Random planar maps: geometry, diffusion, and spectra

Source and verification

This article explains family 211 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

A random planar map can approach a random surface or a random tree. Even after its geometry converges, the clock of a random walk and the spectrum of its Laplacian require additional estimates. The seven companions in family 211 address these different levels of convergence.

🏷️ Preliminaries: maps, measures, and random surfaces

A planar map is a finite connected graph embedded in the sphere, considered up to orientation-preserving change of coordinates. Loops and multiple edges are allowed. A root removes most ambiguities in counting maps. A Fortuin–Kasteleyn (FK) decoration is an edge subset weighted using its number of connected components; the parameter $q$ controls the cluster weight. Here both the graph and the decoration are random. This differs from FK percolation on a fixed square lattice.

A metric-measure space records distances together with a probability measure. For an $n$-edge map, the stationary measure of simple random walk is

$${\mu }_n(v)=\frac{\mathrm{deg}(v)}{2n}.$$

A loop contributes twice to degree. Convergence of these spaces compares distances and masses after suitable rescaling and embedding in a common space.

Liouville quantum gravity (LQG) constructs random area and geometry from a Gaussian free field. The formal density $e^{\gamma h(z)}dz$ needs renormalization because $h$ is a distribution, not a function. The associated metric is an additional construction; it is not obtained by substituting $h$ into an ordinary smooth Riemannian metric. The Brownian continuum random tree (CRT), by contrast, is a random real tree encoded by a Brownian excursion.

🏷️ What the companions claim

The geometric papers treat subcritical FK parameters and spanning-tree weighting, the critical parameter $q=4$, and $q>4$. “Subcritical” here describes the LQG parameter regime, not an arbitrary choice of a subcritical bond density on a fixed graph.

• Conformal geometry: canonical conformal limits for subcritical FK maps, with the embedding and field information retained. (OpenAI, 2026)
• Metric geometry: metric-measure limits for subcritical FK and spanning-tree maps. The critical companion constructs the critical quantum sphere and treats FK geometry at $q=4$. (OpenAI, 2026b) (OpenAI, 2026c)
• Tree geometry: for each fixed $q>4$, rescaled graph distance at order $n^{-1/2}$ and normalized degree measure converge to a Brownian CRT with the specified deterministic scale. The statement runs through all integer sizes and does not give uniformity as $q\downarrow 4$. (OpenAI, 2026a)
• Diffusion: critical FK–Ising maps, with LQG parameter $\sqrt{3}$, and tree-weighted maps, with parameter $\sqrt{2}$, have the stated Liouville Brownian limits with a linear clock. The Ising result retains finitely many conditionally independent stationary walks jointly with the geometry. (OpenAI, 2026d) (OpenAI, 2026a)
• Spectra: the FK–Ising companion claims convergence of eigenvalues with multiplicity and positive-time heat traces, using its stated Brownian and LQG inputs. (OpenAI, 2026f)

The last two conclusions demand more than convergence of spaces: a region of tiny stationary mass can still create a trap or support an eigenfunction with unexpectedly small energy.

🏷️ A worked normalization: why the clock is linear

Let the continuous-time walk jump at rate one by choosing a uniformly random incident half-edge. For its generator $L_n$ and a real function $f$,

$$\begin{array}{c}-⟨f,L_{n}f{⟩}_{{\mu }_n}\\ =\frac{1}{2n}\sum _{e=\{x,y\}}(f(x)-f(y){)}^2.\end{array}$$

The sum counts each unoriented edge once, including multiplicity; loops contribute zero. To verify the formula, expand the inner product and pair the two orientations of each edge. Each orientation carries stationary jump weight $1/(2n)$, and the usual factor $1/2$ in the Dirichlet form removes the double counting.

Accelerating the walk by $n$ gives energy $\frac{1}{2}{\sum }_e(f(x)-f(y){)}^2$. This explains the proposed normalization. It does not prove convergence of forms or exclude an additional effective conductivity. Those are the substantive homogenization questions addressed in the walk companions.

🏷️ How the proposed arguments retain the missing information

The surface arguments keep finite records of primal and dual crossings, incidences, and energies as the discrete objects converge. A crossing and a dual crossing cannot both have arbitrarily cheap energy: a discrete Cauchy–Schwarz estimate relates their conductances. Such records protect information that a bare limiting metric can discard.

For the clock, conditional local limits and isotropy identify a scalar effective conductivity. The primal-dual normalization then fixes that scalar. Circuit and flow estimates supply compactness for functions of bounded energy, including control near sets carrying very little mass. In the spectral argument, this compactness rules out disappearing low-energy modes; convergence of forms can then be upgraded to eigenvalues and heat traces. The source-specific identification and compactness estimates carry the proof, not the elementary stationary-measure calculation.

Above four, the mechanism changes. The manuscript decomposes the map into blocks organized by a conditioned finite-variance branching tree. An excursion limit gives the tree geometry; bounds for the distances contributed by individual blocks and for degree mass transfer the limit to the map itself. This is why a tree can emerge even though every finite object is planar and has cycles.

🏷️ Key steps and proof outlines

Local crossing measurements force a conformal structure

In the conformal companion, a subsequential comparison map $f$ is first shown to be a homeomorphism with controlled distortion. The remaining question is whether its infinitesimal distortion is round. At almost every point where it is differentiable with nonzero Jacobian, define

$$T(z)=\frac{Df(z{)}^{T}Df(z)}{|\det Df(z)|}.$$

This is a positive symmetric matrix with determinant one. Its quadratic form can be recovered from thin-rectangle extremal lengths. If $R_{v,t}$ has length one in direction $v$ and width $t$, a nonsingular linear map $L$ satisfies

$$\frac{(|Lv|-t|L{v}^{\perp }|{)}_{+}^2}{t|\det L|}\le {\mathrm{EL}}_{\leftrightarrow }(L{R}_{v,t})\le \frac{|Lv{|}^2}{t|\det L|}.$$

The upper bound uses straight-line flows; the lower uses a constant test density. Multiply by $t$, let $t\downarrow 0$, and use differentiability on shrinking rectangles. This recovers $v^{T}T(z)v$ from the local crossing data. Three directions determine the matrix.

The source’s local conditional-product and field-germ argument makes this local tensor deterministic. Rotation covariance then gives $T=R^{T}TR$ for a quarter-turn $R$. A symmetric two-by-two matrix with that property is scalar, and determinant one forces $T=I$. Thus the differential is a similarity almost everywhere. The imported quasiconformal characterization makes the limit conformal or anticonformal, with orientation fixed by the construction. This explains why local extremal-length data identify more than the underlying topology. (OpenAI, 2026c)

Relative distance units need a two-sided comparison

The metric companion compares passage rules whose deterministic units $b_i$ may initially vary between scales. Lower crossing and upper surrounding tests are needed in both directions. If at least a quarter of a late window had $b_k>T{b}_i$, annulus transfer would force the base passage rule to dominate the reference metric by a factor too large for its own upper crossing test. If a quarter had $b_k<b_i/T$, the reversed comparison would contradict the base lower test.

The remaining step is elementary but useful. For each base $i$, fewer than a quarter of late-window indices are too large, and fewer than a quarter too small. Windows based at any fixed $i,j$ overlap in all but $O(|i-j|)$ indices. For sufficiently long windows, there is a common index $k$ with

$$b_k\le T{b}_i,b_k\ge b_j/T.$$

Hence $b_j\le T^2{b}_i$; reversing $i,j$ gives the reciprocal bound. This prevents scale normalizations from drifting apart. The shell tests must be evaluated in their actual relative units under the conditional annular laws; knowing only separate marginal distance limits would not prove the comparison. (OpenAI, 2026d)

Critical geometry: rule out too many bad addresses, then calibrate

At $q=4$, the critical companion needs endpoint control for the joint metric, area, and loop construction. Its address argument supposes that many grid times are bad and samples a fixed number $d$ of them. That produces a lower bound on counting mass. Each sampled configuration is then encoded by a multiscale address code. The proof compares the number of possible codes with their probability costs: shell crossings charge directed moves, auxiliary tests charge isolated nodes, and smoothing of equality constraints contributes a rank-dependent gain.

One detail matters for summing all codes. The resulting upper bound contains a negative term proportional to $-(V-2dbN{)}_{+}$, where $V$ is the total node count. It controls codes with arbitrarily large $V$; the proof does not assume the relevant codes have bounded complexity. After choosing the scale tolerances, then $d$ large enough, the upper exponential rate is strictly smaller than the rate forced by the assumed abundance of bad times. This contradiction supplies the endpoint estimate.

A separate bounded calibration removes the residual deterministic metric factor without requiring distance moments. Let $Y_n$ be the discrete distance between two prescribed area-sampled normalization vertices, and let $Z=D_h(0,1)>0$ be its target continuum distance. Set

$$t_{\ast }=\mathbb{E}\tanh Z\in (0,1),\mathbb{E}\tanh (a_n{Y}_n)=t_{\ast }.$$

For large $n$ the latter equation has a unique positive solution. If along an extraction $Y_n/b_n\Rightarrow cZ$, bounded weak convergence gives $\mathbb{E}\tanh (u{Y}_n/b_n)\to \mathbb{E}\tanh (ucZ)$. Strict monotonicity brackets $a_n{b}_n$ around $1/c$. Thus calibration removes the factor $c$ on every subsequence, while retaining the same joint area, embedding, and loop law. (OpenAI, 2026g)

Primal-dual reciprocity fixes the walk’s conductivity

For tree-weighted maps, compactness and isotropy first identify a limiting electrical energy with a deterministic scalar $\lambda$. Isotropy alone does not determine its value. The source obtains one inequality, $\lambda \le 1$, from its variational comparison and proves the reverse using an annulus.

Let $u_m$ be a discrete condenser voltage between the two boundary wells. Construct a dual closed cochain ${\alpha }_m$ with exact unit period around the annulus and asymptotically optimal energy. Pair primal edges with crossing dual edges. Discrete summation by parts gives

$$\left|\sum _{e}d{u}_m(e){\alpha }_m(e^{\ast })\right|=1.$$

Indeed, replace the voltage by an indicator of one side of an intermediate circle. Their difference is supported where the dual cochain is closed, so the pairing is unchanged. The remaining boundary chain winds once, counted with multiplicity, and the period is exactly one.

Cauchy–Schwarz bounds this pairing by the product of condenser and dual angular energies. Their limits give $1\le \lambda C_0\lambda C_0^{-1}={\lambda }^2$. Positivity and the earlier upper bound force $\lambda =1$. This is the normalization step that excludes an extra effective clock constant. (OpenAI, 2026a)

For FK–Ising walks, after the operator and path limit is established, removing exponential waiting times is quantitative. Give each attempted step an independent exponential time of mean $1/n$, and let $S_{n,k}$ be their sum. Doob’s inequality gives

$$\mathbb{E}\underset{k\le nT}{\max }|S_{n,k}-k/n{|}^2\le \frac{4\lfloor nT\rfloor }{n^2}⟶0.$$

Combine this with the tight modulus of continuity of the accelerated walk and the vanishing maximum cable-edge length. The deterministic-step interpolation and continuous-time path then agree uniformly on compact times in probability. Every attempted step is counted, including a loop traversal that returns to the same vertex. (OpenAI, 2026e)

Compact inverse kernels exclude disappearing eigenmodes

The spectral paper establishes finite-rank approximation and strong compactness for bounded-energy functions. After transferring to a common space, this yields Hilbert–Schmidt convergence of centered inverse kernels $K_n\to K$. If ${\kappa }_{n,i}$ and ${\kappa }_i$ are their decreasing positive eigenvalues, the min–max principle gives

$$|{\kappa }_{n,i}-{\kappa }_i|\le \Vert K_n-K{\Vert }_{\mathrm{op}}\le \Vert K_n-K{\Vert }_{\mathrm{HS}}.$$

The continuum inverse is injective on its centered subspace, so each fixed ${\kappa }_i>0$ and reciprocation is legitimate. This yields convergence of generator eigenvalues with multiplicities and the prescribed factor $n$ in the clock.

Heat traces need more than convergence of each fixed eigenvalue. The source also proves a uniform high-index lower bound ${\nu }_{n,i}\ge c_{\ast }{i}^p$ for some $p>0$ in its doubled-energy convention. Consequently, for $t\ge \delta >0$,

$$\sum _{i>N}{e}^{-t{\nu }_{n,i}/2}\le \sum _{i>N}{e}^{-\delta c_{\ast }{i}^p/2}⟶0.$$

The finite initial sum converges uniformly on bounded time intervals, and the tail estimate completes positive-time heat-trace convergence. This is where small regions capable of supporting hidden low-energy modes must already have been controlled. (OpenAI, 2026f)

Above four: transfer both distance and degree mass from the tree

In the tree regime, a block is attached to its parent only at a pole. Excursions into attached submaps return to that same pole, so they cannot shorten distances inside the parent block. Let $g$ be the distance from a block’s pole to the corner selected by a child, and let $G(w)$ sum these increments along the ancestor chain of node $w$. Its positive finite spine mean is ${\beta }_{\mathrm{sp}}$.

The conditioned-tree estimate gives ${\max }_w|G(w)-{\beta }_{\mathrm{sp}}H(w)|=o_{\mathbb{P}}(\sqrt{N})$. If $w_{\ast }$ is the last common ancestor of $w,w^{\prime }$, the difference between their map distance and

$$G(w)+G(w^{\prime })-2G(w_{\ast })$$

is at most twice the maximum offspring number. That maximum is $o_{\mathbb{P}}(\sqrt{N})$. The excursion limit therefore controls all map distances uniformly, including vertices in the largest blocks.

The measure comparison is exact: the $2n$ nonroot tree nodes occupy the $2n$ block corners once each. A map vertex receives exactly its degree many corners, with a loop contributing two. Uniform nonroot-node measure thus pushes forward to $\mathrm{deg}(v)/(2n)$. Including the root changes total variation by at most $1/(2n+1)$. The uniform distance comparison and this measure coupling give the metric-measure CRT limit together. (OpenAI, 2026b)

See Also

• Family 223 studies FK curves on a fixed planar lattice, where the ambient geometry does not fluctuate.
• Family 225 introduces another lattice route to a Gaussian free field.

🏷️ Manuscripts and verification scope

• Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion — pinned manuscript and companion source.
• Spectral convergence for critical FK–Ising planar maps — pinned manuscript and companion source.
• A Linear Clock for Random Walk on Tree-Weighted Planar Maps — pinned manuscript and companion source.
• Canonical conformal limits of subcritical FK planar maps — pinned manuscript and companion source.
• The critical Liouville quantum sphere and geometric limits of FK maps at q=4 — pinned manuscript and companion source.
• Metric-measure limits of subcritical FK and spanning-tree planar maps — pinned manuscript and companion source.
• Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection covers the Brownian CRT limit for fixed q greater than four. It does not document the other six conformal, metric, diffusion, or spectral claims. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A Linear Clock for Random Walk on Tree-Weighted Planar Maps.

🐻  OpenAI 2026b. Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four.

🐻  OpenAI 2026c. Canonical conformal limits of subcritical FK planar maps.

🐻  OpenAI 2026d. Metric-measure limits of subcritical FK and spanning-tree planar maps.

🐻  OpenAI 2026e. Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion.

🐻  OpenAI 2026f. Spectral convergence for critical FK–Ising planar maps.

🐻  OpenAI 2026g. The critical Liouville quantum sphere and geometric limits of FK maps at q=4.