==== TABLE OF CONTENTS ==== Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//blocks.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//corner-substitution.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//history.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//ladders.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//main.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//metric.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//references.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//trees.tex Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//words.tex =========================== ==== BEGIN Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//blocks.tex ==== \section{The critical block tree}\label{block:section} We now pass from the word estimates to an exact representation of the finite map law. A \emph{corner} of a nonempty map is a sector between successive darts at a vertex. Thus a map with $m$ edges has $2m$ corners, including two at a vertex carrying a single loop. \subsection{Decomposition at corners} The decomposition of a rooted map into a root block and corner insertions goes back to Tutte~\cite[Section~6]{Tutte1963}. Addario-Berry makes its even-offspring tree representation explicit for uniform maps~\cite[Section~2 and Proposition~3.1]{AddarioBerry2019}; the block-weighted form belongs to Stufler's enriched-tree framework~\cite[preprint Section~6.1.5]{Stufler2020}. We prove the rooted version here to keep the one-link and one-loop conventions, trivial inserts, and exact finite weights explicit. Our nontrivial \emph{blocks} are the rooted one-loop map and the rooted nonempty connected loopless maps with at least two vertices and no separating vertex. The one-link map is included: deleting either of its vertices leaves a single vertex. Write $b_m$ for the total weight $a(B)$ of the blocks with $m$ edges. In particular, \begin{equation}\label{block:one-edge} b_1=2(1+t)>0. \end{equation} \begin{lemma}[Rooted corner substitution]\label{block:decomposition} Every nontrivial rooted planar map is uniquely obtained from a rooted block by inserting a rooted map, possibly trivial, in each of its corners. A nonempty insert is attached at the origin of its root dart. Moreover, $a$ is multiplicative when two maps are joined at one vertex. Consequently, with \[ T(z)=\sum_{n\ge0}a_nz^n,\qquad \Phi(y)=1+\sum_{m\ge1}b_my^m, \] there is an identity of formal power series \begin{equation}\label{block:equation} T(z)=\Phi\big(zT(z)^2\big). \end{equation} \end{lemma} \begin{proof} Suppose first that the root is not a loop. Take the maximal loopless block containing the root edge. It is unique: the union of two connected subgraphs without a separating vertex that share an edge again has no separating vertex. An exterior path joining two different vertices of this block, with interior disjoint from it, could be added without producing a separating vertex. Thus every component outside the block attaches at one block vertex. Each face boundary of a loopless block visits a given vertex at most once. Indeed, if two distinct sectors at a vertex belonged to the same face, a simple curve through that face joining the sectors and closed at the vertex would separate incident edges on its two sides. Connectivity after deleting the vertex rules this out. The assertion also holds for a one-link block. Planarity therefore places each outside component in a unique corner of its attachment vertex. Extra loops at a block vertex lie in such corners as well. Group all parts in each corner into one insert. If the root is a loop, take that loop as the root block. Its two sides are its two corners, and the parts on either side form the two inserts. In both cases a nonempty insert is rooted at the first outgoing dart in its corner sector, in the oriented cyclic order. Conversely, glue an arbitrary rooted insert at each corner, identifying its root origin with the corner vertex and starting its cyclic order with its root dart. This reconstructs the map and its rotations uniquely. Each insertion meets the root block at only one vertex, so it cannot enlarge that block. These constructions are inverse. Rooting also identifies the corners individually. Indeed, an automorphism fixing a root dart and preserving vertex rotations and edge reversal fixes all darts by connectedness. We may thus choose a deterministic order of the $2m$ corners of every rooted block, with no symmetry divisor. For the weight claim, join $M_1$ and $M_2$ at one vertex and write an edge subset as $A_1\cup A_2$. The number of components and the number of vertices are, respectively, \[ k_{M_1}(A_1)+k_{M_2}(A_2)-1,\qquad |V(M_1)|+|V(M_2)|-1. \] The exponent $2k+|A|-|V|-1$ is therefore the sum of the two exponents. Summing independently over $A_1,A_2$ proves multiplicativity. A size-$m$ root block contributes $b_mz^mT(z)^{2m}$; adding the trivial map gives \eqref{block:equation}. \end{proof} \subsection{An endpoint criterion for finite variance} An $m$-edge block supplies $2m$ child slots, so the first two offspring moments will be controlled by the first two derivatives of the block series. The next lemma extracts the required endpoint identities and bounds from the word coefficient estimate. \begin{lemma}[Quadratic inverse criterion]\label{block:critical} Suppose $T$ and $\Phi$ have nonnegative coefficients and constant term $1$, and satisfy \eqref{block:equation}. Let $u>0$ satisfy $T(u)<\infty$, and suppose that, for some $c>0$, \begin{equation}\label{block:lower-assumption} \sum_{n\ge1}na_nu^nx^n\ge c(1-x)^{-1/2} \qquad\text{for all $x<1$ sufficiently close to $1$}. \end{equation} Put $\tau=T(u)$ and $v=u\tau^2$. Then \begin{equation}\label{block:critical-equations} \Phi(v)=\tau,\qquad 2v\Phi'(v)=\Phi(v),\qquad \Phi''(v)<\infty, \end{equation} where endpoint derivatives denote the corresponding nonnegative series. \end{lemma} \begin{proof} Integrating \eqref{block:lower-assumption} gives, for $z0$, sufficiently near $v$ we would then have $f''(y)\le-K$. Integrate first on an interior interval $[y,r]$ and let $r\uparrow v$, using $f'(v-)=0$, to obtain $f'(y)\ge K(v-y)$. A second integration, using $f(v-)=u$, gives \[ u-f(y)\ge\frac K2(v-y)^2 \] there, contradicting \eqref{block:quadratic} when $K>2C$. Thus $\Phi''(v)<\infty$. \end{proof} By Corollary~\ref{word:catalan}, $T(u)<\infty$. Proposition~\ref{ladder:lower} supplies \eqref{block:lower-assumption}, so Lemma~\ref{block:critical} applies. Notice that it does not require analytic continuation of $\Phi$ across $v$. \subsection{The marked branching law} The exponential tilt below is the standard passage from simply generated trees to a conditioned Galton--Watson law; see Janson~\cite[Section~4]{Janson2012}. The preceding lemma supplies the criticality and moment properties for these particular FK weights. Define a distribution supported on even integers by \begin{equation}\label{block:offspring} p_{2m}=\frac{b_mv^m}{\Phi(v)}\quad(m\ge0),\qquad b_0=1,\qquad p_{2m+1}=0. \end{equation} If $\xi$ has this distribution, then \begin{equation}\label{block:moments} \E\xi=1,\qquad \sigma^2:=\operatorname{Var}(\xi) =1+\frac{4v^2\Phi''(v)}{\Phi(v)}\in(0,\infty). \end{equation} Here normalization, criticality, and finiteness follow from \eqref{block:critical-equations}; positivity also follows from $p_0,p_2>0$. Construct a plane Galton--Watson tree with offspring law \eqref{block:offspring}. Independently at each node with $2m>0$ children, assign a rooted block $B$ of size $m$ with probability $a(B)/b_m$. The children occupy the block's ordered corners. A leaf represents a trivial insert. Iterating the corner substitutions recovers a rooted map. We call the origin of a block's root dart its \emph{pole}; the pole of a trivial insert is its attachment vertex. \begin{proposition}[Exact finite representation]\label{block:finite-law} The marked tree conditioned to have $N=2n+1$ nodes produces exactly the map marginal of \eqref{eq:finite-law}. Every such size is attainable. \end{proposition} \begin{proof} A block of size $m$ gives $2m$ children. Thus a map with $n$ edges corresponds to a tree with $N-1=2n$ child slots. The probability factor for a node carrying a prescribed block $B$ is \[ \frac{b_mv^m}{\Phi(v)}\frac{a(B)}{b_m} =\frac{v^ma(B)}{\Phi(v)}. \] For a leaf it is $1/\Phi(v)$. Multiplication over all nodes and Lemma~\ref{block:decomposition} give the unconditioned probability \[ \Phi(v)^{-N}v^na(M). \] At fixed $N=2n+1$ the first two factors are constant, so the conditional map law is proportional to $a(M)$, as required. Since $p_0,p_2>0$, a full binary tree with $n$ internal nodes and all internal blocks chosen as one-link maps has positive probability. This gives an admissible tree for every $n\ge1$. \end{proof} Figure~\ref{fig:corners} illustrates why trivial inserts must be retained as leaves: they record corners even when they add no map edges. \input{figures/corner-substitution} ==== END Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//blocks.tex ==== ==== BEGIN Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//corner-substitution.tex ==== \begin{figure}[tb] \centering \begin{tikzpicture}[>=Stealth,font=\small] \begin{scope}[xshift=-3.5cm,yshift=0.4cm] \coordinate (u) at (0,0); \coordinate (v) at (1.45,0); \coordinate (w) at (2.9,0); \draw[thick,->] (u)--(v); \draw[thick] (v)--(w); \draw[thick] (u) .. controls (-0.85,1.05) and (0.85,1.05) .. (u); \draw[thick] (w) .. controls (2.05,1.05) and (3.75,1.05) .. (w); \foreach \x in {u,v,w}{\fill (\x) circle (1.8pt);} \node[below=4pt] at (u) {$u$}; \node[below=4pt] at (v) {$v$}; \node[below=4pt] at (w) {$w$}; \node at (1.45,-0.95) {$(\deg(u),\deg(v),\deg(w))=(3,2,3)$}; \node at (1.45,1.45) {map: four edges}; \end{scope} \begin{scope}[xshift=2.8cm,yshift=1.55cm, block/.style={circle,draw,fill=black!10,inner sep=2pt,minimum size=15pt}, leaf/.style={circle,draw,inner sep=1.5pt,minimum size=13pt}] \node[block] (r) at (0,0) {$u$}; \node[block] (a) at (-1.15,-0.85) {$u$}; \node[block] (b) at (1.15,-0.85) {$v$}; \node[leaf] (a1) at (-1.55,-1.65) {$u$}; \node[leaf] (a2) at (-0.75,-1.65) {$u$}; \node[leaf] (b1) at (0.7,-1.65) {$v$}; \node[block] (b2) at (1.6,-1.65) {$w$}; \node[leaf] (c1) at (1.2,-2.45) {$w$}; \node[leaf] (c2) at (2.0,-2.45) {$w$}; \foreach \x/\y in {r/a,r/b,a/a1,a/a2,b/b1,b/b2,b2/c1,b2/c2} {\draw (\x)--(\y);} \node[above=2pt] at (r.north) {link}; \node[left=2pt] at (a.west) {loop}; \node[right=2pt] at (b.east) {link}; \node[right=2pt] at (b2.east) {loop}; \node at (0.1,-3.02) {block tree: nine nodes}; \end{scope} \end{tikzpicture} \caption{Corner substitution and pole multiplicities. Shaded nodes carry one-edge blocks; white nodes are trivial inserts. Node labels give their poles in the map. The eight nonroot nodes have pole counts $3,2,3$, exactly the vertex degrees. The arrow marks the root dart.} \label{fig:corners} \end{figure} ==== END Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//corner-substitution.tex ==== ==== BEGIN Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//history.tex ==== \subsection{History and the ingredients of the proof} The random-cluster representation of Fortuin and Kasteleyn connects percolation and Potts-type spin systems through a weight on edge subsets~\cite{FortuinKasteleyn1972}. On a random planar map, the map itself is also sampled, so the cluster weight changes the underlying geometry. Sheffield's inventory model encodes this joint randomness by two types of burgers, two fixed order types, and flexible orders that take the freshest available burger~\cite{Sheffield2016}. Its parameter $p=\sqrt q/(\sqrt q+2)$ places $q>4$ exactly in the regime $p>1/2$. Sheffield's unconditioned walk theorem identifies the transition at $q=4$: above it, the limiting inventory fluctuations are one-dimensional~\cite[Theorem~2.5]{Sheffield2016}. The finite-volume CRT prediction appears in Sheffield's appendix~\cite{Sheffield2016}. Feng restates it as Conjecture~1.2 and proves an infinite-volume counterpart~\cite{Feng2026}: his Theorem~1.3 concerns an infinite FK map and convergence to an infinite continuum random tree in the \emph{local} Gromov--Hausdorff--Prokhorov topology. Theorem~\ref{thm:main} realizes the finite-volume prediction in the compact topology, with the degree probability measure and all integer edge sizes. It is proved directly under the finite law, not deduced from the infinite-map limit. The contrast with the other FK regimes is geometric. Companion results for $00$ such that, for all real $x<1$ sufficiently close to $1$, \begin{equation} \sum_{n\ge1} n a_n u^n x^n\ge c(1-x)^{-1/2}. \label{ladder:coefficient-bound} \end{equation} \end{proposition} To prove this bound, we first show that the entire seam interval reduces to the empty word with probability bounded below uniformly in $h$. We construct boundaries that give burger-only reductions on the left and orders-only reductions on the right, then compare their type counts. \subsection{Stationary cuts} We index pieces by $\mathbb Z$, with boundary $i$ immediately before piece $i$. Thus the word between boundaries $a\eta\right) \le \frac{\E U_0}{\eta},\qquad \eta>0. \label{ladder:maximal} \end{equation} To prove it, first allow witnessing intervals of length at most $s$. Among violating starts in $1,\ldots,m$, select an interval at the first such start, skip the starts it covers, and repeat. The selected intervals are disjoint, cover every violating start, and lie in $1,\ldots,m+s$. Their sum exceeds $\eta$ times their total length, which is at least the number of violating starts. Taking expectations, dividing by $m$, and then letting $m$ and $s$ tend to infinity proves Equation~\eqref{ladder:maximal}. Apply this inequality to the absolute errors of the finite-window approximations. As their $L^1$ errors tend to zero, the almost-sure convergence of the averages follows. Reversing the indices proves the other direction. Apply the result to the indicator of the boundary set. If $b_k\le k$ is its last boundary before $k$ and $d>0$ is its density, the absence of boundaries between $b_k$ and $k$ gives $d(k-b_k)=o(k)+o(b_k)=o(k)$. The negative-index assertion is identical. \end{proof} \begin{definition} For a two-sided iid sequence of $L$-pieces, a boundary $a$ is an \emph{$L$-cut} if, for every $b>a$, the word between $a$ and $b$ has no unfilled orders when evaluated from an empty supply. For a two-sided iid sequence of $R$-pieces, a boundary $b$ is an \emph{$R$-cut} if, for every $a0$, bounded stopping gives \[ \mu\,\E(N_L\wedge m) =\E\left[\sum_{j=1}^{N_L\wedge m}(1-K_j)\right]\le1. \] The bound holds because the increments are at most $1$ and the walk is stopped on its first hit of $1$. Thus $\E N_L\le1/\mu<\infty$. Replace each parsing step by its reduction and put these reductions back in chronological order. Lemma~\ref{word:reduction} allows this replacement inside any larger word, including with an older supply of burgers. The resulting shortened $L$-piece has only burgers and fixed orders, and its length $\ell$ satisfies \begin{equation} \E\ell =\E\left[\sum_{j=1}^{N_L}(1+K_j)\right] =\E N_L\,\E(1+K)<\infty. \label{ladder:short-length} \end{equation} Here reaching parsing step $j$ depends only on the preceding steps, which justifies the second equality. The two net type counts in a shortened piece are integrable, sum to $1$, and have the same law under exchange of the types. Each therefore has expectation $1/2$. Across successive forward $L$-pieces, the net count of each type has positive drift. Within a shortened piece, the deviation from its initial count is at most its length. Moreover, for iid lengths of finite mean, $\ell_i/i\to0$ almost surely, by the integrable-tail bound and Borel--Cantelli. The running infimum of each type count over all shortened letters is therefore finite almost surely. A sufficiently large deterministic starting supply of both types prevents any unfilled order forever with positive probability. A finite prefix of prescribed single-burger $L$-pieces creates this supply with positive probability, independently of the remaining tail. Starting with that prefix and an empty supply proves positive probability of an $L$-cut at boundary $0$. Only the shortened length in Equation~\eqref{ladder:short-length} was integrated; no moment of the original ladder-piece length is needed. \paragraph{Reversing right pieces.} For an $R$-cut, we will show that a backward scan from a piece endpoint has positive probability of never revealing a surviving burger. Scan the letters backwards from boundary $0$ in a two-sided iid $R$-piece sequence. Let $S_j$ be the burger-minus-order sum of the first $j$ scanned letters. This scan has a different law from ordinary iid backwards sampling. The density of its first $m$ letters relative to ordinary iid backwards sampling is \begin{equation} 2(-S_m)\,\1_{\{S_j<0\text{ for }1\le j\le m\}}. \label{ladder:reverse-density} \end{equation} To see this, use the last $h$ pieces for any $h>m$. Their concatenation is an iid word stopped at its first hit of $-h$. Write $W$ for its forward walk, started at $0$, and $\tau$ for this hitting time. The reversed sums satisfy \[ S_j=W_\tau-W_{\tau-j}=-h-W_{\tau-j},\qquad 1\le j\le m. \] Since $W$ stays above $-h$ before $\tau$, a possible final string read backwards has strictly negative partial sums. If its total is $-r$, its chronological start is at $-h+r$, where $1\le r\le m0$. On the event that $0$ is an $L$-cut, let $B_a(h,k)$ count type-$a$ burgers among the bottom $k$ burgers of the reduction of the $h$ pieces starting at $0$, where $1\le k\le h$. Then, almost surely on this event, \begin{equation} \lim_{k\to\infty}\sup_{h\ge k} \left|\frac{B_a(h,k)}k-\frac12\right|=0, \qquad a=1,2. \label{ladder:left-counts} \end{equation} On the event that $0$ is an $R$-cut, let $C_a(h,k)$ count fixed orders of type $a$ among the terminal $k$ residual orders of the $h$ pieces ending at $0$. Then, almost surely on this event, \begin{equation} \lim_{k\to\infty}\sup_{h\ge k} \left|\frac{C_a(h,k)}k-\frac{1-\rho}{2}\right|=0, \qquad a=1,2. \label{ladder:right-counts} \end{equation} \end{lemma} \begin{proof} Type exchange symmetry and Lemma~\ref{ladder:averaging} give the claimed $L$-frequencies. The event of an $R$-cut at $0$ depends only on pieces before $0$. Independently requiring piece $0$ to be the single letter $F$, an event of probability $p/2>0$, makes boundary $1$ a cut as well and places an $F$ in output slot $0$. Consequently \[ \rho\ge\P(0\text{ is an }R\text{-cut})\frac p2>0. \] Stationary averaging and type exchange symmetry give the remaining $R$-frequencies. For Equation~\eqref{ladder:left-counts}, let $b_k$ be the last $L$-cut at or before $k$, on the event that $0$ is a cut. It satisfies $k-b_k=o(k)$. For every $h\ge k$, the bottom $b_k$ surviving burgers are exactly the stationary output in slots $0,\ldots,b_k-1$: the pieces after the cut $b_k$ require no older supply and cannot remove these burgers, by Lemma~\ref{word:reduction}. Thus the bottom-$k$ count differs from the corresponding stationary prefix count by at most $k-b_k$, uniformly in $h$. This proves Equation~\eqref{ladder:left-counts}. For Equation~\eqref{ladder:right-counts}, take the first $R$-cut $c_k$ at or after $-k$; then $c_k=-k+o(k)$. For every $h\ge k$, the whole-piece word from $-h$ to $c_k$ leaves no burgers, by the definition of the cut at $c_k$. Its residual orders cannot be served by later burgers and do not alter the reduction of the suffix from $c_k$ to $0$. The terminal $-c_k$ orders therefore agree exactly with stationary output in these slots. The possible discrepancy among the terminal $k$ orders is at most $k+c_k=o(k)$, independently of $h$. This proves Equation~\eqref{ladder:right-counts}. \end{proof} \subsection{Matching at a seam} \begin{proposition}[Uniform success at a seam] \label{ladder:uniform} There is $\delta>0$, depending only on $p$, such that for every integer $h\ge1$ the seam interval formed by the backwards hit of $+h$ and the forwards hit of $-h$ reduces to the empty word with probability at least $\delta$. \end{proposition} \begin{proof} Choose \[ \frac{1-\rho}{2}<\alpha<\frac12. \] By Lemmas~\ref{ladder:cuts} and~\ref{ladder:frequencies}, for a sufficiently large deterministic integer $B$, each of the following events has positive probability in its respective stationary piece sequence: \begin{align*} \mathcal G_L={}&\{0\text{ is an }L\text{-cut, and } B_a(h,k)\ge\alpha k \text{ for }a=1,2\text{ and all }h\ge k\ge B\},\\ \mathcal G_R={}&\{0\text{ is an }R\text{-cut, and } C_a(h,k)\le\alpha k \text{ for }a=1,2\text{ and all }h\ge k\ge B\}. \end{align*} Indeed, increasing $B$ makes each event exhaust its positive-probability cut event, up to a null set. Take an integer $D\ge B$ with $\alpha D\ge B$. The event $\mathcal G_R$ is measurable with respect to the past pieces, so requiring the next $D$ pieces, indexed $0,\ldots,D-1$, to be single flexible orders preserves positive probability. Use independent stationary $L$- and $R$-sequences. For the left word take $h$ pieces starting at $0$, and for the right word take $h$ pieces ending at $D$. Unconditionally these have exactly the independent ladder-piece laws of the seam interval for each fixed $h$. The event consisting of $\mathcal G_L$, $\mathcal G_R$, and the prescribed $D$ flexible pieces has probability \[ \delta=\P(\mathcal G_L)\P(\mathcal G_R)(p/2)^D>0, \] independent of $h$. On this event the left word reduces to $h$ burgers. The right word reduces to $h$ orders: it is all flexible if $h\le D$; otherwise it is the orders-only reduction of $h-D$ pieces ending at $0$, followed by $D$ flexible orders. Consider its terminal $k$ orders, with $k\le h$. If $k\le D$, their fixed-type counts vanish. If $k>D$ and $k-D\ge B$, each fixed-type count is at most $\alpha(k-D)\le\alpha k$ by $\mathcal G_R$. If $0m)\le C_0\frac{h+1}{\sqrt m}. \] Consequently the two hitting lengths defining $I_h$, divided by $h^2$, are tight uniformly over $h\ge1$. Choose a deterministic $A<\infty$ large enough that \[ \P\bigl(I_h\text{ reduces to the empty word and }|I_h|\le Ah^2\bigr) \ge\delta/2,\qquad h\ge1. \] This follows by subtracting the length-tail probabilities from the success probability; no independence between length and success is needed. The intervals $I_h$ are distinct as $h$ varies, since the first-hitting endpoints move strictly with the level. There are exactly $2n-1$ deterministic intervals of length $2n$ using letters on both sides of the seam. Every empty-reducing word has even length. By Proposition~\ref{word:identity} and Tonelli's theorem, the expected sum of $x^{|I|/2}$ over all empty-reducing finite intervals through the seam is \[ \sum_{n\ge1}(2n-1)a_nu^n x^n. \] For $H=\lfloor(1-x)^{-1/2}\rfloor$, the distinct intervals $I_1,\ldots,I_H$ therefore give \begin{align*} \sum_{n\ge1}(2n-1)a_nu^n x^n &\ge \frac\delta2\sum_{h=1}^H x^{Ah^2/2}\\ &\ge \frac\delta2 H\,x^{A/(2(1-x))}. \end{align*} The last factor is bounded away from zero as $x\uparrow1$, while $H\asymp(1-x)^{-1/2}$. Since $2n-1\le2n$, this proves Equation~\eqref{ladder:coefficient-bound}. \end{proof} ==== END Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//ladders.tex ==== ==== BEGIN Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//main.tex ==== \documentclass[11pt,a4paper]{article} \usepackage[margin=28mm]{geometry} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm} \usepackage{mathtools} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Brownian continuum random tree limits of finite Fortuin--Kasteleyn maps above four}, pdfauthor={OpenAI}, pdfsubject={Finite-volume graph-metric scaling of critical FK planar maps for q greater than four} } \setlength{\emergencystretch}{2em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewcommand{\P}{\mathbb P} \newcommand{\E}{\mathbb E} \newcommand{\R}{\mathbb R} \newcommand{\N}{\mathbb N} \newcommand{\1}{\mathbf 1} \newcommand{\dd}{\mathrm d} \newcommand{\op}{o_{\P}} \newcommand{\Op}{O_{\P}} \newcommand{\GHP}{\mathrm{GHP}} \title{Brownian continuum random tree limits of finite Fortuin--Kasteleyn maps above four} \author{OpenAI} \date{September 24, 2026} \begin{document} \maketitle \begin{abstract} We prove the finite-volume continuum-random-tree prediction for critical Fortuin--Kasteleyn planar maps at every fixed $q>4$. After rescaling graph distances by a constant times $n^{-1/2}$, an $n$-edge map with normalized degree measure converges to the Brownian continuum random tree in the Gromov--Hausdorff--Prokhorov topology. The convergence holds through all positive integer sizes. \end{abstract} \section{Introduction}\label{sec:introduction} A large planar map can contain many cycles while its distances, viewed on a sufficiently large scale, approach those of a tree. We prove that this happens for critical Fortuin--Kasteleyn maps at every fixed $q>4$: their finite-volume metric scaling limit is the Brownian continuum random tree. Two features of this statement require separate work. The probability law is conditioned on the size of the entire finite map, and the metric uses all its edges. A limit of an unconditioned encoding walk does not supply either conclusion. Our proof identifies an exact marked block tree, proves that its offspring law is critical with finite variance, and controls the graph distances contributed by its blocks uniformly over all ancestral paths. \subsection{The finite model and the main result} A \emph{rooted planar map} is a connected graph embedded in an oriented sphere, up to orientation-preserving homeomorphism, with a distinguished oriented edge, or \emph{dart}. Loops and multiple edges are allowed. For an edge subset $A\subseteq E(M)$, let $k_M(A)$ be the number of components of the spanning graph $(V(M),A)$, counting isolated vertices. For $q>4$ and $n\ge1$, sample a pair $(M_n,A_n)$ with $|E(M_n)|=n$ according to \begin{equation}\label{eq:finite-law} \P\big((M_n,A_n)=(M,A)\big) =\frac{1}{Z_{n,q}}\, q^{\,k_M(A)+(|A|-|V(M)|)/2}. \end{equation} Equivalently, the weight is $q^{\ell(M,A)/2}$, where $\ell(M,A)=2k_M(A)+|A|-|V(M)|$ is the number of FK interface loops. Let $d_n$ be graph distance on $V(M_n)$ using every edge of $M_n$, and put \begin{equation}\label{eq:degree-measure} \mu_n(\{v\})=\frac{\deg_{M_n}(v)}{2n}. \end{equation} A loop contributes two to the degree. Let $e$ be a standard normalized Brownian excursion on $[0,1]$. For $s,t\in[0,1]$, define \[ d_e(s,t)=e(s)+e(t)- 2\min_{r\in[s\wedge t,s\vee t]}e(r). \] The quotient by $d_e=0$, equipped with its induced metric and the pushforward $\mu_e$ of Lebesgue probability measure, is denoted $(\mathcal T_e,d_e,\mu_e)$. We use this normalization throughout. \begin{theorem}\label{thm:main} For every fixed real $q>4$, there is a deterministic constant $c(q)\in(0,\infty)$ such that, as $n\to\infty$ through all positive integers, \[ \big(V(M_n),c(q)n^{-1/2}d_n,\mu_n\big) \ \Longrightarrow\ (\mathcal T_e,d_e,\mu_e) \] in the Gromov--Hausdorff--Prokhorov topology on compact metric probability spaces modulo measure-preserving isometry. \end{theorem} The root and FK decoration are forgotten in this convergence. We recall the correspondence estimate for this topology in Section~\ref{metric:section}, where the metric and measure comparison is completed. \input{sections/history} \subsection{Proof roadmap} Section~\ref{word:section} expresses the total weight of $n$-edge maps, up to an explicit exponential factor, as the probability that an independent-letter inventory word of length $2n$ matches completely. Ignoring burger types turns such a word into a nonnegative simple-walk excursion. The resulting Catalan upper bound shows that the map generating function is finite at the parameter selected by the word law. For a complementary lower bound, fix a boundary between two letters. Scan to its left until the number of burgers minus orders first reaches $h$, and to its right until that number first reaches $-h$. Section~\ref{ladder:section} uses strict inventory drift and comparisons of burger supply with order demand to prove that the whole interval matches with probability bounded below uniformly in $h$. Counting these intervals, together with a random-walk hitting-time estimate, gives a square-root lower bound for the derivative of the map series. Section~\ref{block:section} decomposes a map into nonseparable pieces, called blocks, joined at vertices. A corner is the sector between successive incident darts; each block corner is an ordered attachment slot, with an empty insertion recorded by a leaf. The exact substitution equation for these attachments combines with the two word estimates to force criticality and finite offspring variance, without analytic continuation beyond the convergence endpoint. The resulting marked Galton--Watson tree has exactly $2n+1$ nodes for an $n$-edge map. Finally, Section~\ref{tree:section} combines the joint conditioned-tree coding limit with uniform laws for the distance marks accumulated along ancestral paths. Reversing the child order at every node gives a second depth-first walk; the two walks control large marks using only the second offspring moment. Section~\ref{metric:section} converts this path estimate into a uniform comparison of map and tree distances. The corners push uniform nonroot-node mass to the degree measure exactly, which completes the metric probability-space comparison. All constants may depend on the fixed parameter $q$. No uniform statement as $q$ approaches $4$ is asserted. \input{sections/words} \input{sections/ladders} \input{sections/blocks} \input{sections/trees} \input{sections/metric} \input{references} \end{document} ==== END Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//main.tex ==== ==== BEGIN Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//metric.tex ==== \section{From the block tree to the map}\label{metric:section} The mean block-distance comparison follows the general strategy for subcritical graph classes~\cite[Section~5]{PanagiotouStuflerWeller2016} and tree-like maps~\cite[preprint Section~6.8]{Stufler2020}. Here Theorem~\ref{tree:paths} supplies the required uniformity under only finite offspring variance. We also retain the exact corner-mass calculation, as in~\cite[Section~9]{Stufler2026}, because the conclusion specifies the degree probability measure. Let $\mathsf T_N$ be the conditioned marked tree of Proposition~\ref{block:finite-law}, where $N=2n+1$. Its map has the required marginal law of $M_n$. Write $\pi(w)$ for the pole of a tree node $w$. For a strict ancestor with $k$ children, block mark $B$, and selected child $i$, put \[ g(k,i,B)=d_B\big(\text{pole of }B,\text{vertex of corner }i\big), \qquad \beta=\E_{\rm sp}g. \] The expectation is under the spine law \eqref{tree:spine-law}, and $d_B$ uses all block edges. Since a block has $k/2$ edges, $0\le g\le k$. Therefore \begin{equation}\label{metric:beta} 0<\beta<\infty. \end{equation} Finiteness follows from $\E_{\rm sp}k=\E\xi^2<\infty$. For positivity, a one-link block and its corner at the endpoint opposite the pole have positive spine probability, and their distance is one. Let $G(w)$ be the sum of these $g$ labels over the strict ancestors of $w$. By Theorem~\ref{tree:paths} and the coding estimates in Section~\ref{tree:section}, with $\nu=\sigma^2/2$, \begin{equation}\label{metric:uniform-poles} \max_{w\in\mathsf T_N} \left|G(w)-\beta H(w)\right|=\op(\sqrt N),\qquad \max_{w\in\mathsf T_N} \left|G(w)-\frac{\beta}{\nu}W(w)\right|=\op(\sqrt N). \end{equation} \subsection{Uniform comparison of distances} \begin{lemma}[Distances between poles]\label{metric:poles} Let $w_*$ be the last common ancestor of two tree nodes $w,w'$. If $\Delta$ is the maximum offspring number, then \[ \left|d_{M_n}\big(\pi(w),\pi(w')\big) -\big(G(w)+G(w')-2G(w_*)\big)\right|\le2\Delta. \] The map $\pi$ from tree nodes to map vertices is onto. \end{lemma} \begin{proof} Every inserted map meets the preceding part at its pole only. An excursion from a block into an attached submap must return to the same vertex, so attachments cannot shorten distances within a block. Moving from an ancestor pole to a descendant pole therefore adds the distances $g$ along their chain of blocks. If one of $w,w'$ is their last common ancestor, this proves the displayed comparison with zero error. Otherwise let $x,x'$ be the two selected corner vertices in the block of $w_*$. The route through the pole of that block has length $d_B(x,\pi(w_*))+d_B(\pi(w_*),x')$ within the block, whereas the shortest route uses $d_B(x,x')$. Their difference lies between zero and twice the diameter of $B$, and that diameter is at most its number of edges, hence at most $\Delta$. Outside the common block the distances add exactly as above. The same argument allows coincident poles and loop blocks. Every vertex of the nonempty map is incident to at least one edge, and hence to a corner in one of its blocks. That corner is a child slot, whose child's pole is the given vertex. Thus every map vertex is the pole of a nonroot node. \end{proof} Index the nodes by $j=0,\ldots,N-1$ in preorder. Combining Lemma~\ref{metric:poles}, \eqref{metric:uniform-poles}, and Lemma~\ref{tree:lca}, and using $\Delta=\op(\sqrt N)$, gives \begin{equation}\label{metric:walk-comparison} \max_{0\le j\le j'0$, and a coupling of their probability measures is supported on the correspondence. Their Gromov--Hausdorff--Prokhorov distance is at most $\varepsilon$. If one marginal of that coupling differs from the desired measure by total variation at most $\eta$, the bound is $\varepsilon+\eta$. \end{lemma} \begin{proof} Join corresponding points by links of length $\varepsilon$, and use the induced shortest-chain distance on the disjoint union. This preserves the distances within each original space. Indeed, a segment leaving one space and returning to it can be replaced by a segment inside that space: the two linking lengths are $2\varepsilon$, whereas the difference of the distances between the corresponding endpoints is at most $\varepsilon$. Every point is within $\varepsilon$ of the other space, giving the Hausdorff bound. The coupling places every pair at distance at most $\varepsilon$, which gives the Prokhorov bound directly from its defining neighborhood inequalities. Changing one marginal by total variation $\eta$ increases these inequalities, and hence the Prokhorov bound, by at most $\eta$. \end{proof} For a nonnegative continuous function $f$ on $[0,1]$ with $f(0)=f(1)=0$, define $d_f$ by the same formula as $d_e$, and denote its quotient metric probability space by $\mathcal T_f$. The interval-minimum formula gives the triangle inequality for $d_f$. The quotient projection is continuous, since $d_f(s,t)$ is at most twice the oscillation of $f$ on the interval between $s$ and $t$. Thus $\mathcal T_f$ is compact. Relating equal-time representatives in $\mathcal T_f$ and $\mathcal T_{\tilde f}$ gives a correspondence of distortion at most $4\|f-\tilde f\|_\infty$ and a coupling from one uniform time. Lemma~\ref{metric:ghp} therefore proves continuity of \[ f\longmapsto \mathcal T_f \] from the uniform norm to the Gromov--Hausdorff--Prokhorov topology. \subsection{Proof of the main theorem} \begin{proof}[Proof of Theorem~\ref{thm:main}] Define \begin{equation}\label{metric:constant} c(q)=\frac{\nu}{\sqrt2\,\beta\sigma} =\frac{\sigma}{2\sqrt2\,\beta}. \end{equation} Equations~\eqref{block:moments} and \eqref{metric:beta} show that this constant is deterministic, positive, and finite for every fixed $q>4$. Let $f_N$ interpolate $W_j/(\sigma\sqrt N)$ at $j/N$ for $0\le j4$}, Probab. Theory Related Fields \textbf{194} (2026), 299--331. \href{https://doi.org/10.1007/s00440-025-01424-2}{doi:10.1007/s00440-025-01424-2}. \bibitem{FortuinKasteleyn1972}C.~M.~Fortuin and P.~W.~Kasteleyn, \emph{On the random-cluster model. I. Introduction and relation to other models}, Physica \textbf{57} (1972), no.~4, 536--564. \href{https://doi.org/10.1016/0031-8914(72)90045-6}{doi:10.1016/0031-8914(72)90045-6}. \bibitem{Janson2012}S.~Janson, \emph{Simply generated trees, conditioned Galton--Watson trees, random allocations and condensation}, Probability Surveys \textbf{9} (2012), 103--252. \href{https://doi.org/10.1214/11-PS188}{doi:10.1214/11-PS188}. \bibitem{LyonsPemantlePeres1995}R.~Lyons, R.~Pemantle, and Y.~Peres, \emph{Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes}, Annals of Probability \textbf{23} (1995), no.~3, 1125--1138. \href{https://arxiv.org/abs/math/0404083}{arXiv:math/0404083}. \bibitem{MarckertMokkadem2003}J.-F.~Marckert and A.~Mokkadem, \emph{The depth first processes of Galton--Watson trees converge to the same Brownian excursion}, Annals of Probability \textbf{31} (2003), no.~3, 1655--1678. \bibitem{Mullin1967}R.~C.~Mullin, \emph{On the enumeration of tree-rooted maps}, Canad. J. Math. \textbf{19} (1967), 174--183. \href{https://doi.org/10.4153/CJM-1967-010-x}{doi:10.4153/CJM-1967-010-x}. \bibitem{SubcriticalConformal2026}OpenAI, \emph{Canonical conformal limits of subcritical FK planar maps}, OpenAI Math Release preprint \href{https://github.com/openai/math/blob/main/preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/main.pdf}{OAI:Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026}, 2026. \bibitem{SubcriticalMetric2026}OpenAI, \emph{Metric-measure limits of subcritical FK and spanning-tree planar maps}, OpenAI Math Release preprint \href{https://github.com/openai/math/blob/main/preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026/main.pdf}{OAI:Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026}, 2026. \bibitem{CriticalFK2026}OpenAI, \emph{The critical Liouville quantum sphere and geometric limits of FK maps at $q=4$}, OpenAI Math Release preprint \href{https://github.com/openai/math/blob/main/preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026/main.pdf}{OAI:The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026}, 2026. \bibitem{PanagiotouStuflerWeller2016}K.~Panagiotou, B.~Stufler, and K.~Weller, \emph{Scaling limits of random graphs from subcritical classes}, Annals of Probability \textbf{44} (2016), no.~5, 3291--3334. \href{https://doi.org/10.1214/15-AOP1048}{doi:10.1214/15-AOP1048}. \bibitem{Sheffield2016}S.~Sheffield, \emph{Quantum gravity and inventory accumulation}, Ann. Probab. \textbf{44} (2016), no.~6, 3804--3848. \href{https://doi.org/10.1214/15-AOP1061}{doi:10.1214/15-AOP1061}. \bibitem{Stufler2020}B.~Stufler, \emph{Limits of random tree-like discrete structures}, Probability Surveys \textbf{17} (2020), 318--477. \href{https://doi.org/10.1214/19-PS338}{doi:10.1214/19-PS338}. Preprint: \href{https://arxiv.org/abs/1612.02580v2}{arXiv:1612.02580v2}. \bibitem{Stufler2026}B.~Stufler, \emph{Non-bijective scaling limits and phase transitions of planar maps}, \href{https://arxiv.org/abs/2608.21063v1}{arXiv:2608.21063v1} (2026). \bibitem{Tutte1963}W.~T.~Tutte, \emph{A census of planar maps}, Canad. J. Math. \textbf{15} (1963), 249--271. \href{https://doi.org/10.4153/CJM-1963-029-x}{doi:10.4153/CJM-1963-029-x}. \end{thebibliography} ==== END Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//references.tex ==== ==== BEGIN Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//trees.tex ==== \section{Conditioned trees and uniform path sums} \label{tree:section} We isolate the probabilistic statement needed to pass from the block tree to graph distances. Conditioned-tree height limits originate in \cite{Aldous1993}; their stable-domain extension is developed in \cite{Duquesne2003}. Joint depth-first process convergence to the same excursion was proved by Marckert and Mokkadem under stronger moment conditions~\cite{MarckertMokkadem2003}. We invoke the finite-variance, attainable-size formulation of Broutin and Marckert~\cite{BroutinMarckert2014} for the tree geometry. We then prove a uniform law of large numbers for marked ancestral paths, using only a finite second offspring moment. Let $\xi$ have distribution $(p_k)_{k\geq0}$ supported on the even nonnegative integers, with \begin{equation} p_0p_2>0,\qquad \E\xi=1,\qquad 0<\sigma^2:=\operatorname{Var}(\xi)<\infty. \label{tree:assumptions} \end{equation} Write $\mathcal T$ for the corresponding plane Galton--Watson tree and $\mathcal T_N$ for its law conditional on having $N$ nodes. Its attainable sizes are exactly the odd integers: every tree satisfies $N-1=\sum_v\deg^+(v)$, and every odd size can be realized using only offspring numbers zero and two. All limits in this section run through these odd integers. Given the offspring numbers, attach independent marks $A_v$ to the nodes, with a fixed law $\mathcal L_k$ at a node with $k$ children. Index the nodes in preorder by $v_0,\ldots,v_{N-1}$. Let $H(v)$ denote depth, put $H_j=H(v_j)$ for $j0$. The exponential estimate for bounded independent variables gives, uniformly for $h\leq C\sqrt N$, \[ \P_{\mathrm{sp}}\left( \left|\sum_{r=1}^h b(Z_r)-\mu_bh\right|>\varepsilon\sqrt N \right)\leq C' e^{-c'\sqrt N}. \] For short paths the event may be empty; otherwise the usual bound $2\exp(-c\varepsilon^2N/h)$ gives the displayed estimate. The probability under $\mathcal T_N$ that any such path violates the desired bound is at most \[ \pi_N^{-1}\sum_{h\leq C\sqrt N}C'e^{-c'\sqrt N}=o(1), \] by the unconditioned identity \eqref{tree:spine-identity} and Lemma~\ref{tree:conditioning}. Now let $C$ increase, using maximum-depth tightness from Corollary~\ref{tree:maxdegree}. \end{proof} \subsection{Labels bounded by the offspring number} We now allow labels that grow with the offspring number. The two exploration orders control the sum of offspring numbers along every ancestral path. Subtracting its bounded truncation will control the remaining tail uniformly. \begin{theorem}[Uniform marked path sums] \label{tree:paths} Let $b=b(k,i,A)$ be a fixed measurable label with $0\leq b\leq k$. Then $\mu_b<\infty$ and \begin{equation} \max_{v\in\mathcal T_N}|B_b(v)-\mu_bH(v)|=o_{\P}(\sqrt N), \qquad \max_{v\in\mathcal T_N} \left|B_b(v)-\frac{\mu_b}{\nu}W(v)\right|=o_{\P}(\sqrt N). \label{tree:marked-sums} \end{equation} \end{theorem} \begin{proof} The exact identities \eqref{tree:waiting-slots} give \begin{equation} \sum_{\operatorname{path}(v)}k=W(v)+\widehat W(v)+H(v). \label{tree:total-label} \end{equation} By \eqref{tree:walk-height-estimate} and \eqref{tree:mirror-estimate}, this is $(2\nu+1)H(v)+o_{\P}(\sqrt N)$ uniformly in $v$. For each fixed $R$, Lemma~\ref{tree:bounded-paths} applies to the bounded label $k\1_{\{k\leq R\}}$. Subtracting its estimate yields \begin{equation} \sum_{\operatorname{path}(v)}k\1_{\{k>R\}} =m_RH(v)+o_{\P}(\sqrt N) \quad\hbox{uniformly in }v, \qquad m_R:=\E_{\mathrm{sp}}[K\1_{\{K>R\}}]. \label{tree:tail-estimate} \end{equation} The error statement here is for each fixed $R$. Integrability of $K$ gives $m_R\to0$, and maximum-depth tightness consequently implies \begin{equation} \lim_{R\to\infty}\limsup_{N\to\infty} \P\left(\max_v\sum_{\operatorname{path}(v)} k\1_{\{k>R\}}>\varepsilon\sqrt N\right)=0 \quad(\varepsilon>0). \label{tree:uniform-tail} \end{equation} Apply Lemma~\ref{tree:bounded-paths} to $b\1_{\{k\leq R\}}$. The discarded path sum is bounded by the tail in \eqref{tree:uniform-tail}, and the discarded spine mean is bounded by $m_R$. Letting first $N$ and then $R$ tend to infinity proves the first assertion. The second follows from \eqref{tree:walk-height-estimate} and $\nu>0$. \end{proof} This truncation uses $\E_{\mathrm{sp}}K=\E\xi^2<\infty$; it does not require a second moment of the size-biased variable $K$, and hence does not impose a third offspring moment. We finish with a deterministic comparison for later use. \begin{lemma}[Common ancestors and walk minima] \label{tree:lca} If $j\leq j'$ and $v_*$ is the lowest common ancestor of $v_j,v_{j'}$, then \begin{equation} W(v_*)\leq\min_{j\leq r\leq j'}W_r\leq W(v_*)+\Delta_N. \label{tree:lca-walk} \end{equation} \end{lemma} \begin{proof} Every intervening node is a descendant of $v_*$. Its walk value is at least $W(v_*)$ by \eqref{tree:waiting-slots}. If $v_*=v_j$, the upper bound is immediate. Otherwise the interval contains the child of $v_*$ leading toward $v_{j'}$; its walk value exceeds $W(v_*)$ by at most the offspring number of $v_*$. \end{proof} ==== END Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//trees.tex ==== ==== BEGIN Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four//words.tex ==== \section{Words and strict drift}\label{word:section} Fix $q>4$, and put \begin{equation}\label{word:parameters} t=\sqrt q,\qquad p=\frac{t}{t+2},\qquad u=\frac{1-p}{16}. \end{equation} Thus $1/2