\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[a4paper,margin=32mm]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{enumitem}
\usepackage{microtype}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,calc,decorations.pathreplacing,patterns}
\usepackage{url}
\usepackage[numbers,sort&compress]{natbib}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\hypersetup{
pdftitle={Spectral convergence for critical FK-Ising planar maps},
pdfauthor={OpenAI},
pdfsubject={Spectral convergence to Liouville Brownian motion},
pdfkeywords={FK-Ising planar maps, Liouville quantum gravity, spectral convergence}
}
\urlstyle{same}
\Urlmuskip=0mu plus 1mu
\setlist{topsep=4pt,itemsep=2pt,parsep=0pt}
\setlength{\emergencystretch}{2em}
\numberwithin{equation}{section}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\R}{\mathbb R}
\newcommand{\N}{\mathbb N}
\newcommand{\C}{\mathbb C}
\newcommand{\Sph}{\mathbb S^2}
\newcommand{\PP}{\mathbb P}
\newcommand{\EE}{\mathbb E}
\newcommand{\ind}{\mathbf 1}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\ip}[2]{\langle #1,#2\rangle}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\abs}[1]{\lvert #1\rvert}
\newcommand{\HS}{\mathrm{HS}}
\DeclareMathOperator{\EL}{EL}
\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\osc}{osc}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\Law}{Law}
\DeclareMathOperator{\Id}{Id}
\title{Spectral convergence for critical FK--Ising planar maps}
\author{OpenAI}
\date{October 2026}
\begin{document}
\maketitle
\begin{abstract}
The Fortuin--Kasteleyn random-cluster model at cluster weight two is the
Ising model. On a critical random planar map, decorated by the model,
there is a natural walk with total attempt rate one and stationary corner
measure. We prove that this walk, viewed in accelerated time, converges
jointly to Liouville Brownian motion on the ordinary unit-area
$\sqrt 3$-Liouville quantum gravity sphere. Three objects converge
together: the metric-measure space with deterministic diameter
normalization, all ordered eigenvalues with multiplicities accelerated by
the number of edges, and the operator heat trace at strictly positive
times. The clock is exactly the number of edges. The proof combines a
local electrical argument that fixes the energy as the unit conformal
Dirichlet integral, a compactness estimate at all remaining scales from
the companion address analysis, and an abstract compactness lemma
upgrading four-point Green kernel limits to Hilbert--Schmidt convergence
of the centered inverse operators.
\end{abstract}
\section*{How to read this document}
\paragraph{The picture, informally.}
Build a random surface by gluing polygons randomly along edges. You get a
random sphere with bumps. Paint each edge ``closed'' or ``open'' by a
random rule that favors clusters; this is the Fortuin--Kasteleyn (FK)
model. At cluster weight two it is the Ising model, and at self-dual
temperature it is critical. On the surface, put a walker. It waits for
an exponential time of mean one, then picks one of the half-edges at its
corner uniformly and moves to the vertex across it. Loops keep it in
place.
When the surface is very large, the picture simplifies. The surface,
suitably scaled to unit diameter, converges to a random fractal measure
space: Liouville quantum gravity (LQG) at $\gamma = \sqrt 3$. The random
walk converges to a diffusion on that space, Liouville Brownian motion
(LBM). This paper proves the limit at the level of eigenvalues and heat
traces: if we accelerate the walk by the number of edges, its eigenvalues
converge, in order and with multiplicities, to the eigenvalues of the
LBM generator on the LQG sphere of area one, and its heat traces at
strictly positive time converge. The three convergences -- geometry,
spectrum, heat -- happen jointly, on the same limit surface.
\paragraph{What makes the proof hard.}
Two difficulties organize the argument.
\begin{enumerate}
\item \emph{Identifying the energy.} The discrete walk has an electrical
energy $\mathcal E_n^0$, the sum of squared voltage differences across
the edges. In the limit this energy must be a scalar multiple $k$ of
the round conformal Dirichlet integral. Nothing forces $k=1$ a priori,
and every spectral conclusion would be off by $k$. The paper fixes
$k=1$ by a primal--dual fixed-point argument: the discrete model has a
duality $(M,A)\leftrightarrow(M^*,A^*)$ preserving its law, and the
complementary mixed-boundary conductances of a square multiply to one.
This is the Dykhne fixed point in a discrete, rigorously justified form.
\item \emph{Compactness at every scale.} Knowing the energy on each fixed
annulus does not identify the spectrum. One also needs a uniform
quadratic estimate at every small scale, controlling functions of
bounded energy whose mass can concentrate. The companion
metric-measure paper's address estimate gives this for dyadic block
averages of the walk listed in exploration order.
Section~\ref{sec:compactness} turns it into an all-scale approximation
theorem.
\end{enumerate}
\paragraph{The three ingredients of the spectral limit.}
Once the energy is identified as the unit conformal Dirichlet integral,
the four-point voltage of a unit current converges to the round Green
kernel. That identifies the kernel of the limiting inverse operator.
The all-scale approximation theorem supplies compactness of
bounded-energy functions. An abstract compactness lemma then upgrades
four-point convergence of kernels to Hilbert--Schmidt convergence of the
centered inverses, from which eigenvalues and heat traces follow. The
joint metric-measure statement is patched on by the two companion papers'
geometric convergence.
\paragraph{Notation.}
Throughout, $\Sph$ is the target sphere; $d_n$ is graph distance on
$M_n$ using every map edge, loops included; $\mu_n$ is the corner
measure; $\lambda_{n,j}$ are the eigenvalues of $-L_n$ on $L^2(\mu_n)$
in increasing order; $H_n$ is the discrete heat trace. On the continuum
side, $\mu_h$ is the quantum area, $D_h$ is the LQG metric, $A_h$ is the
LBM generator, $\Lambda_j(h)$ are its eigenvalues, $H_h$ is its heat
trace. The unhalved electrical energy carries a superscript $0$; the
halved form appearing in the continuum generator does not.
\tableofcontents
\newpage
\section{Introduction}\label{sec:intro}
Random planar maps are connected finite multigraphs embedded in the oriented
two-sphere, taken up to orientation-preserving homeomorphisms fixing a
root. They are a natural discrete model of a random surface. On a given
map, the Ising model can be put into its Fortuin--Kasteleyn random-cluster
form, which thinks of the model as a law on an edge subset $A$ weighted by
the number of clusters of $A$ and of its complementary dual $A^*$. At
cluster weight two it is the Ising model at its critical self-dual
temperature. The joint object -- a random map together with a
random-cluster state -- is the natural discrete random surface on which the
Ising model of statistical mechanics is critical.
Sheffield's \emph{inventory accumulation} \cite{Sheffield2016Inventory}
encodes an FK-decorated finite map by a random word over a small alphabet,
and uses it to identify the infinite-volume contour limit with a correlated
Brownian motion. The refinements
\cite{GwynneMaoSun2019ConeTimes,GwynneSun2017Local,GwynneSun2015Finite}
identify contour and loop data at finer scales. The mating-of-trees
theory \cite{DuplantierMillerSheffield2021Mating,MillerSheffield2019SphereMating}
identifies the relevant continuum Brownian sewing with a Liouville
quantum gravity (LQG) surface. At the FK-Ising parameter $q=2$, the
resulting LQG parameter is $\gamma=\sqrt 3$. The two companion papers
\cite{C} and \cite{P} turn this encoding into the conformal invariance
and metric--measure convergence of the FK-Ising map itself. We use their
geometric and probabilistic conclusions as inputs; the electrical and
spectral conclusions are the new content of the present paper.
On the continuum side, Liouville quantum gravity is built by exponentiating
a planar Gaussian free field. Its quantum area $\mu_h$ is a Gaussian
multiplicative chaos \cite{DuplantierSheffield2011KPZ}; its intrinsic metric
$D_h$ was shown to be tight, to exist, to be unique up to a deterministic
constant, and to be conformally covariant in
\cite{DingDubedatDunlapFalconet2020Tightness,GwynneMiller2021Metric,GwynneMiller2021ConformalMetric}.
Liouville Brownian motion (LBM) is the diffusion obtained by time-changing
round Brownian motion by $\mu_h$. It was constructed in
\cite{Berestycki2015Diffusion,GarbanRhodesVargas2016LBM} and its Dirichlet
form, heat kernel, and spectral theory were developed further in
\cite{GarbanRhodesVargas2014Dirichlet,AndresKajino2016Heat,MaillardRhodesVargasZeitouni2016Heat}.
Berestycki and Gwynne \cite{BerestyckiGwynne2022MatedCRTLBM} proved a
functional limit theorem for walks on mated-CRT maps that also goes to LBM;
our discrete object is not the same but the spirit is close.
The question in the title of the section is this. On the critical
FK-Ising map, the natural vertex walk has total attempt rate one and
stationary corner measure. Does it converge to LBM on the LQG sphere, and
does the spectral content -- eigenvalues, multiplicities, and heat trace --
converge as well? We answer yes, and in fact the three convergences
(geometry, spectrum, heat) hold jointly, on the same limit surface. The
clock that makes the heat traces match is exactly $n$, the number of edges
of the discrete map. What pins $n$ is the normalization of the electrical
energy and a conductivity identity proved below.
\subsection{The model and the theorem}\label{sec:model-and-theorem}
For $n\ge1$, let $\mathcal F_n$ be the finite set of pairs $(M,A)$, where
$M$ is a connected planar multigraph with $n$ edges embedded in the oriented
sphere and rooted at an oriented edge, and $A\subseteq E(M)$.
Equivalence is by orientation-preserving homeomorphisms preserving the root
and $A$. Loops and multiple edges are allowed. Write $M^*$ for the dual
map and $A^*=\{e^*:e\notin A\}$. Let $k(A)$ denote the number of connected
components of the spanning subgraph $(V(M),A)$, including isolated vertices,
and put
\[
\ell(M,A)=k(A)+k(A^*)-1=2k(A)+|A|-|V(M)|.
\]
The second identity is Euler's formula applied to the embedded spanning
subgraph. Define the critical FK-Ising probability
\begin{equation}\label{eq:fk-law}
\PP\bigl((M_n,A_n)=(M,A)\bigr)
=\frac{2^{\ell(M,A)/2}}{Z_n},
\qquad
Z_n=\sum_{(M,A)\in\mathcal F_n}2^{\ell(M,A)/2}.
\end{equation}
Let $V_n=V(M_n)$, let $d_n$ be graph distance using every original map
edge with length one, and set
\begin{equation}\label{eq:corner-measure}
\mu_n=\frac{1}{2n}\sum_{v\in V_n}\deg(v)\delta_v.
\end{equation}
A loop contributes two to the degree, so $\sum_{v}\deg(v)=2n$ and
$\mu_n$ is a probability measure. It is the law of the vertex where a
uniformly chosen oriented half-edge starts; we call it the \emph{corner
measure}.
\emph{The vertex walk.} The walk waits for an exponential time of mean one,
then chooses a uniformly random incident half-edge $a$, and moves to its
opposite endpoint $\operatorname{opp}(a)$. A chosen loop leaves the
position unchanged. Its generator on vertex functions is
\begin{equation}\label{eq:walk-generator}
(L_nf)(v)=\frac{1}{\deg(v)}
\sum_{\substack{a\ \mathrm{half\hbox{-}edge}\\ a\ \mathrm{incident\ to}\ v}}
\bigl(f(\operatorname{opp}(a))-f(v)\bigr).
\end{equation}
The operator $-L_n$ is nonnegative and self-adjoint on $L^2(\mu_n)$.
List its eigenvalues with multiplicity in increasing order,
\[
0=\lambda_{n,0}\le\lambda_{n,1}\le\cdots
\le\lambda_{n,|V_n|-1},
\]
and put $\lambda_{n,j}=+\infty$ for $j\ge|V_n|$. For $t>0$, define the
\emph{discrete heat trace}
\begin{equation}\label{eq:discrete-heat-trace}
H_n(t)=\Tr e^{ntL_n}
=\sum_{j=0}^{|V_n|-1}e^{-nt\lambda_{n,j}}.
\end{equation}
This is the operator trace, counted once per mode.
\emph{Continuum target.} Let $(\Sph,h)$ be the ordinary unit-area
$\gamma$-Liouville quantum gravity sphere at $\gamma=\sqrt3$: the
ordinary quantum-sphere measure of Duplantier, Miller, and Sheffield,
disintegrated at quantum area one, with its original marks forgotten
\cite[Definition~4.21(ii)]{DuplantierMillerSheffield2021Mating}.
Write $\mu_h$ for its quantum area probability measure and $D_h$ for its
intrinsic LQG metric in one fixed deterministic normalization. We work
in a conformal coordinate of the round sphere. On smooth functions define
\begin{equation}\label{eq:continuum-energy}
\mathcal E_h(f,g)=\frac12\int_{\Sph}
\ip{\nabla f}{\nabla g}_{\mathrm{round}}\dd\mathrm{vol}_{\mathrm{round}}.
\end{equation}
The proof (Section~\ref{sec:continuum}) will establish that the closure of
this form on $L^2(\mu_h)$ is the LBM Dirichlet form and has compact
resolvent. Let $A_h$ be the nonnegative self-adjoint operator of this form
and
\[
0=\Lambda_0(h)\le\Lambda_1(h)\le\Lambda_2(h)\le\cdots
\]
its eigenvalues with multiplicity. Define the \emph{continuum heat trace}
\begin{equation}\label{eq:continuum-heat-trace}
H_h(t)=\Tr e^{-tA_h}=\sum_{j\ge0}e^{-t\Lambda_j(h)},\qquad t>0.
\end{equation}
The factor $1/2$ in \eqref{eq:continuum-energy} fixes the Brownian clock.
The two-dimensional Dirichlet integral is conformally invariant, so pushing
$\mu_h$ forward under a change of conformal coordinate gives a unitarily
equivalent operator.
\emph{Distance normalization.} For a finite real random variable $Y$,
let $\mathfrak m(Y)$ be the unique $r$ with
$\EE[\arctan(Y-r)]=0$. Put
\[
\Delta_n=\max\{1,\diam(V_n,d_n)\},\qquad
\Delta=\diam(\Sph,D_h),
\]
and define the deterministic scale
\begin{equation}\label{eq:distance-normalization}
a_n=\exp\{\mathfrak m(\log\Delta)-\mathfrak m(\log\Delta_n)\}.
\end{equation}
Both centers use unconditional laws, so $a_n$ is deterministic. Lemma
\ref{lem:space-normalization} below verifies that $a_n$ agrees
asymptotically with the spatial normalization of the companion geometric
limit.
\begin{theorem}[Joint Liouville spectral limit]\label{thm:main}
The form \eqref{eq:continuum-energy} is closable on $L^2(\mu_h)$.
Its closure has compact resolvent, has the constants as its one-dimensional
kernel, and satisfies $H_h(t)<\infty$ for every $t>0$.
As $n\to\infty$ through all positive integers,
\begin{equation}\label{eq:main-limit}
\Bigl(
(V_n,a_nd_n,\mu_n),\;
(n\lambda_{n,j})_{j\ge0},\;
H_n
\Bigr)
\ \Longrightarrow\
\Bigl(
(\Sph,D_h,\mu_h),\;
(\Lambda_j(h))_{j\ge0},\;
H_h
\Bigr).
\end{equation}
The first coordinate has the Gromov--Hausdorff--Prokhorov topology on
isometry classes of compact metric probability spaces. The second has the
product topology on $[0,\infty]^{\N_0}$. The third has
$C_{\mathrm{loc}}((0,\infty))$, uniform convergence on each
$[\delta,T]$ with $0<\delta<T<\infty$. All three limiting coordinates
are constructed from the same quantum sphere $h$.
\end{theorem}
The clock is exactly $n$. The reason is the normalization of the
electrical energy together with a conductivity identity proved below.
Formula~\eqref{eq:distance-normalization} fixes only the spatial
convention; we use no power-law formula for $a_n$.
\emph{Inputs.} The proof uses the canonical conformal companion \cite{C}
and the metric-measure companion \cite{P}, the latter in its revision of
October 3, 2026. Their encoding, protected local laws, quantitative
address estimates, finite-view estimates, and geometric limits are
probabilistic inputs. The electrical and spectral conclusions of
Theorem~\ref{thm:main} are the new content of this paper. In the
companion titles, \emph{subcritical} refers to the LQG range $\gamma<2$;
the FK model itself is at its self-dual critical point.
\subsection{Proof strategy}\label{sec:strategy}
Two different questions must be resolved, and the paper is organized around
them. The first is to identify the macroscopic electrical behavior of the
map. This determines the candidate limiting operator and its coefficient
$k$. The second is to control functions whose variation is concentrated in
regions of vanishing corner mass. Only the second question makes it
possible to pass from the identification of a candidate operator to its
actual spectral content: eigenvalues, multiplicities, heat trace.
\paragraph{Energy.}
Protected incidence records of the canonical companion \cite{C} retain a
finite graph, together with its ports and confinement, in a form that
depends measurably but locally on the underlying word and its limit. They
also retain \emph{numerical} quantities of this graph, such as the effective
resistance between two ports, provided the readout is a measurable function
of that finite record and its exact parameter instruction.
Section~\ref{sec:locality} extends the companion's conditional-law
construction from combinatorial outputs to such numerical outputs. Its
conclusion is a conditional kernel: given the continuum limit up to that
chart, the discrete readout is distributed according to a specified
probability law that is a measurable function of the local continuum data.
The companion's aligned-annulus argument then rules out collapse of the
raw edge extremal length on any fixed annulus. The mechanism is a
primal--dual obstruction. A hypothetical collapse would produce a cheap
crossing of the annulus at almost every position and in almost every
direction, for both networks simultaneously. Transverse primal and dual
crossings must cross on a paired edge. Cauchy--Schwarz for the product of
their expected edge traffics then gives $1 \le \sqrt{C_+C_-}\rho_{j_k}$
with $\rho_{j_k} \to 0$, a contradiction. The resulting
Theorem~\ref{thm:network-bounds} turns the barrier into fixed-region
electrical estimates: cutoffs of bounded energy across every fixed collar,
surrounding circuit flows of bounded squared traffic, path laws of bounded
squared traffic inside any fixed corridor. The bounds are scale-invariant
for one fixed relative template. They are therefore usable at every radius,
though each individual fixed test only holds after a threshold depending on
the test and on the sample.
The fixed-annulus estimates are then upgraded by a relaxation over
locally uniformly converging functions
(Definition~\ref{def:relaxation}). The infimum of discrete energies over
vertex functions that $\varepsilon$-approximate a fixed continuous $f$ in
an inner open set converges to a local quadratic energy $\mathcal R(f,U)$.
Its matrix density $A_{ij}$ is read from the energy measures of the
coordinate functions $x_i, x_j$. Cutoff gluing makes $\mathcal R$ a local
quadratic integral; circuit bounds bound it above and below by the ordinary
Sobolev energy. The matrix density is a local germ, in the sense of the
companion's conditional-independence and field-germ lemma
\cite[Lemma~8.3]{C}: it is a function of arbitrarily small field data at a
point. Ordinary germs are deterministic a.e.\ by that lemma. Determinism
plus isotropy under rotation forces $A = kI$ for a scalar $k>0$ that may
depend on the extraction and on the choice primal or dual.
The primal and dual scalars agree by the exact FK duality, because the
two rooted uniformizations are the same or are complex conjugates of one
another on the canonical sphere, and the Dirichlet integral is preserved
by both. Dykhne's fixed-point idea (see \cite{Dykhne1971}, and the finite
reciprocity in \cite[Remark~2.11]{BinderPechersky2025}) then pins the
common value to one. In a finite planar square with complementary
mixed-boundary conditions, the primal left--right and dual bottom--top
conductances satisfy $C_{\rm p}C_{\rm d} = 1$ exactly
(Lemma~\ref{lem:mixed-duality}). Approximating squares in the discrete
network with $C_m^{\rm p} \to k_{\rm p}$ and $C_m^{\rm d} \to k_{\rm d}$
gives $k_{\rm p}k_{\rm d} = 1$. Since $k_{\rm p}=k_{\rm d}$, the common
value is one, and Theorem~\ref{thm:conductivity} identifies the relaxation
as exactly the unit conformal Dirichlet integral on every ordinary chart.
A local minimization argument then identifies the unit-current voltage
between two distinct macroscopic points with the round Green voltage
$-(2\pi)^{-1}\log\operatorname{ch}$ (Theorem~\ref{thm:four-point}). This
is a four-point statement: the difference of a difference of the discrete
inverse kernel converges to the same combination of the round Green
kernel.
\paragraph{Compactness.}
The address estimates of \cite{P} control the ranks of graphs formed by
exploration return relations. Section~\ref{sec:compactness} derives a
version of the address estimate with one marked base over a wide range of
scales, and combines it with the protected circuit flows and a packing
estimate on reachable supports. These bound the oscillation of a vertex
function on a connected union of circuit boundaries, and also bound how
long the tour can stay inside one circuit disk. Comparing values associated
with parent and child blocks of a dyadic partition, and telescoping through
to singleton cuts, gives the \emph{all-level block approximation}
(Theorem~\ref{thm:block-approximation}).
To make the estimate usable, we list the $2n$ corner occurrences of a
given rooted exploration in exploration order and represent a vertex
function $f$ by its values on the corresponding $2n$ equal intervals of
$[0,1]$. This identifies $L^2(\mu_n)$ isometrically with a subspace of
$L^2([0,1])$. On this representation the all-level block approximation
becomes a sequence of dyadic projection estimates with the same constants.
It supplies linear maps $Q_{n,j}$ of rank $O(2^j)$ with quadratic
approximation error $O(2^{-pj})$, $p=7/10$, in the discrete energy norm
(Lemma~\ref{lem:energy-compactness}). This gives both strong compactness
of bounded-energy functions and a polynomial lower bound on high
eigenvalues, $\nu_{n,i}\ge c_*i^p$.
\paragraph{Spectral limit.}
Section~\ref{sec:spectrum} combines the four-point voltage limit with the
block approximation. The former identifies double differences of the
centered inverse kernel away from coincident points
(Lemma~\ref{lem:inverse-kernel-compactness}, hypothesis (iii)). The
compactness from $p>1/2$ makes the inverse kernels precompact in
Hilbert--Schmidt norm, so a limit exists. The operator argument
(Lemma~\ref{lem:inv-kernel-compact}) then identifies the limit with the
centered inverse of the continuum generator $A_h$ on the quantum area
space. Ordereed eigenvalues and heat traces follow abstractly: the
spectral theorem gives $\kappa_{n,i}\to\kappa_i$ hence
$n\lambda_{n,i}\to\Lambda_i(h)$, and the polynomial eigenvalue bound
gives uniform $C_{\mathrm{loc}}((0,\infty))$ convergence of the heat
traces. The metric-measure coupling to the same limit sphere is patched
onto the same extraction using the companion geometric convergence and
the diameter normalization of \eqref{eq:distance-normalization}.
The final spectral argument is an abstract compactness principle, related
to Kuwae and Shioya's compact spectral convergence theory
\cite{KuwaeShioya2003}: it uses finite-rank energy approximations and
four-point kernel limits, and does not depend on the specific FK encoding.
The FK-specific content of the paper is the electrical identification
(Sections~\ref{sec:locality}--\ref{sec:conductivity}) and the block
approximation at all remaining scales (Section~\ref{sec:compactness}).
\medskip
\noindent\emph{Structure of the rest of the paper.}
Section~\ref{sec:model-geometry} records the exact relation between the
walk and the electrical network, and fixes the geometry of the common
surface used throughout. Section~\ref{sec:locality} extends the
conditional-law machinery to numerical observations. Section~\ref{sec:network}
proves the fixed-region electrical estimates. Section~\ref{sec:conductivity}
proves unit conductivity for the relaxation. Section~\ref{sec:green}
proves four-point convergence of voltages. Section~\ref{sec:continuum}
constructs the continuum form and its compact inverse. Section~\ref{sec:compactness}
proves the all-level block approximation. Section~\ref{sec:spectrum}
finishes the proof of Theorem~\ref{thm:main}.
\section{The discrete energy and the common geometric limit}\label{sec:model-geometry}
This section has three jobs. First, it records the exact relation between
the walk and the electrical network: how the graph Laplacian and the
unhalved edge energy control the walk. Second, it fixes a geometric
surface on which every vertex, edge, and face of the FK map can be drawn,
together with its dual. Third, it states the joint geometric convergence
from the two companion papers, in exactly the form we consume later on.
Every later argument refers to vertices, edges, and faces drawn in this
one coordinate, so the bookkeeping established here is essential.
\subsection{The raw electrical energy}\label{sec:raw-energy}
For a finite map $M$, choose an orientation of each unoriented edge only
for writing differences, and put
\[
d_e f=f(e^+)-f(e^-),\qquad
\mathcal E_M^0(f,g)=\sum_{e\in E(M)} d_e f\, d_e g.
\]
The value is independent of the chosen orientations. Each parallel edge
is a separate summand, and a loop has zero difference. Write
$\mathcal E_n^0=\mathcal E_{M_n}^0$. For a set $F$ of edges,
$\mathcal E_n^0(f;F)=\sum_{e\in F}(d_e f)^2$.
\begin{lemma}[Energy normalization]\label{lem:normalization}
Let $Q_n$ be the unit-conductance graph Laplacian of $M_n$, with loop
terms equal to zero, and set $B_n=-2nL_n$. On $L^2(\mu_n)$,
\[
B_n=\operatorname{diag}(\mu_n)^{-1}Q_n,\qquad
\ip{f}{B_n g}_{L^2(\mu_n)}=\mathcal E_n^0(f,g).
\]
The kernel of $B_n$ consists exactly of the constant functions. In
particular, its ordered eigenvalues are
\(\nu_{n,j}=2n\lambda_{n,j}\).
\end{lemma}
\begin{proof}
For a nonloop edge with endpoints $u,v$, the two incident half-edge
contributions to $-\ip{f}{L_n g}_{L^2(\mu_n)}$ are
\[
\frac1{2n}
\bigl(f(u)(g(u)-g(v))+f(v)(g(v)-g(u))\bigr)
=\frac{d_e f\,d_e g}{2n}.
\]
The two half-edges of a loop contribute zero. Summing over edges gives
both operator identities. A function of zero energy agrees across every
nonloop edge; since the map is connected, it is constant.
\end{proof}
The superscript \(0\) will always indicate this unhalved electrical
energy. The continuum energy $\mathcal E_h^0=2\mathcal E_h$ has operator
$B_h=2A_h$. Accordingly, conductivity one for the raw energies gives the
clock in Theorem~\ref{thm:main}: $B_n/2=-nL_n$ converges to $A_h$.
\subsection{Flags, corner occurrences, and exploration order}\label{sec:flags}
The point of the flag construction is to place the primal map, the dual
map, the corner measure, and the conformal coordinate on the same
topological sphere. An incidence $(v,e,F)$ between a vertex, a primal
edge, and a face of the embedded map is the natural local chart in which
primal and dual edges cross transversally.
We use the inventory encoding of the exact FK law in
\cite[Proposition~3.1 and Lemma~3.2]{C}. At \(q=2\), its flexible-order
parameter is
\[
b_{\mathrm F}=\frac{\sqrt2}{2+\sqrt2}=\sqrt2-1.
\]
The length-\(2n\) word conditioned to reduce to the empty word produces
precisely the probability law \eqref{eq:fk-law}. Its resolved contours
carry the LQG parameter \(\gamma=\sqrt3\), and the limiting duration-one
surface is the ordinary unit-area quantum sphere
\cite[Theorem~3.4]{C}. The convention \(m=2n\) then gives \(m\) letters
per unit of quantum-area time.
\emph{Corner occurrences.} The finite exploration assigns a primal corner
vertex \(v_n(i)\) to each cut $i\in\{0,\ldots,m-1\}$. These are the \(m\)
corner occurrences, including repetitions of the same vertex.
Consecutive cut vertices agree or are joined by a primal edge. Set
\(v_n(m)=v_n(0)\) only to close the tour; the closing cut receives no
mass. A vertex \(v\) occurs \(\deg(v)\) times, so
\begin{equation}\label{eq:corner-pushforward}
\frac1m\sum_{i=0}^{m-1}\delta_{v_n(i)}=\mu_n.
\end{equation}
In particular, a vertex function is represented isometrically in
$L^2([0,1],\dd t)$ by the function equal to $f(v_n(i))$ on
\([i/m,(i+1)/m)\). This representation is used in the spectral
argument (Section~\ref{sec:spectrum}).
\emph{The flag surface.} For every occurrence of a vertex $v$, an edge
$e$, and an incident face $F$, form an equilateral triangle with
vertices labelled $(v,e,F)$. Glue sides according to the cyclic
incidences of the embedded map. There are $4n$ triangles; repeated
incidences are distinct triangles, even for a loop or a bridge. The
resulting incidence subdivision $X_n$ is a topological sphere with its
piecewise Euclidean conformal structure. Write $m_n$ for normalized
triangle area; each flag has mass $1/(4n)$.
If the interior of each flag is sent to its primal vertex, then
\begin{equation}\label{eq:flag-pushforward}
(\pi_n)_*m_n=\mu_n.
\end{equation}
Indeed, $v$ belongs to $2\deg(v)$ flag occurrences, and this is also
the law of the tail of a uniformly chosen oriented root edge conditional
on the unrooted decorated map. The counting argument is valid in the
presence of automorphisms because we count rooted occurrences before
taking a quotient \cite[Lemma~2.2]{C}.
\emph{The canonical coordinate.} Choose three independent
$m_n$-distributed points. The unique orientation-preserving
uniformization \(\phi_n:X_n\to\Sph\) sending them to $0,1,\infty$ is the
canonical coordinate of \cite[Section~2]{C}. We identify $\Sph$ with the
Riemann sphere and write
\[
z_n(v)=\phi_n(v),\qquad v\in V_n.
\]
The points used to normalize \(\phi_n\) are distinct from exploration
roots and from any auxiliary chart marks used below.
\begin{lemma}[Duality in the flag surface]\label{lem:dual-geometry}
The map \((M,A)\mapsto(M^*,A^*)\), with a consistent directed dual-root
convention, is a bijection of the $n$-edge rooted decorated maps and
preserves their FK-Ising probabilities. The primal and dual flag
surfaces are canonically isometric as unoriented piecewise Euclidean
surfaces. Under this identification, a primal edge and its paired dual
edge meet at their common edge midpoint, with alternating incident rays;
a primal edge meets no other dual edge. The same three physical
flag-area samples may be used in the two canonical uniformizations.
Their comparison is conformal or anticonformal according to the
orientation convention.
\end{lemma}
\begin{proof}
Taking complements across dual edges exchanges $A$ and $A^*$, and
\[
\ell(M,A)=k(A)+k(A^*)-1=\ell(M^*,A^*).
\]
The dual operation is invertible, including its root convention, and
preserves the number of edges. It therefore preserves the normalized
finite law. The flag correspondence is
$(v,e,F)\leftrightarrow(F,e^*,v)$; it preserves equilateral triangles
and their incidence gluings, including repeated occurrences. Drawing a
primal edge along its vertex-to-midpoint sides and a dual edge along its
face-to-midpoint sides proves the crossing assertion in each flag
neighborhood. Conditional on the common metric surface, the same three
points are independent uniform area samples in both marginal
experiments. Genus-zero three-point normalization then gives the last
assertion, allowing complex conjugation if the surface orientation is
reversed.
\end{proof}
\subsection{Continuity of the diameter normalization}\label{sec:diam-cont}
We record one normalization fact used both in
Proposition~\ref{prop:geometry} and in the final metric-measure
convergence of Theorem~\ref{thm:main}.
\begin{lemma}[Continuity of the diameter normalization]\label{lem:space-normalization}
The center \(\mathfrak m\) is well defined, satisfies
\(\mathfrak m(Y+c)=\mathfrak m(Y)+c\), and is continuous under
convergence in distribution of finite real random variables. If
$b_n>0$, $b_n\to0$, and $b_n\Delta_n$ converges in distribution to a
finite strictly positive random variable $\Delta$, then the
normalization \eqref{eq:distance-normalization} satisfies
$a_n/b_n \to 1$.
\end{lemma}
\begin{proof}
Bounded convergence shows that $F_Y(r)=\EE[\arctan(Y-r)]$ is continuous
with limits $\pi/2$ and $-\pi/2$ at the two ends of the real line. It is
strictly decreasing because the integrand is strictly decreasing. The
zero exists and is unique, and the shift identity is a direct
consequence. If $Y_j\to Y$ in distribution, $F_{Y_j}(r)\to F_Y(r)$ for
every fixed $r$. For each $\varepsilon>0$, the two values at
$\mathfrak m(Y)\pm\varepsilon$ have opposite strict signs; eventually
the same is true for $F_{Y_j}$. Its zero lies between these two points,
proving continuity.
Apply this to $Y_n=\log(b_n\Delta_n)$. The continuous mapping theorem
applies because $\Delta>0$. Shift equivariance gives
\[
\log\frac{b_n}{a_n}
=\mathfrak m\bigl(\log(b_n\Delta_n)\bigr)-\mathfrak m(\log\Delta)
\longrightarrow 0.
\]
\end{proof}