities such that
${\varlimsup_{n}}\;\Phi^{\prime}_{n}(\beta,B)>\widehat{\Phi}$. To do so, we
rely on our next observation, that the supremum in (5.27) over $\underline{b}$
with at least two coordinates strictly inside
$(b_{\operatorname{f}},b_{\operatorname{w}})$, is smaller than
$\widehat{\Phi}$.

######  Lemma 5.7.

For $(\beta,B)\in{\sf R}_{c}$ and any $\delta>0$ there exists
$\varepsilon_{o}\in(0,\delta)$ such that

$\sup_{\underline{b}\in\Lambda_{\delta}}\\{\log(\Psi^{{\rm
sym}}(\underline{b}))\\}<\widehat{\Phi}-\varepsilon_{o},$

where

$\Lambda_{\delta}:=\left\\{\underline{b}\in[b_{\operatorname{f}},b_{\operatorname{w}}]^{d}:\left|\left\\{i:b_{i}\in[b_{\operatorname{f}}+\delta,b_{\operatorname{w}}-\delta]\right\\}\right|\geq
2\right\\}.$ (5.28)

###### Proof.

Since $\Lambda_{\delta}$ is compact and $\Psi^{{\rm sym}}(\cdot)$ is
continuous, the supremum over $\Lambda_{\delta}$ is achieved at some
$\underline{b}^{\prime}\in\Lambda_{\delta}$ and it suffices to show that
$\log(\Psi^{{\rm sym}}(\underline{b}^{\prime}))<\widehat{\Phi}$. Now, suppose
that $\underline{b}^{\circ}\in[b_{\operatorname{f}},b_{\operatorname{w}}]^{d}$
is such that $\log(\Psi^{{\rm sym}}(\underline{b}^{\circ}))=\widehat{\Phi}$
and $b_{i}^{\circ}\in(b_{\operatorname{f}},b_{\operatorname{w}})$ for some
$i\in[d]$. Since both $\Phi^{{\rm vx}}$ and $\Phi^{{\rm e},{\rm sym}}$ are
affine in each $b_{j}$, $j\in[d]$, and $\underline{b}^{\circ}$ is a maximizer,
the function must be constant as we vary the $i$-th coordinate of
$\underline{b}^{\circ}$. This, in particular implies that if we replace
$b_{i}^{\circ}$ with either $b_{\operatorname{f}}$ or $b_{\operatorname{w}}$
then the new vector continues be a maximizer of $\log\Psi^{{\rm sym}}(\cdot)$.
In particular, if $\log(\Psi^{{\rm
sym}}(\underline{b}^{\prime}))=\widehat{\Phi}$ for $\underline{b}^{\prime}$
with at least two coordinates $i\neq j\in[d]$ such that
$b^{\prime}_{i},b^{\prime}_{j}\in(b_{\operatorname{f}},b_{\operatorname{w}})$,
then by the preceding argument there exists
$\underline{b}^{\prime\prime}\in\\{b_{\operatorname{f}},b_{\operatorname{w}}\\}^{d}$,
still a maximizer, with the property
$2\leq|\\{i:b^{\prime\prime}_{i}=b_{\operatorname{f}}\\}|\leq d-2$. However,
the proof of [6, Proposition 4.2] shows that such a
$\underline{b}^{\prime\prime}$ can never be a maximizer, yielding a
contradiction. ∎

Combining Lemma 5.4(c) with Lemma 5.7, we proceed to show that if a local weak
limit point of rcm-s puts a positive mass on
$\mathcal{Q}_{\star}\backslash\mathcal{Q}_{\rm m}$ then it must also put a
positive mass on those $\varphi\in\mathcal{Q}_{\star}$ for which
$\log\Psi^{{\rm sym}}(s_{\partial o}^{t,r})$ is strictly smaller than
$\widehat{\Phi}$, for large $t<r$, with non-negligible probability.

######  Lemma 5.8.

For $\varepsilon>0$ and $t<r\in\mathbb{N}$ set

$\mathscr{B}_{\varepsilon}^{t,r}:=\left\\{\log\Psi^{{\rm sym}}(s_{\partial
o}^{t,r})<\widehat{\Phi}-\varepsilon\right\\}\,.$ (5.29)

If
$\mathfrak{m}_{\operatorname{b}}(\mathcal{Q}_{\star}\backslash\mathcal{Q}_{{\rm
m}})>0$ for a local weak limit point $\mathfrak{m}_{\operatorname{b}}$ of
$\\{\varphi_{n}^{\beta,B}\\}$, then there exist $\xi>0$,
$t^{\prime}\in\mathbb{N}$, and $r^{\prime}(t)<\infty$, such that
$\int\varphi(\mathscr{B}_{\xi}^{t,r})d\mathfrak{m}_{\operatorname{b}}\geq\xi$
for any $t\geq t^{\prime}$ and $r\geq r^{\prime}(t)$.

###### Proof.

With $\mathfrak{m}_{\operatorname{b}}$ supported on the convex set
$\mathcal{Q}_{\star}$ (see Lemma 2.9(d)), also
$\bar{\varphi}(\cdot):=\int\varphi(\cdot)d\mathfrak{m}_{\operatorname{b}}\in\mathcal{Q}_{\star}$.
Further, for any non-negative $F\in C_{b}(\mathbb{R}^{d})$ and
$\varphi\in\mathcal{Q}_{\star}$, it follows from (5.2) that

$\lim_{t\to\infty}\lim_{r\to\infty}\varphi(F(s_{\partial
o}^{t,r}))=\varphi(F(s_{\partial o}(\varphi)))\,,$

hence by definition of $\bar{\varphi}$ and Fatou’s lemma,

$\bar{\varphi}(F(s_{\partial
o}(\bar{\varphi})))=\lim_{t\to\infty}\lim_{r\to\infty}\bar{\varphi}(F(s_{\partial
o}^{t,r}))=\lim_{t\to\infty}\lim_{r\to\infty}\int\varphi(F(s_{\partial
o}^{t,r}))d\mathfrak{m}_{\operatorname{b}}\geq\int\varphi(F(s_{\partial
o}(\varphi))d\mathfrak{m}_{\operatorname{b}}\,.$ (5.30)

Considering $F(\underline{s}):=\sum_{i=1}^{d}f_{\varepsilon}(s_{i})$ for
continuous $f_{\varepsilon}$ such that $0\leq
f_{\varepsilon}(\cdot)\uparrow{\bf
1}_{(b_{\operatorname{f}},b_{\operatorname{w}})}(\cdot)$ as
$\varepsilon\downarrow 0$, we deduce from (5.30) by monotone convergence and
Fatou’s lemma, that in view of (5.6),

$\sum_{u\in\partial o}\bar{\varphi}(s_{u\to
o}(\bar{\varphi})\neq\\{b_{\operatorname{f}},b_{\operatorname{w}}\\})\geq\int\sum_{u\in\partial
o}\varphi(s_{u\to
o}(\varphi)\neq\\{b_{\operatorname{f}},b_{\operatorname{w}}\\})d\mathfrak{m}_{\operatorname{b}}\geq\int\varphi((\mathscr{A}_{\operatorname{f}}\cup\mathscr{A}_{\operatorname{w}})^{c})d\mathfrak{m}_{\operatorname{b}}\,.$
(5.31)

Recall Lemma 5.4(c) that
$\varphi((\mathscr{A}_{\operatorname{f}}\cup\mathscr{A}_{\operatorname{w}})^{c})>0$
whenever $\varphi\in\mathcal{Q}_{\star}\backslash\mathcal{Q}_{\rm m}$. Hence,
our assumption that
$\mathfrak{m}_{\operatorname{b}}(\mathcal{Q}_{\star}\backslash\mathcal{Q}_{{\rm
m}})>0$ implies the strict positivity of the rhs of (5.31) and thereby the
same holds for the lhs. By definition of the local weak convergence, the law
$\mathfrak{m}_{\operatorname{b}}$ must be invariant to our choice of the root
of ${\sf T}_{d}$. Hence, with ${\sf T}_{d}$ a regular tree, necessarily
$\bar{\varphi}$ is invariant under any automorphism of ${\sf T}_{d}$. In
particular, fixing $u_{1}\in\partial o$ and setting

$H_{\varepsilon}:=\big{\\{}\min_{\ddagger\in\\{\operatorname{f},\operatorname{w}\\}}|s_{u_{1}\to
o}(\bar{\varphi})-b_{\ddagger}|>\varepsilon\big{\\}},$

we deduce that $\bar{\varphi}(H_{0})>0$. This in turn allows us to fix
$\varepsilon>0$ small enough so that
$\bar{\varphi}(H_{\varepsilon})\geq\varepsilon$. Next, fix $\ell:=\lceil
3/\varepsilon\rceil$ and a non-random path $o=:v_{1},v_{2},\ldots,v_{\ell+1}$
in ${\sf T}_{d}$ such that $v_{i+1}\in\partial{\sf T}_{d}(i)$. Then, fixing
for each $1\leq i\leq\ell$, some $w_{i}\in\partial v_{i}$, $w_{i}\neq v_{i+1}$
(which as $d\geq 3$, is always possible), we have by the automorphism
invariance of $\bar{\varphi}$ that

$\bar{\varphi}(A_{i})=\bar{\varphi}({\sf
H}_{\varepsilon})\geq\varepsilon\geq\frac{3}{\ell}\quad\text{for }\quad
A_{i}:=\Big{\\{}\min_{\ddagger\in\\{\operatorname{f},\operatorname{w}\\}}|s_{w_{i}\to
v_{i}}(\bar{\varphi})-b_{\ddagger}|>\varepsilon\Big{\\}}\,.$ (5.32)

The smooth map $\widehat{\sf BP}(\cdot;B):[0,1]^{d-1}\to[0,1]$ of (5.4) is
coordinate-wise strictly increasing, having
$(b_{\ddagger},\ldots,b_{\ddagger})$ as its fixed points (see (5.5) and Lemma
5.4(b)). Thus, for some $\delta_{0}=\delta_{0}(\varepsilon,\beta,B,q,d)>0$, it
follows from (5.5) and (5.6) that $s_{v_{j}\to
v_{j-1}}(\bar{\varphi})\in(b_{\operatorname{f}}+\delta_{0},b_{\operatorname{w}}-\delta_{0})$
on the event $A_{j}$. Iterating this argument, while reducing $\delta_{0}$ to
$\delta=\delta(\delta_{0},\ell,\beta,B,q,d)\in(0,\varepsilon)$, it follows
that on each $A_{j}$, $j\leq\ell$,

$\\{s_{v_{i+1}\to v_{i}}(\bar{\varphi}),1\leq
i<j\\}\subset(b_{\operatorname{f}}+\delta,b_{\operatorname{w}}-\delta)\,.$

So, if $A_{i}\cap A_{j}$ occurs for $i<j$, then $s_{v_{i+1}\to
v_{i}}(\bar{\varphi}),s_{w_{i}\to
v_{i}}(\bar{\varphi})\in(b_{\operatorname{f}}+\delta,b_{\operatorname{w}}-\delta)$
and in particular $s_{\partial v_{i}}\in\Lambda_{\delta}$ of (5.28). Namely,
any pair from $\\{A_{1},\ldots,A_{\ell}\\}$ results with $s_{\partial
v_{i}}(\bar{\varphi})\in\Lambda_{\delta}$ for some $i<\ell$. Consequently, by
the union bound and the automorphism invariance property of $\bar{\varphi}$,

$\displaystyle\ell\bar{\varphi}(s_{\partial
o}(\bar{\varphi})\in\Lambda_{\delta})$
$\displaystyle\geq\bar{\varphi}\big{(}\bigcup_{i=1}^{\ell}\\{s_{\partial
v_{i}}(\bar{\varphi})\in\Lambda_{\delta}\\}\big{)}\geq\bar{\varphi}\big{(}\sum_{i=1}^{\ell}{\bf
I}_{A_{i}}\geq
2\big{)}\geq\frac{1}{\ell}\bar{\varphi}\big{(}\sum_{i=1}^{\ell}{\bf
I}_{A_{i}}-2\big{)}\geq\frac{1}{\ell}$ (5.33)

(where the right-most inequality is due to (5.32)). In view of Lemma 5.7 we
deduce from (5.33) that for some $\varepsilon_{o}>0$,

$\bar{\varphi}(\log\Psi^{{\rm sym}}(s_{\partial
o}(\bar{\varphi}))<\widehat{\Phi}-\varepsilon_{o})\geq\frac{1}{\ell^{2}}\,.$

With $\log\Psi^{{\rm sym}}\in C_{b}([0,1]^{d})$, it follows from (5.2) that
$\bar{\varphi}$-a.e. $\log\Psi^{{\rm sym}}(s_{\partial
o}^{t,r})\to\log\Psi^{{\rm sym}}(s_{\partial o}(\bar{\varphi}))$ as
$r\to\infty$ and then $t\to\infty$. Thus, by the preceding

$\lim_{t\to\infty}\lim_{r\to\infty}\bar{\varphi}(\mathscr{B}_{\varepsilon_{o}}^{t,r})\geq\frac{1}{\ell^{2}}\,,$

yielding our claim for $\xi=\varepsilon_{o}\wedge(2\ell^{2})^{-1}$. ∎

To prove Theorem 1.13 we keep moving from a finite graph ${\sf G}$ to a graph
${\sf G}^{-}(w)$ without some $w\in V({\sf G})$ and all the edges to it,
thereby re-connecting the neighbors $x_{1},x_{2},\ldots,x_{d}$ of $w$ to form
for $\pi\in\mathfrak{S}_{d}$ the graph ${\sf G}^{\pi}(w)$ as ${\sf G}^{-}(w)$
with the additional edges $(x_{\pi(2i-1)},x_{\pi(2i)})$, $i=1\ldots,d/2$. Key
to the proof is thus to estimate the ratios of contributions to the partition
functions from rcm-s on ${\sf G}$ versus ${\sf G}^{-}(w)$, and on ${\sf
G}^{\pi}(w)$ versus ${\sf G}^{-}(w)$, when fixing certain bonds. To this end,
recall $\beta_{e}^{\star}$ of (2.1) and $p_{e}$, ${\sf C}(\cdot)$ of
Definition 1.7, where the un-normalized Edwards-Sokal probability mass

$\widehat{\varpi}_{{\sf
G}^{\star}}^{\beta,B}(\underline{\sigma},\underline{\eta}):=\prod_{e\in
E^{\star}}e^{\beta^{\star}_{e}}\left[(1-p_{e})({1-\eta_{e}})+p_{e}{\eta_{e}}\delta_{e}(\underline{\sigma})\right]\cdot\delta_{\sigma_{v^{\star}},1},\quad\underline{\sigma}\in[q]^{V^{\star}},\;\underline{\eta}\in\\{0,1\\}^{E^{\star}}$
(5.34)

(see Definition 2.1), has total mass matching the Potts partition function of
${\sf G}$, as in (2.1). Namely,

$\sum_{\underline{\sigma},\underline{\eta}}\widehat{\varpi}_{{\sf
G}^{\star}}^{\beta,B}(\underline{\sigma},\underline{\eta})=Z_{{\sf
G}}(\beta,B)\,.$

More generally, restricting (5.34) to bond values
$\underline{y}_{W^{\star}}\in\\{0,1\\}^{E(W^{\star})}$ on the edges of
$W\subset{\sf G}$ (or, to only $\underline{y}_{W}\in\\{0,1\\}^{E(W)}$ on non-
ghosted edges), yields the rcm-restricted partition functions

$\displaystyle\mathcal{Z}_{{\sf G},W^{\star}}(\underline{y}_{W^{\star}})$
$\displaystyle:={\frac{1}{q}}\sum_{\underline{\eta}:\underline{\eta}_{W^{\star}}=\underline{y}_{W^{\star}}}q^{|{\sf
C}(\underline{\eta})|}\prod_{e\in
E^{\star}}e^{\beta^{\star}_{e}}p_{e}^{\eta_{e}}(1-p_{e})^{1-\eta_{e}}=\sum_{\underline{\eta}:\underline{\eta}_{W^{\star}}=\underline{y}_{W^{\star}}}\sum_{\underline{\sigma}}\widehat{\varpi}_{{\sf
G}^{\star}}^{\beta,B}(\underline{\sigma},\underline{\eta})\,,$ (5.35)
$\displaystyle\mathcal{Z}_{{\sf G},W}(\underline{y}_{W})$
$\displaystyle:={\frac{1}{q}}\sum_{\underline{\eta}:\underline{\eta}_{W}=\underline{y}_{W}}q^{|{\sf
C}(\underline{\eta})|}\prod_{e\in
E^{\star}}e^{\beta^{\star}_{e}}p_{e}^{\eta_{e}}(1-p_{e})^{1-\eta_{e}}=\sum_{\underline{\eta}:\underline{\eta}_{W}=\underline{y}_{W}}\sum_{\underline{\sigma}}\widehat{\varpi}_{{\sf
G}^{\star}}^{\beta,B}(\underline{\sigma},\underline{\eta})\,.$ (5.36)

For any $v\in V({\sf G})$ with $\partial v=(u_{i})_{i=1}^{d}$,
$\pi\in{\mathfrak{S}}_{d}$ and $\underline{y}\in\\{0,1\\}^{{\sf
B}_{v}(r)\backslash{\sf B}_{v}(t)}$, $t<r$, we set the functions

$\displaystyle\Psi^{{\rm vx}}_{t,r,{\sf G}}(v,\underline{y})$
$\displaystyle:=\varphi^{\beta,B}_{{\sf G}^{-}(v)}\big{[}\Psi^{{\rm
vx}}(\mathfrak{s}^{t,r}_{\partial
v})\mid\eta_{v,t}^{r}=\underline{y}\,\big{]},\quad\Psi^{{\rm e},{\rm
sym}}_{t,r,{\sf
G}}(v,\underline{y}):=\frac{1}{d!}\sum_{\pi\in{\mathfrak{S}}_{d}}\Psi^{{\rm
e},\pi}_{t,r,{\sf G}}(v,\underline{y})\,,$ (5.37) $\displaystyle\Psi^{{\rm
e},\pi}_{t,r,{\sf G}}(v,\underline{y})$
$\displaystyle:=\varphi^{\beta,B}_{{{\sf G}^{-}(v)}}\big{[}\Psi^{{\rm
e}}(\mathfrak{s}^{t,r}_{u_{\pi(1)}\to
v},\ldots,\mathfrak{s}^{t,r}_{u_{\pi(d)}\to
v})\mid\eta_{v,t}^{r}=\underline{y}\,\big{]}.$

Utilizing Lemma 5.6 and (5.34)-(5.36), we proceed to estimate the ratios of
various rcm-restricted partition functions, in terms of these functions.

######  Lemma 5.9.

Fix $B>0$ and $\beta\geq 0$. For some
$r_{o}^{\prime}=r_{o}^{\prime}(\varepsilon,t)$, if $o\in V(\widehat{\sf G})$
and ${\sf B}_{o}(r)\cong{\sf T}_{d}(r)$, $r\geq r_{o}^{\prime}$, then for any
${\sf B}_{o}(r)\subset{\sf G}\subset\widehat{\sf G}$, we get upon setting
$\widehat{W}:={\sf G}\backslash{\sf B}_{o}(t)$ and $W_{r}:={\sf
B}_{o}(r)\backslash{\sf B}_{o}(t)$, that

$\displaystyle\max_{\underline{y}_{\widehat{W}}}\Big{\\{}\big{|}\frac{\mathcal{Z}_{{\sf
G},\widehat{W}}(\underline{y}_{\widehat{W}})}{\mathcal{Z}_{{\sf
G}^{-}(o),\widehat{W}}(\underline{y}_{\widehat{W}})}-\Psi^{{\rm
vx}}_{t,r,\widehat{\sf G}}(o,\underline{y}_{W_{r}})\big{|}\Big{\\}}$
$\displaystyle\leq\varepsilon\,,$ (5.38)
$\displaystyle\max_{\pi\in{\mathfrak{S}}_{d}}\max_{\underline{y}_{\widehat{W}}}\Big{\\{}\big{|}\frac{\mathcal{Z}_{{\sf
G}^{\pi}(o),\widehat{W}}(\underline{y}_{\widehat{W}})}{\mathcal{Z}_{{\sf
G}^{-}(o),\widehat{W}}(\underline{y}_{\widehat{W}})}-\Psi^{{\rm
e},\pi}_{t,r,\widehat{\sf G}}(o,\underline{y}_{W_{r}})\big{|}\Big{\\}}$
$\displaystyle\leq\varepsilon.$ (5.39)

######  Remark 5.10.

In view of (5.36), the bounds (5.38)-(5.39) actually hold also for any
subgraph $\widehat{W}$ of ${\sf G}\backslash{\sf B}_{o}(t)$ which contains
${\sf B}_{o}(r)\backslash{\sf B}_{o}(t)$.

###### Proof.

With ${\sf G}\cap{\sf B}_{o}(r)=\widehat{\sf G}\cap{\sf B}_{o}(r)\cong{\sf
T}_{d}(r)$ and $W_{r}=\widehat{W}\cap{\sf B}_{o}(r)$, we first examine the
special case ${\sf G}=\widehat{\sf G}={\sf B}_{o}(r)$. Specifically, using
${\sf B}^{-}_{o}(r):={\sf G}^{-}(o)$ and ${\sf B}^{\pi}_{o}(r):={\sf
G}^{\pi}(o)$ when ${\sf G}={\sf B}_{o}(r)$, we show that for any
$\underline{y}\in\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t}^{\star}$ and
$\pi\in{\mathfrak{S}}_{d}$,

$\displaystyle\widehat{\mathcal{Z}}_{r}(\underline{y})$
$\displaystyle:=\frac{\mathcal{Z}_{{\sf
B}_{o}(r),W_{r}^{\star}}(\underline{y})}{\mathcal{Z}_{{\sf
B}^{-}_{o}(r),W_{r}^{\star}}(\underline{y})}=\Psi^{{\rm
vx}}(\mathfrak{s}_{\partial o}^{t,r}(\underline{y})),$ (5.40)
$\displaystyle\widehat{\mathcal{Z}}^{\pi}_{r}(\underline{y})$
$\displaystyle:=\frac{\mathcal{Z}_{{\sf
B}^{\pi}_{o}(r),W_{r}^{\star}}(\underline{y})}{\mathcal{Z}_{{\sf
B}^{-}_{o}(r),W_{r}^{\star}}(\underline{y})}=\Psi^{{\rm
e}}(\mathfrak{s}^{t,r}_{u_{\pi(1)}\to
o}(\underline{y}),\ldots,\mathfrak{s}^{t,r}_{u_{\pi(d)}\to
o}(\underline{y})).$ (5.41)

Indeed, it follows from (5.35) that

$\displaystyle\mathcal{Z}_{{\sf
B}^{-}_{o}(r),W_{r}^{\star}}(\underline{y})=\\!\\!\\!\\!\\!\\!\\!\sum_{\\{\underline{\eta}^{\prime}:\eta_{o,t}^{\star,r}=\underline{y}\\}}\sum_{\underline{\sigma}^{\prime}}\widehat{\varpi}_{{\sf
B}^{-}_{o}(r)^{\star}}^{\beta,B}(\underline{\sigma}^{\prime},\underline{\eta}^{\prime}).$

Further, for any $e\in E^{\star}$ and $\underline{\sigma}\in[q]^{V^{\star}}$,

$\sum_{\eta_{e}\in\\{0,1\\}}e^{\beta^{\star}_{e}}\left[(1-p_{e})({1-\eta_{e}})+p_{e}{\eta_{e}}\delta_{e}(\underline{\sigma})\right]=1+(e^{\beta^{\star}_{e}}-1)\delta_{e}(\underline{\sigma})\,.$

Thus, with $\underline{\eta}_{{\sf
B}_{o}^{\star}(r)}=(\underline{\eta}^{(1)},\underline{\eta}^{\prime})$ for
$\underline{\eta}^{(1)}:=(\eta_{e})_{e\in{\sf
B}_{o}(1)\cup\\{(o,v^{\star})\\}}$, we similarly get from (5.34)-(5.35) that

$\displaystyle\mathcal{Z}_{{\sf B}_{o}(r),W_{r}^{\star}}(\underline{y})$
$\displaystyle=\\!\\!\\!\sum_{\\{\underline{\eta}^{\prime}:\underline{\eta}^{\prime}_{W_{r}^{\star}}=\underline{y}\\}}\sum_{\underline{\sigma}^{\prime}}\sum_{\sigma_{o},\underline{\eta}^{(1)}}\widehat{\varpi}_{{\sf
B}^{\star}_{o}(r)}^{\beta,B}((\sigma_{o},\underline{\sigma}^{\prime}),(\underline{\eta}^{(1)},\underline{\eta}^{\prime}))=\\!\\!\\!\sum_{\\{\underline{\eta}^{\prime}:\eta_{o,t}^{\star,r}=\underline{y}\\}}\sum_{\underline{\sigma}^{\prime}}f(\sigma^{\prime}_{\partial
o})\widehat{\varpi}_{{\sf
B}^{-}_{o}(r)^{\star}}^{\beta,B}(\underline{\sigma}^{\prime},\underline{\eta}^{\prime}),$

for some function $f(\cdot)$, where by enumerating over the $[q]$-valued
$\sigma_{o}$, one verifies that

$f(\sigma_{1},\ldots,\sigma_{d})=\sum_{k=1}^{q}e^{B\delta_{k,1}}\prod_{i=1}^{d}\big{(}1+(e^{\beta}-1)\delta_{\sigma_{i},k}\big{)}\,.$

Consequently,

$\widehat{\mathcal{Z}}_{r}(\underline{y})=\varpi_{{\sf
B}^{-}_{o}(r)^{\star}}^{\beta,B}\Big{[}f(\sigma_{\partial
o})\big{|}\eta_{o,t}^{\star,r}=\underline{y}\Big{]}.$ (5.42)

By the same reasoning, now with
$\underline{\eta}^{(1)}:=\underline{\eta}_{{\sf B}_{o}^{\pi}(1)\backslash{\sf
B}_{o}^{-}(1)}$, we find that, for any $\pi\in{\mathfrak{S}}_{d}$,

$\displaystyle\widehat{\mathcal{Z}}^{\pi}_{r}(\underline{y})$
$\displaystyle=\varpi_{{\sf
B}^{-}_{o}(r)^{\star}}^{\beta,B}\Big{[}f^{\pi}(\sigma_{\partial
o})\big{|}\eta_{o,t}^{\star,r}=\underline{y}\Big{]},$ (5.43) $\displaystyle
f^{\pi}(\sigma_{1},\ldots,\sigma_{d})$
$\displaystyle:=\prod_{i=1}^{d/2}\big{(}1+(e^{\beta}-1)\delta_{\sigma_{\pi(2i-1)},\sigma_{\pi(2i)}}\big{)}\,.$

Since ${\sf B}_{o}(r)\cong{\sf T}_{d}(r)$, the spins $(\sigma_{u},u\in\partial
o)$ are mutually independent under $\varpi_{{\sf
B}^{-}_{o}(r)^{\star}}^{\beta,B}(\cdot\mid\eta_{o,t}^{\star,r})$, with each of
them uniformly distributed on $\\{2,\ldots,q\\}$. Thus, upon setting for
$i=1,\ldots,d$,

$b_{i}:=\frac{1}{1-1/q}\varpi_{{\sf
B}^{-}_{o}(r)^{\star}}^{\beta,B}\big{(}\sigma_{u_{i}}=1\big{|}\eta_{o,t}^{\star,r}=\underline{y}\big{)}-\frac{1/q}{1-1/q}\,,$

a straightforward computation shows that the rhs of (5.42) and (5.43) are
given by $\Psi^{\rm vx}(\underline{b})$ and $\Psi^{{\rm
e}}(b_{\pi(1)},\ldots,b_{\pi(d)})$, respectively. Finally, the identification
$\underline{b}=\mathfrak{s}^{t,r}_{\partial o}$ follows from Lemma 2.2,
analogously to the derivation of (5.11).

Armed with (5.40), we proceed to prove (5.38). To this end, writing
$\underline{y}=\underline{y}_{W^{\star}}$ and
$\underline{y}^{o}=\underline{y}_{W}$, we get by following the derivation of
(5.42), that for any ${\sf G}$ with $|\partial o|=d$ and $W_{r}\subset
W\subset{\sf G}\backslash{\sf B}_{o}(t)$,

$\displaystyle\frac{\mathcal{Z}_{{\sf
G},W^{\star}}(\underline{y})}{\mathcal{Z}_{{\sf
G}^{-}(o),W^{\star}}(\underline{y})}$ $\displaystyle=\varpi_{{\sf
G}^{-}(o)^{\star}}^{\beta,B}\Big{[}f(\sigma_{\partial
o})\big{|}\underline{\eta}_{W^{\star}}=\underline{y}\Big{]}\text{ and
}\frac{\mathcal{Z}_{{\sf G},W}(\underline{y}^{o})}{\mathcal{Z}_{{\sf
G}^{-}(o),W}(\underline{y}^{o})}=\varpi_{{\sf
G}^{-}(o)^{\star}}^{\beta,B}\Big{[}f(\sigma_{\partial
o})\big{|}\underline{\eta}_{W}=\underline{y}^{o}\Big{]}\,.$

Consequently,

$\displaystyle\frac{\mathcal{Z}_{{\sf
G},W}(\underline{y}^{o})}{\mathcal{Z}_{{\sf G}^{-}(o),W}(\underline{y}^{o})}$
$\displaystyle=\sum_{\underline{y}\backslash\underline{y}^{o}}\frac{\mathcal{Z}_{{\sf
G},W^{\star}}(\underline{y})}{\mathcal{Z}_{{\sf
G}^{-}(o),W^{\star}}(\underline{y})}\varphi^{\beta,B}_{{\sf
G}^{-}(o)}({\underline{\eta}_{W^{\star}}=\underline{y}}\mid{\underline{\eta}_{W}=\underline{y}^{o}}).$
(5.44)

Further, from (5.34)-(5.35) we have that for some
$\bar{c}=\bar{c}(q,\beta,B,d)>0$,

$\frac{\mathcal{Z}_{{\sf
G}^{-}(o),{W}^{\star}}(\underline{y})}{\mathcal{Z}_{{\sf
G},W^{\star}}(\underline{y})}=\frac{1}{q}\varphi^{\beta,B}_{{\sf
G}}(\underline{\eta}^{(1)}\equiv
0\mid\underline{\eta}_{W^{\star}}=\underline{y})\geq\bar{c}\,.$ (5.45)

Comparing (5.40) with (5.45) for $({\sf B}_{o}(r),W_{r})$, we deduce that

$\frac{\mathcal{Z}_{{\sf G},W^{\star}}(\underline{y})}{\mathcal{Z}_{{\sf
G}^{-}(o),{W}^{\star}}(\underline{y})}=\frac{\varphi^{\beta,B}_{{\sf
B}_{o}(r)}(\underline{\eta}^{(1)}\equiv
0\mid\eta_{o,t}^{\star,r}=\underline{y}_{W_{r}^{\star}})}{\varphi^{\beta,B}_{{\sf
G}}(\underline{\eta}^{(1)}\equiv
0\mid{\underline{\eta}_{W^{\star}}=\underline{y}})}\Psi^{{\rm
vx}}(\mathfrak{s}_{\partial o}^{t,r}(\underline{y}_{W_{r}^{\star}}))\,,$
(5.46)

which, in view of (5.21) equals to $\Psi^{{\rm vx}}(\mathfrak{s}_{\partial
o}^{t,r}(\underline{y}_{W_{r}^{\star}}))$ whenever
$\underline{y}_{W_{r}^{\star}}\in\mathscr{S}_{t,r}$. Utilizing this
observation when plugging (5.46) into (5.44), then using the uniform lower
bound of (5.45) and the triangle inequality, we find that for $r\geq
r_{0}(\varepsilon\bar{c}/6,t)$ of Lemma 5.6,

$\displaystyle\Big{|}\frac{\mathcal{Z}_{{\sf
G},W}(\underline{y}_{W})}{\mathcal{Z}_{{\sf
G}^{-}(o),W}(\underline{y}_{W})}-\varphi^{\beta,B}_{{\sf G}^{-}(o)}(\Psi^{{\rm
vx}}(\mathfrak{s}_{\partial
o}^{t,r})\mid{\underline{\eta}_{W}=\underline{y}_{W}})\Big{|}$
$\displaystyle\leq(2/\bar{c})\varphi^{\beta,B}_{{\sf
G}^{-}(o)}(\mathscr{S}_{t,r}^{c}\mid{\underline{\eta}_{W}=\underline{y}_{W}})\leq\frac{\varepsilon}{3}\,,$
(5.47)

where in the last inequality we have employed (5.22) for
$\varphi^{\beta,B}_{{\sf G}^{-}(o)}$.

Analogously to the derivation of (5.45), except for now using (5.36), we get
for any $W_{r}\subset W\subset\widehat{W}$,

$\frac{\mathcal{Z}_{{\sf G}^{-}(o),W}(\underline{y}_{W})}{\mathcal{Z}_{{\sf
G},W}(\underline{y}_{W})}=\frac{1}{e^{B}+q-1}\varphi^{\beta,B}_{{\sf
G}}(\underline{\eta}_{{\sf B}_{o}(1)}\equiv
0\mid\underline{\eta}_{W}=\underline{y}_{W})\,.$

Hence, by (5.23), if $\widehat{\sf G}\supset{\sf G}$ matches ${\sf G}$ on
their respective $r$ balls around $o$ and $r\geq r_{0}(\varepsilon/9,t)$, then

$\max_{\underline{y}_{\widehat{W}}}\Big{|}\frac{\mathcal{Z}_{\widehat{\sf
G},W_{r}}(\underline{y}_{W_{r}})}{\mathcal{Z}_{\widehat{\sf
G}^{-}(o),W_{r}}(\underline{y}_{W_{r}})}\frac{\mathcal{Z}_{{\sf
G}^{-}(o),\widehat{W}}(\underline{y}_{\widehat{W}})}{\mathcal{Z}_{{\sf
G},\widehat{W}}(\underline{y}_{\widehat{W}})}-1\Big{|}\,\leq\frac{\varepsilon}{3}.$
(5.48)

We combine (5.48) with (5.47) at $(\widehat{\sf G},W_{r})$, and recall (5.37)
that $\Psi^{{\rm vx}}_{t,r,\widehat{\sf
G}}(o,\cdot)=\varphi^{\beta,B}_{\widehat{\sf G}^{-}(o)}(\Psi^{{\rm
vx}}(\mathfrak{s}_{\partial o}^{t,r})|{\underline{\eta}_{W_{r}}})$, to arrive
at (5.38).

Upon changing $f$ to $f^{\pi}$ and $\underline{\eta}^{(1)}$ to
$\underline{\eta}_{{\sf B}_{o}^{\pi}(1)\backslash{\sf B}_{o}^{-}(1)}$ (as in
the derivation of (5.43)), the proof of (5.39) out of (5.41) is the same,
hence omitted. ∎

Hereafter we fix the sequence $\\{{\sf G}_{n}\\}$, using for $v\in[n]$ the
shorthand $\varphi_{n}^{v}:=\varphi_{{\sf G}_{n}^{-}(v)}^{\beta,B}$ and define

$\displaystyle\Psi^{{\rm sym}}_{t,r,n}(v,\underline{y})$
$\displaystyle:=\Psi^{{\rm vx}}_{t,r,n}(v,\underline{y})/\Psi^{{\rm e},{\rm
sym}}_{t,r,n}(v,\underline{y}),$ (5.49)

for $\Psi^{{\rm vx}}_{t,r,n}:=\Psi^{{\rm vx}}_{t,r,{\sf G}_{n}}$ and
$\Psi^{{\rm e},{\rm sym}}_{t,r,n}:=\Psi^{{\rm e},{\rm sym}}_{t,r,{\sf G}_{n}}$
of (5.37). For any $t<r$, $v\in[n]$ and $\varepsilon>0$, we further define,
analogously to (5.29), the bond events

$\mathcal{A}_{\varepsilon,v,n}^{t,r}:=\Big{\\{}\underline{y}:\log\Psi_{t,r,n}^{{\rm
sym}}(v,\underline{y}_{{\sf B}_{v}(r)\backslash{\sf
B}_{v}(t)})<\widehat{\Phi}-\varepsilon\Big{\\}}\,.$ (5.50)

Aiming to later utilize Lemma 5.9, we first adapt Lemma 5.8 to lower bound the
average over $v$ of $\varphi^{\beta,B}_{{\sf
G}_{n}}(\mathcal{A}_{\varepsilon,v,n}^{t,r})$, when $\varphi^{\beta,B}_{n}$
converges locally weakly to $\mathfrak{m}_{\operatorname{b}}$. Out of this we
produce vertex subsets $\mathscr{V}_{n}$, having well-spaced, tree-like
neighborhoods in ${\sf G}_{n}$, at which with non-negligible probability a
positive fraction of the events (5.50) hold.

######  Lemma 5.11.

Suppose
$\mathfrak{m}_{\operatorname{b}}(\mathcal{Q}_{\star}\backslash\mathcal{Q}_{{\rm
m}})>0$.
The following then holds for some $\delta_{o}>0$, $t^{\prime}$, large
$r^{\prime}(t)<\infty$, and small enough $\zeta(\delta_{o},r)>0$.
(a). If ${\sf G}_{n}\stackrel{{\scriptstyle loc}}{{\longrightarrow}}{\sf
T}_{d}$ and $\\{\varphi_{n}:=\varphi_{{\sf G}_{n}}^{\beta,B}\\}$ converges
locally weakly to $\mathfrak{m}_{\operatorname{b}}$, then for $t\geq
t^{\prime}$ and $r\geq r^{\prime}(t)$,

$\liminf_{n\to\infty}\frac{1}{n}\sum_{v=1}^{n}\varphi_{n}(\mathcal{A}_{\delta_{o},v,n}^{t,r})\geq
4\delta_{o}\,.$

(b). For any $n\geq n^{\prime}(\zeta)$, there exist
$\mathscr{V}_{n}\subset[n]$ of size $|\mathscr{V}_{n}|=\zeta n$, such that

$\displaystyle\inf_{v\neq v^{\prime}\in\mathscr{V}_{n}}\\{{\rm dist}_{{\sf
G}_{n}}(v,v^{\prime})\\}>3r,\quad$ $\displaystyle\quad{\sf B}_{v}(3r)\cong{\sf
T}_{d}(3r),\;\;\forall v\in\mathscr{V}_{n}\,,$ (5.51)
$\displaystyle\varphi_{n}\Big{(}\frac{1}{|\mathscr{V}_{n}|}\sum_{v\in\mathscr{V}_{n}}{\bf
1}_{\mathcal{A}^{t,r}_{\delta_{o},v,n}}$
$\displaystyle\geq\delta_{o}\Big{)}\geq\delta_{o}.$ (5.52)

###### Proof.

(a). Note that for some finite $C=C(q,d,\beta,B)$, uniformly over $n$ and
$v\in[n]$ with $|\partial v|\leq d$,

$C^{-1}\varphi_{n}(\underline{\eta}_{{{\sf
G}_{n}^{-}(v)}})\leq\varphi_{n}^{v}(\underline{\eta}_{{\sf G}_{n}^{-}(v)})\leq
C\varphi_{n}(\underline{\eta})\qquad\forall\underline{\eta}\in\\{0,1\\}^{E_{n}^{\star}}\,.$

Then, from ${\sf G}_{n}\stackrel{{\scriptstyle loc}}{{\longrightarrow}}{\sf
T}_{d}$ and Lemma 5.8, we have that for any $\varepsilon\leq\xi/(2C)$, $t\geq
t^{\prime}$ and $r\geq r^{\prime}(t)$,

$\liminf_{n\to\infty}\frac{1}{n}\sum_{v=1}^{n}{\bf 1}_{\\{{\sf
B}_{v}(r)\cong{\sf
T}_{d}(r)\\}}\big{(}\varphi_{n}^{v}(\mathscr{B}_{2C\varepsilon,v,n}^{t,r})-\varepsilon\big{)}\geq
C^{-1}\int\varphi(\mathscr{B}_{2C\varepsilon}^{t,r})d\mathfrak{m}_{\operatorname{b}}-\varepsilon\geq\varepsilon\,,$
(5.53)

where $\mathscr{B}_{\varepsilon,v,n}^{t,r}:=\\{\log\Psi^{{\rm
sym}}(\mathfrak{s}_{\partial v}^{t,r})<\widehat{\Phi}-\varepsilon\\}$.
Further, for some $c=c(q,d,\beta,B)>0$,

$\Psi^{{\rm vx}}(\underline{b}),\Psi^{{\rm e},{\rm
sym}}(\underline{b})\in[1,c^{-1}],\qquad\forall\underline{b}\in[0,1]^{d},$
(5.54)

and we now set for any $v\in[n]$,

$Y_{v,n}(\underline{y}):=\varphi_{n}^{v}\left[\Psi^{{\rm e},{\rm
sym}}(\mathfrak{s}_{\partial v}^{t,r}){\bf
1}_{\mathscr{B}^{t,r}_{2C\varepsilon,v,n}}\mid\eta_{v,t}^{r}=\underline{y}\right]-c\varepsilon\Psi_{t,r,n}^{{\rm
e},{\rm sym}}(v,\underline{y}).$ (5.55)

Recall from (5.20) that $\mathfrak{s}_{\partial
v}^{t,r}\in[b_{\operatorname{f}}-\varepsilon^{4},b_{\operatorname{w}}+\varepsilon^{4}]^{d}$
whenever $t>\kappa_{o}(\varepsilon^{4})$ and ${\sf B}_{v}(r)\cong{\sf
T}_{d}(r)$. For such $v$ we have by (5.27) and the Lipschitz continuity of
$\log\Psi^{{\rm sym}}(\cdot)$, that for some $\varepsilon^{\prime}>0$ and all
$\varepsilon\leq\varepsilon^{\prime}$,

$\sup_{r>t>\kappa_{o}(\varepsilon^{4})}\sup_{\underline{y}^{\prime}\in\mathbb{B}_{r}^{\star}\backslash\mathbb{B}_{t}^{\star}}\\{\log\Psi^{{\rm
sym}}\big{(}\mathfrak{s}_{\partial
v}^{t,r}(\underline{y}^{\prime})\big{)}\\}\leq\widehat{\Phi}+\varepsilon^{3}.$
(5.56)

In view of (5.37), we have for any $t>\kappa_{o}(\varepsilon^{4})$ that if
${\sf B}_{v}(r)\cong{\sf T}_{d}(r)$ and $Y_{v,n}(\underline{y})\geq 0$, then

$\displaystyle\Psi^{{\rm vx}}_{t,r,n}$
$\displaystyle(v,\underline{y})={\varphi_{n}^{v}}\Big{[}\Psi^{{\rm
vx}}(\mathfrak{s}_{\partial v}^{t,r}){\bf
1}_{\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n}}|\eta_{v,t}^{r}=\underline{y}\Big{]}+{\varphi_{n}^{v}}\left[\Psi^{{\rm
vx}}(\mathfrak{s}_{\partial v}^{t,r}){\bf
1}_{(\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n})^{c}}\mid\eta_{v,t}^{r}=\underline{y}\right]$
$\displaystyle<e^{\widehat{\Phi}-2{C}\varepsilon}{\varphi_{n}^{v}}\left[\Psi^{{\rm
e},{\rm sym}}(\mathfrak{s}_{\partial v}^{t,r}){\bf
1}_{\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n}}\mid\eta_{v,t}^{r}=\underline{y}\right]+e^{\widehat{\Phi}+\varepsilon^{3}}{\varphi_{n}^{v}}\left[\Psi^{{\rm
e},{\rm sym}}(\mathfrak{s}_{\partial v}^{t,r}){\bf
1}_{(\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n})^{c}}\mid\eta_{v,t}^{r}=\underline{y}\right]$
$\displaystyle=e^{\widehat{\Phi}-2{C}\varepsilon}\left(\Psi^{{\rm e},{\rm
sym}}_{t,r,n}(v,\underline{y})+(e^{2{C}\varepsilon+\varepsilon^{3}}-1){\varphi_{n}^{v}}\big{[}\Psi^{{\rm
e},{\rm sym}}(\mathfrak{s}_{\partial v}^{t,r}){\bf
1}_{(\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n})^{c}}\mid\eta_{v,t}^{r}=\underline{y}\,\big{]}\right)$
$\displaystyle\leq
e^{\widehat{\Phi}-2{C}\varepsilon}(1+(e^{2{C}\varepsilon+\varepsilon^{3}}-1)(1-c\varepsilon))\Psi^{{\rm
e},{\rm sym}}_{t,r,n}(v,\underline{y})\leq
e^{\widehat{\Phi}-c\varepsilon^{2}}\Psi^{{\rm e},{\rm
sym}}_{t,r,n}(v,\underline{y}),$

where the first inequality uses the definition of
$\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n}$ and the uniform bound (5.56), the
second one holds for $Y_{v,n}(\underline{y})\geq 0$ (see (5.55)), and the last
one applies when $\varepsilon\in(0,\varepsilon^{\prime\prime})$. That is, in
view of (5.49), for any
$\varepsilon\leq\varepsilon^{\prime}\wedge\varepsilon^{\prime\prime}$ if
$r>t>\kappa_{o}(\varepsilon^{4})$ and $\delta\leq c\varepsilon^{2}$, then

$\\{{\sf B}_{v}(r)\cong{\sf T}_{d}(r)\text{ and }Y_{v,n}(\underline{y})\geq
0\\}\Longrightarrow\\{\log\Psi_{t,r,n}^{{\rm
sym}}(v,\underline{y})<\widehat{\Phi}-\delta\\}\,.$ (5.57)

As $Y_{v,n}\leq 1/c$, clearly ${\varphi_{n}^{v}}(Y_{v,n}\geq 0)\geq
c{\varphi_{n}^{v}}[Y_{v,n}]$. Further, by (5.55), (5.37) and (5.54),

$\displaystyle{\varphi_{n}^{v}}[Y_{v,n}(\eta_{v,t}^{r})]$
$\displaystyle={\varphi_{n}^{v}}\big{[}\Psi^{{\rm e},{\rm
sym}}(\mathfrak{s}_{\partial v}^{t,r}){\bf
1}_{\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n}}\big{]}-c\varepsilon{\varphi_{n}^{v}}\big{[}\Psi^{{\rm
e},{\rm sym}}(\mathfrak{s}_{\partial
v}^{t,r})\big{]}\geq{\varphi_{n}^{v}}({\mathscr{B}^{t,r}_{2{C}\varepsilon,v,n}})-\varepsilon\,.$
(5.58)

Combining (5.50), (5.57) and (5.58), we find that

$\varphi_{n}(\mathcal{A}_{\delta,v,n}^{t,r})\geq C^{-1}{\bf 1}_{\\{{\sf
B}_{v}(r)\cong{\sf T}_{d}(r)\\}}{\varphi_{n}^{v}}(Y_{v,n}(\eta_{v,t}^{r})\geq
0)\geq\frac{c}{C}{\bf 1}_{\\{{\sf B}_{v}(r)\cong{\sf
T}_{d}(r)\\}}\big{(}{\varphi_{n}^{v}}(\mathscr{B}_{2{C}\varepsilon,v,n}^{t,r})-\varepsilon\big{)}\,.$

The preceding inequality, together with (5.53), completes our proof (with
$\delta_{o}:=\delta\wedge{c\varepsilon/(4C)}$).

(b). We establish the existence of $\mathcal{V}_{n}$ via a probabilistic
construction based on the auxiliary iid, uniform over $[n]$, samples
$\\{w_{i},i\leq 2\zeta n\\}$. Specifically, set
$\mathscr{U}:=\mathscr{U}_{0}\backslash\bar{\mathscr{U}}$ where

$\mathscr{U}_{0}:=\\{i:{\sf B}_{w_{i}}(3r)\cong{\sf
T}_{d}(3r)\\},\quad\bar{\mathscr{U}}:=\\{i\in\mathscr{U}_{0}:\exists j\neq
i\text{ such that }{\rm dist}_{{\sf G}_{n}}(w_{i},w_{j})\leq 3r\\}\,.$

Observe that $n^{-1}\mathbb{E}|\mathscr{U}_{0}|\to 2\zeta$ since ${\sf
G}_{n}\stackrel{{\scriptstyle loc}}{{\longrightarrow}}{\sf T}_{d}$. Moreover,

$\mathbb{E}|\bar{\mathscr{U}}|\leq\sum_{i\neq j}\mathbb{P}(w_{j}\in{\sf
B}_{w_{i}}(3r)\cong{\sf T}_{d}(3r))\leq 2\zeta|{\sf
T}_{d}(3r)|\,\mathbb{E}|\mathscr{U}_{0}|\,.$

Thus, $\mathbb{E}|\mathscr{U}|\geq(1-\sqrt{\zeta})\mathbb{E}|\mathscr{U}_{0}|$
for any $\zeta\leq\zeta_{o}(r)$, in which case

$\liminf_{n\to\infty}\mathbb{P}\big{(}|\mathscr{U}|\geq 2\zeta
n(1-3\sqrt{\zeta})\big{)}\geq\liminf_{n\to\infty}\frac{\mathbb{E}|\mathscr{U}|}{2\zeta
n}-(1-3\sqrt{\zeta})\geq 2\sqrt{\zeta}.$ (5.59)

With $q_{n}(v):=\varphi_{n}(\mathcal{A}_{\delta_{o},v,n}^{t,r})$, recall from
part (a) that the iid $[0,1]$-valued $\\{q_{n}(w_{i})\\}$ satisfy

$\liminf_{n\to\infty}\mathbb{E}[q_{n}(w_{i})]=\liminf_{n\to\infty}\frac{1}{n}\sum_{v=1}^{n}q_{n}(v)\geq
4\delta_{o}\,.$

Hence, by Chebychev’s inequality

$\lim_{n\to\infty}\mathbb{P}(\frac{1}{2\zeta n}\sum_{i=1}^{2\zeta
n}q_{n}(w_{i})\geq 3\delta_{o})=1$ (5.60)

and in view of (5.59), at any $n\geq n^{\prime}(\zeta)$ we have with
probability $\sqrt{\zeta}$ that $|\mathscr{U}|\geq 2\zeta n(1-3\sqrt{\zeta})$
and the event in (5.60) holds. For
$3\sqrt{\zeta}\leq\delta_{o}\leq\frac{1}{2}$, it then follows that

$\frac{1}{|\mathscr{U}|}\sum_{i\in\mathscr{U}}q_{n}(w_{i})\geq\frac{1}{2\zeta
n}\sum_{i\in\mathscr{U}}q_{n}(w_{i})\geq\frac{1}{2\zeta n}\sum_{i=1}^{2\zeta
n}q_{n}(w_{i})-3\sqrt{\zeta}\geq 3\delta_{o}-3\sqrt{\zeta}\geq 2\delta_{o}\,.$

Take for $\mathscr{V}_{n}$ the $[\zeta n]$ elements of largest $q_{n}(\cdot)$
in $\\{w_{i}:i\in\mathscr{U}\\}$. Then, (5.51) holds and moreover

$\frac{1}{|\mathscr{V}_{n}|}\sum_{v\in\mathscr{V}_{n}}\varphi_{n}(\mathcal{A}_{\delta_{o},v,n}^{t,r})\geq
2\delta_{o}\,,$

from which (5.52) immediately follows. ∎

###### Proof of Theorem 1.13.

Fix $(\beta,B)\in{\sf R}_{c}$, $B>0$, $d\in 2\mathbb{N}$, $d\geq 3$ and ${\sf
G}_{n}\stackrel{{\scriptstyle loc}}{{\longrightarrow}}{\sf T}_{d}$, further
assuming wlog that $\varphi_{n}=\varphi_{{\sf G}_{n}}^{\beta,B}$ converges
locally weakly to $\mathfrak{m}_{\operatorname{b}}$. In view of [6, Theorem
1], postulating further that
$\mathfrak{m}_{\operatorname{b}}(\mathcal{Q}_{\star}\backslash\mathcal{Q}_{\rm
m})>0$, it suffices to produce ${\sf G}^{\prime}_{n}\stackrel{{\scriptstyle
loc}}{{\longrightarrow}}{\sf T}_{d}$ such that
${\varlimsup_{n}}\;\Phi^{\prime}_{n}(\beta,B)>\widehat{\Phi}$. To this end,
for $\delta_{o}>0$ of Lemma 5.11 and $c>0$ from (5.54), set

$\varepsilon_{\star}:=-\frac{1}{3}\log\Big{(}1-(1-e^{-\delta_{o}})\frac{\delta_{o}c^{2}}{3}\Big{)}.$
(5.61)

Similarly to the derivation of (5.56), with $\log\Psi^{\rm vx}(\cdot)$ and
$\log\Psi^{{\rm e},{\rm sym}}(\cdot)$ Lipschitz on $[0,1]^{d}$, taking
$\kappa_{o}=\kappa_{o}(\varepsilon_{\star}^{2})$ as in Remark 5.5 (and
shrinking $\delta_{o}$ as needed for $\varepsilon_{\star}$ to be small
enough), guarantees by (5.27), (5.37), and (5.49) that for $r>t>\kappa_{o}$
and any finite graph ${\sf G}_{n}$,

$\sup_{\underline{y}\in\mathbb{B}_{r}\backslash\mathbb{B}_{t}}\log\Psi_{t,r,n}^{{\rm
sym}}(v,\underline{y})\leq\widehat{\Phi}+\varepsilon_{\star}$ (5.62)

provided $v\in V({\sf G}_{n})$ is such that ${\sf B}_{v}(r)\cong{\sf
T}_{d}(r)$. Hereafter we further take $r>r^{\prime}(t)$ and $t>t^{\prime}$ as
needed in Lemma 5.11, possibly increasing $r^{\prime}(t)$ till Lemma 5.9 holds
with $\varepsilon=\frac{\varepsilon_{\star}}{4}$. Then, for $\mathscr{V}_{n}$
of Lemma 5.11(b), set the disjoint union
$W_{n}:=\cup_{v\in\mathscr{V}_{n}}({\sf B}_{v}(r)\backslash{\sf B}_{v}(t))$
and for $\mathcal{A}_{v}=\mathcal{A}_{\delta_{o},v,n}^{t,r}$ of (5.50) let

$\mathcal{Y}_{n}:=\Big{\\{}\underline{y}\in\\{0,1\\}^{E(W_{n})}:\frac{1}{|\mathscr{V}_{n}|}\sum_{v\in\mathscr{V}_{n}}{\bf
1}_{\mathcal{A}_{v}}\geq\delta_{o}\Big{\\}}.$ (5.63)

Our graph decomposition starts at ${\sf G}_{n}^{(0)}={\sf G}_{n}$ and
iteratively has ${\sf G}_{n}^{(\ell)}:={\sf
G}_{n}^{(\ell-1),\pi_{\ell}}(v_{\ell})$ for

$\displaystyle v_{\ell}$
$\displaystyle:=\operatorname*{argmax}_{v\in\mathscr{V}_{n}\backslash\\{v_{1},\ldots,v_{\ell-1}\\}}\Big{\\{}\sum_{\underline{y}\in\mathcal{Y}_{n}}\mathcal{Z}_{{\sf
G}_{n}^{(\ell-1)},W_{n}}(\underline{y}){\bf
1}_{\mathcal{A}_{v}}(\underline{y})\Big{\\}},$ $\displaystyle\pi_{\ell}$
$\displaystyle:=\operatorname*{argmax}_{\pi\in\mathfrak{S}_{d}}\Big{\\{}\sum_{\underline{y}\in\mathcal{Y}_{n}}\mathcal{Z}_{{\sf
G}_{n}^{(\ell-1),\pi}(v_{\ell}),W_{n}}(\underline{y})\Big{\\}}.$

We will show in the sequel that for any
$\ell\leq\ell_{n}:=\frac{\delta_{o}}{3}\zeta n$,

$\Delta\mathcal{Z}_{\ell}:=\log\Big{(}\sum_{\underline{y}\in\mathcal{Y}_{n}}\mathcal{Z}_{{\sf
G}_{n}^{(\ell-1)},W_{n}}(\underline{y})\Big{)}-\log\Big{(}\sum_{\underline{y}\in\mathcal{Y}_{n}}\mathcal{Z}_{{\sf
G}_{n}^{(\ell)},W_{n}}(\underline{y})\Big{)}\leq\widehat{\Phi}-\varepsilon_{\star}.$
(5.64)

We claim that ${\sf G}_{n}^{\prime}:={\sf G}_{n}^{(\ell_{n})}$ then have too
large free densities $\Phi^{\prime}_{n}(\beta,B)$. Indeed, by (5.34)-(5.35),
and the fact that the Potts measure is the spin marginal of the Edwards-Sokal
measure, we have that $Z_{{\sf
G}_{n}}(\beta,B)=\sum_{\underline{y}}\mathcal{Z}_{{\sf
G}_{n},W_{n}}(\underline{y})$. Thus, by the lhs of (5.52) and (5.64),

$\displaystyle\log Z_{{\sf
G}^{\prime}_{n}}(\beta,B)\geq\log\bigg{(}\sum_{\underline{y}\in\mathcal{Y}_{n}}\mathcal{Z}_{{\sf
G}^{(\ell_{n})}_{n},W_{n}}(\underline{y})\bigg{)}$
$\displaystyle=\log\bigg{(}\sum_{\underline{y}\in\mathcal{Y}_{n}}\mathcal{Z}_{{\sf
G}_{n},W_{n}}(\underline{y})\bigg{)}-\sum_{\ell=1}^{\ell_{n}}\Delta\mathcal{Z}_{\ell}$
$\displaystyle\geq\log Z_{{\sf
G}_{n}}(\beta,B)+\log\delta_{o}-(\widehat{\Phi}-\varepsilon_{\star})\ell_{n}\,.$

With ${\sf G}_{n}\stackrel{{\scriptstyle loc}}{{\longrightarrow}}{\sf T}_{d}$
we have from [6, Theorem 1] that $\Phi_{n}(\beta,B)\to\widehat{\Phi}$, so by
the preceding,

$\liminf_{n\to\infty}\frac{1}{n}\log Z_{{\sf
G}^{\prime}_{n}}(\beta,B)\geq\widehat{\Phi}(1-\frac{\delta_{o}}{3}\zeta)+\frac{\delta_{o}}{3}\zeta\varepsilon_{\star}.$
(5.65)

Note that $|V({\sf
G}^{\prime}_{n})|=n-\ell_{n}=(1-\frac{\delta_{o}}{3}\zeta)n$ and the degree in
${\sf G}^{\prime}_{n}$ of any $w\in V({\sf G}^{\prime}_{n})$ matches its
degree in ${\sf G}_{n}$. Moreover, the sets ${\sf B}_{v_{\ell}}(1)$ on which
${\sf G}^{\prime}_{n}$ and ${\sf G}_{n}$ differ, are $2r$-separated on ${\sf
G}^{\prime}_{n}$ and if $(u,u^{\prime})\in E({\sf G}^{\prime}_{n})$ within
such a changed set ${\sf B}_{v}(1)$, then ${\rm dist}_{{\sf
G}_{n}}(u,u^{\prime})\leq 2$. Consequently, any cycle of length $k$ in ${\sf
G}_{n}^{\prime}$ must have been obtained from a cycle of length at most
$k/(1+(2r)^{-1})$ in ${\sf G}_{n}$. The last two observations together imply
that ${\sf G}_{n}^{\prime}\stackrel{{\scriptstyle loc}}{{\longrightarrow}}{\sf
T}_{d}$ as $n\to\infty$. Hence, by [6, Theorem 1]

$\limsup_{n\to\infty}\frac{1}{n}\log Z_{{\sf
G}_{n}^{\prime}}(\beta,B)\leq\widehat{\Phi}(1-\frac{\delta_{o}}{3}\zeta),$

in contradiction to (5.65).

We thus complete the proof of Theorem 1.13 upon establishing (5.64). To this
end, we shall apply Lemma 5.9 with $\widehat{{\sf G}}={\sf G}_{n}$, ${\sf
G}={\sf G}^{(\ell-1)}_{n}$, and $o=v_{\ell}\in\mathcal{V}_{n}$ (so ${\sf
B}_{o}(r)\cong{\sf T}_{d}(r)$ in view of (5.51)), where as $\inf_{i<\ell}{\rm
dist}_{{\sf G}_{n}}(v_{\ell},v_{i})>3r$ (see (5.51)), indeed ${\sf
G}^{(\ell-1)}_{n}$ coincides with ${\sf G}_{n}$ on its $r$-ball around
$v_{\ell}$. Further, Remark 5.10 allows us to do so for $\widehat{W}=W_{n}$
(which by construction contains the relevant annuli ${\sf
B}_{v_{\ell}}(r)\backslash{\sf B}_{v_{\ell}}(t)$ while excluding ${\sf
B}_{v_{\ell}}(t)$). Specifically, using the shorthand

$\widehat{\mathcal{Z}}_{\ell}(\underline{y}):=\mathcal{Z}_{{\sf
G}_{n}^{(\ell)},W_{n}}(\underline{y}),\quad\widehat{\mathcal{Z}}^{-}_{\ell}(\underline{y}):=\mathcal{Z}_{{\sf
G}_{n}^{(\ell-1),-}(v_{\ell}),W_{n}}(\underline{y}),\quad\widehat{\mathcal{Z}}^{\pi}_{\ell}(\underline{y}):=\mathcal{Z}_{{\sf
G}_{n}^{(\ell-1),\pi}(v_{\ell}),W_{n}}(\underline{y}),$

and $\Gamma_{\ell}:=\sum_{\underline{y}\in\mathcal{Y}_{n}}\Psi^{{\rm e},{\rm
sym}}_{t,r,n}(v_{\ell},\underline{y})\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})$,
we have by Lemma 5.9 and our choice of $r$ that,

$\displaystyle\Delta\mathcal{Z}_{\ell}$
$\displaystyle=\log\bigg{(}\sum_{\underline{y}\in\mathcal{Y}_{n}}\frac{\widehat{\mathcal{Z}}_{\ell-1}(\underline{y})}{\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})}\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})\bigg{)}-\log\bigg{(}\max_{\pi}\sum_{\underline{y}\in\mathcal{Y}_{n}}\frac{\widehat{\mathcal{Z}}_{\ell}^{\pi}(\underline{y})}{\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})}\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})\bigg{)}$
$\displaystyle\leq\varepsilon_{\star}+\log\bigg{(}\sum_{\underline{y}\in\mathcal{Y}_{n}}\Psi^{{\rm
vx}}_{t,r,n}(v_{\ell},\underline{y})\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})\bigg{)}-\log\bigg{(}\max_{\pi}\sum_{\underline{y}\in\mathcal{Y}_{n}}\Psi^{{\rm
e},\pi}_{t,r,n}(v_{\ell},\underline{y})\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})\bigg{)}$
$\displaystyle\leq\varepsilon_{\star}+\log\bigg{(}\sum_{\underline{y}\in\mathcal{Y}_{n}}\Psi^{{\rm
vx}}_{t,r,n}(v_{\ell},\underline{y})\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})\bigg{)}-\log\Gamma_{\ell}\,.$
(5.66)

To simplify the rhs of (5.2), note that by our choice of $v_{\ell}$,

$\displaystyle\sum_{\underline{y}\in\mathcal{Y}_{n}}\widehat{\mathcal{Z}}_{\ell-1}(\underline{y}){\bf
1}_{\mathcal{A}_{v_{\ell}}}(\underline{y})$
$\displaystyle\geq\frac{1}{|\mathscr{V}_{n}|}\sum_{v\in\mathscr{V}_{n}\backslash\\{v_{1},\ldots,v_{\ell-1}\\}}\sum_{\underline{y}\in\mathcal{Y}_{n}}\widehat{\mathcal{Z}}_{\ell-1}(\underline{y}){\bf
1}_{\mathcal{A}_{v}}(\underline{y})$
$\displaystyle\geq\frac{1}{|\mathscr{V}_{n}|}\sum_{\underline{y}\in\mathcal{Y}_{n}}\widehat{\mathcal{Z}}_{\ell-1}(\underline{y})\bigg{(}\sum_{v\in\mathscr{V}_{n}}{\bf
1}_{\mathcal{A}_{v}}(\underline{y})-(\ell-1)\bigg{)}\geq\frac{2\delta_{o}}{3}\sum_{\underline{y}\in\mathcal{Y}_{n}}\widehat{\mathcal{Z}}_{\ell-1}(\underline{y}),$
(5.67)

with the right-most inequality due to (5.63) and the size of
$\mathscr{V}_{n}$. Moreover, by Lemma 5.9 and (5.54)

$\displaystyle\sum_{\underline{y}\in\mathcal{Y}_{n}}\Psi^{{\rm e},{\rm
sym}}_{t,r,n}(v_{\ell},\underline{y})\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y}){\bf
1}_{\mathcal{A}_{v_{\ell}}}(\underline{y})\geq\frac{3c}{4}\sum_{\underline{y}\in\mathcal{Y}_{n}}\widehat{\mathcal{Z}}_{\ell-1}(\underline{y}){\bf
1}_{\mathcal{A}_{v_{\ell}}}(\underline{y})\geq\frac{\delta_{o}c}{2}\sum_{\underline{y}\in\mathcal{Y}_{n}}\widehat{\mathcal{Z}}_{\ell-1}(\underline{y})\geq\frac{\delta_{o}c^{2}}{3}\Gamma_{\ell}$

where in the penultimate step we have used (5.67). Substituting the latter
inequality in (5.2), upon recalling the definitions of
$\mathcal{A}_{v_{\ell}}$ and $\Psi^{{\rm sym}}_{t,r,n}(\cdot,\underline{y})$,
and using (5.62) we get

$\displaystyle e^{\Delta\mathcal{Z}_{\ell}-\varepsilon_{\star}}$
$\displaystyle\leq\frac{1}{\Gamma_{\ell}}\sum_{\underline{y}\in\mathcal{Y}_{n}}\exp\big{(}\widehat{\Phi}+\varepsilon_{\star}-\delta_{o}{\bf
1}_{\mathcal{A}_{v_{\ell}}}(\underline{y})\big{)}\Psi^{{\rm e},{\rm
sym}}_{t,r,n}(v_{\ell},\underline{y})\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y})$
$\displaystyle\leq
e^{\widehat{\Phi}+\varepsilon_{\star}}\big{(}1-\frac{1-e^{-\delta_{o}}}{\Gamma_{\ell}}\sum_{\underline{y}\in\mathcal{Y}_{n}}\Psi^{{\rm
e},{\rm
sym}}_{t,r,n}(v_{\ell},\underline{y})\widehat{\mathcal{Z}}_{\ell}^{-}(\underline{y}){\bf
1}_{\mathcal{A}_{v_{\ell}}}(\underline{y})\big{)}$ $\displaystyle\leq
e^{\widehat{\Phi}+\varepsilon_{\star}}\big{(}1-(1-e^{-\delta_{o}})\frac{\delta_{o}c^{2}}{3}\big{)}=e^{\widehat{\Phi}-2\varepsilon_{\star}}$

(with the last equality by our choice (5.61) of $\varepsilon_{\star}$). This
is (5.64), which thus completes our proof. ∎

##  Appendix A Infinite volume Potts measures and Bethe fixed points

We prove here various properties of the phase diagram associated to the Potts
measures on ${\sf T}_{d}$. To this end, for any $(i,j)\in E({\sf T}_{d})$ let
${\sf T}_{i\to j}$ be the tree rooted at $i$ obtained upon deleting $(i,j)$
from $E({\sf T}_{d})$ and with ${\sf T}_{i\to j}(t)$ denoting for
$t\in\mathbb{N}$ the ball ${\sf B}_{i}(t)$ in ${\sf T}_{i\to j}$, we first
prove Lemma 3.3.

###### Proof of Lemma 3.3.

Summing in (3.6) over $\sigma_{j}$ results with (3.7). The proof of (3.6) in
case $q=2$ is well known (e.g. see [5, Lemma 4.1]), and the extension for
$q\geq 3$ (which we now give), follows a similar scheme. With
$\mu^{\beta,B}_{\ddagger}$ a translation invariant measure, it suffices to
consider $i=o$ and $j=1$ a specific neighbor of the root $o$ of ${\sf T}_{d}$.
Fixing $t\in\mathbb{N}$, as there are no cycles in ${\sf T}_{d}(t)$, we have
under $\mu_{\ddagger,t}^{\beta,B}$ of Definition 1.2, that for some
$\widehat{\nu}_{t}^{1},\widehat{\nu}_{t}^{o}\in\mathcal{P}([q])$

$\mu_{\ddagger,t}^{\beta,B}(\sigma_{o},\sigma_{1})\propto
e^{\beta\delta_{\sigma_{o},\sigma_{1}}}\widehat{\nu}^{1}_{t}(\sigma_{1})\widehat{\nu}^{o}_{t}(\sigma_{o})\,,\qquad\forall\sigma_{o},\sigma_{1}\in[q].$

Further, both $T_{o\rightarrow 1}$ and $T_{1\rightarrow o}$ are infinite
$(d-1)$-ary trees (i.e. each vertex has $(d-1)$ children). Hence, by induction
we deduce from Definitions 1.2 and 1.3, that $\widehat{\nu}_{t}^{o}$ and
$\widehat{\nu}_{t}^{1}$ are the probability measures obtained by running the
${\sf BP}$ recursion $t$ and $(t-1)$ times, respectively, starting from
$\nu_{\ddagger}$ (i.e. starting at the uniform measure if
$\ddagger=\operatorname{f}$ and at Dirac at $1$ when $\ddagger=1$). By
definition the limit of these recursions is $\nu_{\ddagger}^{\beta,B}$ and
(3.6) follows. ∎

######  Remark A.1.

By a similar reasoning one finds also that for any $\beta,B\geq 0$ and
$\ddagger\in\\{\operatorname{f},1\\}$, all finite dimensional marginals of
$\mu_{\ddagger,t}^{\beta,B}$ converge as $t\to\infty$. Furthermore, it can be
checked that these marginals are consistent, thus implying the existence of
$\mu_{\ddagger}^{\beta,B}$ for $\ddagger\in\\{\operatorname{f},1\\}$. The
existence of $\mu_{i}^{\beta,0}$ for all $i\in[q]$ is proved similarly (now
starting the ${\sf BP}$ recursion at Dirac at $i$).

Next we prove that $\nu_{\operatorname{f}}$ and $\nu_{1}$, when viewed as
functions of $\beta$ and $B$, are continuously differentiable except on
$\partial{\sf R}_{\neq}^{+}$ and $\partial{\sf R}_{\neq}^{\operatorname{f}}$,
respectively.

###### Proof of Lemma 3.2.

(i) Since $\nu_{1}^{\beta,B}$ is a fixed point of the ${\sf BP}$ recursion of
(1.5), starting from the probability measure on $[q]$ that is Dirac at $1$,
necessarily $\nu_{1}^{\beta,B}(\sigma)=\nu_{1}^{\beta,B}(\sigma^{\prime})$ for
all $\sigma,\sigma^{\prime}\in[q]\backslash\\{1\\}$ and as
$\nu_{1}^{\beta,B}\in\mathcal{P}([q])$, it suffices to show that
$(\beta,B)\mapsto\nu_{1}^{\beta,B}(1)$ is continuously differentiable at any
fixed $(\beta_{0},B_{0})\notin\partial{\sf R}_{\neq}^{\operatorname{f}}$.
Further, after some algebra we deduce from (1.5) that

$r_{1}(\beta,B):=\log\frac{\nu_{1}^{\beta,B}(1)}{\nu_{1}^{\beta,B}(2)}=\log\frac{(1-q)\nu_{1}^{\beta,B}(1)}{1-\nu_{1}^{\beta,B}(1)}\,,$

satisfies the fixed point equation

$F(r;\beta,B)=r,$ (A.1)

where

$F(r;\beta,B):=B+(d-1)\log\left(\frac{e^{\beta+r}+q-1}{e^{r}+e^{\beta}+q-2}\right).$

Using this representation it suffices to show that $(\beta,B)\mapsto
r_{1}(\beta,B)$ is continuously differentiable, and this follows by an
application of the implicit function theorem, once we have verified that

$\frac{\partial}{\partial r}F(r;\beta,B)\neq 1,\qquad\text{ for
}(r,\beta,B)=(r_{1}(\beta_{0},B_{0}),\beta_{0},B_{0}).$ (A.2)

To obtain (A.2) we borrow results from [7]. From the proof of [7, Lemma 4.6]
it follows that for $d\geq 3$ there is some $\beta_{-}\in(0,\infty)$ such that
for $\beta<\beta_{-}$ there does not exist any solution to the equation

$\frac{\partial}{\partial r}F(r;\beta,B)=\frac{\partial}{\partial
r}F(r;\beta,0)=1.$ (A.3)

In contrast, for $\beta\geq\beta_{-}$ there exist solutions
$\rho_{-}(\beta)\leq\rho_{+}(\beta)$ of the equation (A.3), with equality if
and only if $\beta=\beta_{-}$. Thus, to complete the proof of (A.2) we need to
show that if $(\beta_{0},B_{0})\in{\sf R}_{\neq}\backslash\partial{\sf
R}_{\neq}^{\operatorname{f}}$ (and therefore $\beta_{0}>\beta_{-}$, see [7,
Lemma 4.6]) then $r_{1}(\beta_{0},B_{0})\neq\rho_{\pm}(\beta_{0})$.

If possible, let us assume that $r_{1}(\beta_{0},B_{0})=\rho_{+}(\beta_{0})$.
Denoting

$B_{\pm}(\beta):=\rho_{\mp}(\beta)-F(\rho_{\mp}(\beta);\beta,0),$ (A.4)

as $r_{1}(\beta,B)$ is a fixed point of the equation (A.1), we note that
$B_{0}=B_{-}(\beta_{0})$. From the proof of [7, Theorem 1.11(a)] we have that
the map $B\mapsto\beta_{\operatorname{f}}(B)$ is the inverse of the map
$\beta\mapsto B_{-}(\beta)$. So, $\beta_{0}=\beta_{\operatorname{f}}(B_{0})$
implying that $(\beta_{0},B_{0})\in\partial{\sf R}_{\neq}^{\operatorname{f}}$.
As $(\beta_{0},B_{0})\notin\partial{\sf R}^{\operatorname{f}}_{\neq}$ we
arrive at a contradiction.

To rule out the other possibility that
$r_{1}(\beta_{0},B_{0})=\rho_{-}(\beta_{0})$ we again proceed by
contradiction. As $\beta_{0}>\beta_{-}$ we have that
$\rho_{-}(\beta_{0})<\rho_{+}(\beta_{0})$. Now note

$B_{+}(\beta)-B_{-}(\beta)=\int_{\rho_{-}(\beta)}^{\rho_{+}(\beta)}\left[\frac{\partial}{\partial
r}F(r;\beta,0)-1\right]dr.$

It can be checked that $\frac{\partial^{2}}{\partial r^{2}}F(r;\beta,0)$ is
negative for sufficiently large $r$ and has a single change of sign. As
$\rho_{\pm}(\beta)$ are the solutions of (A.3) one therefore have that
$\frac{\partial}{\partial r}F(r;\beta,0)>1$ for
$r\in(\rho_{-}(\beta),\rho_{+}(\beta))$. Hence, it follows from above that
$B_{-}(\beta_{0})<B_{+}(\beta_{0})$. Thus

$F(\rho_{+}(\beta_{0});\beta_{0},B_{0})=\rho_{+}(\beta_{0})+B_{0}-B_{-}(\beta_{0})=\rho_{+}(\beta_{0})+B_{+}(\beta_{0})-B_{-}(\beta_{0})>\rho_{+}(\beta_{0}),$
(A.5)

where in the penultimate step we have used the fact that the assumption
$r_{1}(\beta_{0},B_{0})=\rho_{-}(\beta_{0})$ implies that
$B_{0}=B_{+}(\beta_{0})$.

Noting that $\lim_{r\to\infty}F(r;\beta_{0},B_{0})<\infty$, (A.5) implies that
there exists some $r_{\star}\in(\rho_{+}(\beta_{0}),\infty)$ such that
$r_{\star}=F(r_{\star};\beta_{0},B_{0})$. However, this contradicts the fact
that $r_{1}(\beta_{0},B_{0})$ is the largest root of that fixed point equation
(A.1).

(ii) As in part (i), it now suffices to show that $(\beta,B)\mapsto
r_{\operatorname{f}}(\beta,B)$ is continuously differentiable for
$(\beta,B)\in{\sf R}_{\neq}\backslash\partial{\sf R}_{\neq}^{+}$, where
$r_{\operatorname{f}}(\beta,B):=\log\frac{(1-q)\nu_{\operatorname{f}}^{\beta,B}(1)}{1-\nu_{\operatorname{f}}^{\beta,B}(1)}$
for $\nu_{\operatorname{f}}^{\beta,B}(\cdot)$ of Definition 1.3. To this end,
fixing $(\beta_{0},B_{0})\in{\sf R}_{\neq}\backslash\partial{\sf
R}_{\neq}^{+}$ we only need to show that (A.3) fails at
$(r,\beta,B)=(r_{\operatorname{f}}(\beta_{0},B_{0}),\beta_{0},B_{0})$ which
analogously to part (i) be a direct consequence of having that
$r_{\operatorname{f}}(\beta_{0},B_{0})\neq\rho_{\pm}(\beta_{0})$.

Proceeding to do this task, the same argument as in part (i) shows that
$r_{\operatorname{f}}(\beta_{0},B_{0})\neq\rho_{-}(\beta_{0})$, but a slightly
different argument is needed for ruling out the other choice. Specifically, we
claim that $r_{\operatorname{f}}(\beta_{0},B_{0})=\rho_{+}(\beta_{0})$ would
imply $\rho_{-}(\beta_{0})>0$. To see this, from the definition of
$B_{\pm}(\beta)$ we find that $B_{0}=B_{-}(\beta_{0})$. From the proof of [7,
Theorem 1.11] we also have that the map $B\mapsto\beta_{\operatorname{f}}(B)$
is the inverse of the map $\beta\mapsto B_{-}(\beta)$. Moreover, the
assumption $(\beta_{0},B_{0})\in{\sf R}_{\neq}\backslash\partial{\sf
R}_{\neq}^{+}$ implies that $B_{0}\in[0,B_{+})$. Therefore,

$\beta_{0}=\beta_{\operatorname{f}}(B_{0})<\beta_{+}(B_{0})\leq\beta_{+}(0)=:\beta_{+},$
(A.6)

where the first inequality follows from the fact that
$\beta_{\operatorname{f}}(B)<\beta_{+}(B)$ for $B\in[0,B_{+})$ and the second
inequality follows from the fact that the map $\beta_{+}(\cdot)$ is decreasing
in $B$. Using [7, Eqn. (4.10)] we thus deduce that $\rho_{-}(\beta_{0})>0$
whenever $r_{\operatorname{f}}(\beta_{0},B_{0})=\rho_{+}(\beta_{0})$. Now,
similarly to part (i), if both $\rho_{-}(\beta_{0})>0$ and
$r_{\operatorname{f}}(\beta_{0},B_{0})=\rho_{+}(\beta_{0})$, then there exists
a solution $r_{\star}\in[0,r_{\operatorname{f}}(\beta_{0},B_{0}))$ of (A.1).
From Definition 1.3 we know that $r_{\operatorname{f}}(\beta_{0},B_{0})$ is
the smallest nonnegative solution of (A.1), so having arrived at a
contradiction, the proof of the lemma is thus complete. ∎

###### Proof of Proposition 1.5.

From [7, Theorem 1.11(a)] we have the existence of the smooth curves
$\beta_{\operatorname{f}}(B)$ and $\beta_{+}(B)$ with the desired properties.
It also follows from there that for $(\beta,B)\in[0,\infty)^{2}\backslash{\sf
R}_{\neq}$ one has $\nu_{\operatorname{f}}^{\beta,B}=\nu_{1}^{\beta,B}$. This
together with Lemma 3.3 implies that all two dimensional marginals of
$\mu_{\operatorname{f}}^{\beta,B}$ and $\mu_{1}^{\beta,B}$ coincide. Further,
as described in Remark A.1, any finite dimensional marginal of
$\mu_{\operatorname{f}}^{\beta,B}$ then coincides with that of
$\mu_{1}^{\beta,B}$, thereby yielding our claim (1.8) from which (1.9) also
follows (since at $B=0$ the Potts measures are invariant with respect to
permutations on $[q]$). Since $r_{\operatorname{f}}^{\beta,B}<r_{1}^{\beta,B}$
for $(\beta,B)\in{\sf R}_{\neq}$ our claim that
$\nu_{\operatorname{f}}^{\beta,B}(1)<\nu_{1}^{\beta,B}(1)$ follows from the
fact that map $r\mapsto e^{r}/(e^{r}+q-1)$ is strictly increasing on
$\mathbb{R}$.

Now, for the existence of smooth $B\to\beta_{c}(B)$ of the required
properties, it suffices to verify that

$\displaystyle\frac{\partial}{\partial\beta}\Phi(\nu_{\operatorname{f}}^{\beta,B})<\frac{\partial}{\partial\beta}\Phi(\nu_{1}^{\beta,B}),\qquad$
$\displaystyle\text{ for all }(\beta,B)\in{\rm Int}({\sf R}_{\neq})\,,$ (A.7)
$\displaystyle\Phi(\nu_{\operatorname{f}}^{\beta,B})>\Phi(\nu_{1}^{\beta,B}),\qquad$
$\displaystyle\text{ for }(\beta,B)\in\partial{\sf
R}_{\neq}^{\operatorname{f}},$ (A.8)
$\displaystyle\Phi(\nu_{\operatorname{f}}^{\beta,B})<\Phi(\nu_{1}^{\beta,B}),\qquad$
$\displaystyle\text{ for }(\beta,B)\in\partial{\sf R}_{\neq}^{+}.$ (A.9)

Indeed, (A.7)-(A.9) imply that (1.11) holds for some $B\mapsto\beta_{c}(B)$
such that $\beta_{\operatorname{f}}(B)<\beta_{c}(B)<\beta_{+}(B)$ for
$B\in[0,B_{+})$, with smoothness of $\beta_{c}(\cdot)$ due to the implicit
function theorem (where
$\frac{\partial}{\partial\beta}\Phi(\nu_{\operatorname{f}}^{\beta,B})\neq\frac{\partial}{\partial\beta}\Phi(\nu_{1}^{\beta,B})$
throughout ${\rm Int}({\sf R}_{\neq})$ thanks to (A.7)).

Turning to prove (A.7), we set $\bar{\nu}:=\frac{1-\nu}{q-1}$ and note that
the map

$\nu\mapsto\frac{\bar{\nu}(1+\nu-\bar{\nu})}{\bar{\nu}^{2}+(q-1)\nu^{2}}$

is strictly decreasing on $[\frac{1}{q},1)$ (having positive numerator and
denominator, whose derivatives are strictly negative and positive,
respectively, on the interval $(\frac{1}{q},1)$). Using Lemma 3.3 and the fact
that $\nu_{\ddagger}(i)=\nu_{\ddagger}(2)$ for
$\ddagger\in\\{\operatorname{f},1\\}$ and all $i\in[q]\backslash\\{1\\}$, we
also have at any $i\in\partial o$,

$\mu_{\ddagger}^{\beta,B}(\sigma_{o}=\sigma_{i})=\frac{e^{\beta}\left[\nu_{\ddagger}(1)^{2}+(q-1)\nu_{\ddagger}(2)^{2}\right]}{e^{\beta}\left[\nu_{\ddagger}(1)^{2}+(q-1)\nu_{\ddagger}(2)^{2}\right]+2(q-1)\nu_{\ddagger}(1)\nu_{\ddagger}(2)+(q-1)(q-2)\nu_{\ddagger}(2)^{2}}\,.$

With $\frac{1}{q}\leq\nu_{\operatorname{f}}^{\beta,B}(1)<\nu_{1}^{\beta,B}(1)$
at any $(\beta,B)\in{\sf R}_{\neq}$, combining the last two observations
results with

$\sum_{i\in\partial
o}\mu_{\operatorname{f}}^{\beta,B}(\sigma_{o}=\sigma_{i})<\sum_{i\in\partial
o}\mu_{1}^{\beta,B}(\sigma_{o}=\sigma_{i}),\qquad\qquad\forall\,(\beta,B)\in{\sf
R}_{\neq}$

from which (A.7) follows upon using (3.10).

Moving next to the proof of (A.8), for any
$\beta\in[\beta_{-},\beta_{\operatorname{f}}]$ let

$\Psi_{-}(\beta):=\Phi^{\beta,B_{-}(\beta)}(r_{1}(\beta,B_{-}(\beta)))-\Phi^{\beta,B_{-}(\beta)}(r_{\operatorname{f}}(\beta,B_{-}(\beta)))\,,$

with the convention that
$\Phi^{\beta,B}(r)=\Phi^{\beta,B}(r(\nu)):=\Phi^{\beta,B}(\nu)$ for
$r(\nu):=\log(\nu(1)/\nu(2))$ and any $\nu\in\mathcal{P}([q])$ such that
$\nu(2)=\cdots=\nu(q)$. Fix any $(\beta_{0},B_{0})\notin\partial{\sf
R}_{\neq}^{+}$. By Lemma 3.2(ii), proceeding as in the steps leading to
(3.10), and using the definition of $r(\nu)$ we obtain that

$\frac{\partial}{\partial\beta}\Phi^{\beta_{0},B_{0}}(r_{\operatorname{f}}(\beta_{0},B_{0}))=h_{1}(r_{\operatorname{f}}(\beta_{0},B_{0});\beta_{0})\text{
and }\frac{\partial}{\partial
B}\Phi^{\beta_{0},B_{0}}(r_{\operatorname{f}}(\beta_{0},B_{0}))=h_{2}(r_{\operatorname{f}}(\beta_{0},B_{0});\beta_{0}),$
(A.10)

where

$h_{1}(r;\beta):=\frac{d}{2}\cdot\frac{e^{\beta}(e^{2r}+q-1)}{e^{r}(e^{\beta+r}+q-1)+(q-1)(e^{r}+e^{\beta}+q-2)}$

and

$h_{2}(r;\beta):=\frac{e^{r}(e^{\beta+r}+q-1)}{e^{r}(e^{\beta+r}+q-1)+(q-1)(e^{r}+e^{\beta}+q-2)}.$

By (A.4) we see that $\rho_{+}(\beta_{0})$ is a fixed point of (A.1) for
$(\beta,B)=(\beta_{0},B_{-}(\beta_{0}))$. Recall that $\rho_{+}(\beta)$ is the
largest solution of (A.3) and since $r_{1}(\beta,B)$ is the largest solution
of (A.1), we deduce that
$\rho_{+}(\beta_{0})=r_{1}(\beta_{0},B_{-}(\beta_{0}))$ for
$\beta_{0}\in[\beta_{-},\beta_{\operatorname{f}}]$. From (A.3) it can be
further checked that $\rho_{+}(\beta)$ is the $\log$ of the largest solution
of a quadratic equation with coefficients smooth in $\beta$ that does not
admit a double root for $\beta>\beta_{-}$. Therefore, the map
$\beta\mapsto\rho_{+}(\beta)$ is differentiable on
$(\beta_{-},\beta_{\operatorname{f}}]$. Hence, repeating a computation as in
the steps leading to (3.10), we find that (A.10) continues to hold when
$B_{0}$ is replaced by $B_{-}(\beta_{0})$ and
$r_{\operatorname{f}}(\beta_{0},B_{0})$ is replaced by
$r_{1}(\beta_{0},B_{-}(\beta_{0}))$. Moreover, by (A.4) we have that
$\partial_{\beta}B_{-}(\beta_{0})=-\partial_{\beta}F(\rho_{+}(\beta_{0});\beta_{0},0)$
for $\beta_{0}\in(\beta_{-},\beta_{\operatorname{f}}]$. Thus, using the chain
rule of differentiation, we find that

$\Psi_{-}^{\prime}(\beta_{0})=\Delta h_{1}(\beta_{0})-\Delta
h_{2}(\beta_{0})\cdot\frac{\partial}{\partial\beta}F(\rho_{+}(\beta_{0});\beta_{0},0)\\\
=\int_{r_{\operatorname{f}}(\beta_{0},B_{-}(\beta_{0}))}^{r_{1}(\beta_{0},B_{-}(\beta_{0}))}\left(\frac{\partial}{\partial
r}h_{1}(r;\beta_{0})-\frac{\partial}{\partial
r}h_{2}(r,\beta_{0})\cdot\frac{\partial}{\partial\beta}F(\rho_{+}(\beta_{0});\beta_{0},0)\right)dr,$
(A.11)

where

$\Delta
h_{i}(\beta_{0}):=h_{i}(r_{1}(\beta_{0},B_{-}(\beta_{0}));\beta_{0})-h_{i}(r_{\operatorname{f}}(\beta_{0},B_{-}(\beta_{0}));\beta_{0}),\quad\text{
for }i=1,2.$

Next denoting

$g_{\beta_{0}}(r):=\frac{\frac{\partial}{\partial
r}h_{1}(r;\beta_{0})}{\frac{\partial}{\partial
r}h_{2}(r;\beta_{0})}=d\cdot\frac{e^{2r}+(q-2)e^{r}-(q-1)}{e^{2r}+2(e^{\beta_{0}}+q-2)e^{r}+(q-1)(1+e^{-\beta_{0}}(q-2))}$

we find that
$\operatorname{sgn}(g^{\prime}_{\beta_{0}}(r))=\operatorname{sgn}(2e^{\beta_{0}}+q-2)\cdot\operatorname{sgn}(Q_{\beta_{0}}(e^{r}))$,
where
$Q_{\beta_{0}}(t):=t^{2}+2(q-1)e^{-\beta_{0}}+(q-1)(1+(q-2)e^{-\beta_{0}})$.
Since the roots of $Q_{\beta_{0}}$ are either negative or complex conjugates
of each other we deduce that $g^{\prime}_{\beta_{0}}(r)>0$ for all
$r\in\mathbb{R}$. This entails that

$g_{\beta_{0}}(r)<g_{\beta_{0}}(r_{1}(\beta_{0},B_{-}(\beta_{0})))=\frac{\partial}{\partial\beta}F(\rho_{+}(\beta_{0});\beta_{0},0)\text{
for }r\in[0,r_{1}(\beta_{0},B_{-}(\beta_{0}))),$ (A.12)

where to obtain the right most equality we use the fact that
$\rho_{+}(\beta_{0})=r_{1}(\beta_{0},B_{-}(\beta_{0}))$ satisfies (A.3). So,
noticing that $\frac{\partial}{\partial r}h_{2}(\cdot;\beta_{0})$ is strictly
positive, by (A.12), we find that the integrand in the rhs of (A.11) is
strictly negative for
$r\in[r_{\operatorname{f}}(\beta_{0},B_{-}(\beta_{0}),r_{1}(\beta_{0},B_{-}(\beta_{0})))$,
and consequently that $\Psi_{-}^{\prime}(\beta_{0})<0$ for
$\beta_{0}\in(\beta_{-},\beta_{\operatorname{f}}]$. By definition of the non-
uniqueness regime, necessarily $\Psi_{-}(\beta_{-})=0$ and (A.8) follows.

The proof of (A.9) is similar. Indeed, noting that
$r_{\operatorname{f}}(\beta,B)=\rho_{-}(\beta)$ for any
$(\beta,B)\in\partial{\sf R}_{\neq}^{+}$, denoting
$\Psi_{+}(\beta):=\Phi^{\beta,B_{+}(\beta)}(r_{1}(\beta,B_{+}(\beta)))-\Phi^{\beta,B_{+}(\beta)}(r_{\operatorname{f}}(\beta,B_{+}(\beta)))$
for any $\beta\in[\beta_{-},\beta_{+}]$, and proceeding as in the steps
leading to (A.11) we find that (A.11) continues to hold for
$\Psi_{+}^{\prime}(\beta_{0})$ at any $\beta_{0}\in(\beta_{-},\beta_{+}]$,
provided we replace there $\rho_{+}(\beta_{0})$ and $B_{-}(\beta_{0})$ by
$\rho_{-}(\beta_{0})$ and $B_{+}(\beta_{0})$, respectively. Moreover, a same
reasoning as in (A.12) yields that

$g_{\beta_{0}}(r)>g_{\beta_{0}}(r_{\operatorname{f}}(\beta_{0},B_{+}(\beta_{0})))=\frac{\partial}{\partial\beta}F(\rho_{-}(\beta_{0});\beta_{0},0)\text{
for }r\in(r_{\operatorname{f}}(\beta_{0},B_{+}(\beta_{0})),\infty).$

Repeating now the rest of the argument in the proof of (A.8) we obtain (A.9)
and thereby complete the proof of the proposition. ∎

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