Subjects

Probability and statistical mechanics

29 results from OpenAI’s math release. 29 explained so far. OpenAI has published Lean proofs for 19: 10 of the main theorem, 9 of part of the claim.

See these in the full list, with search and filters.

• 211 The geometric phase diagram, diffusion, and spectra of random planar maps

  Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $0\lt q\le4$ and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.

  7 papers ▶Episode, 4:25 OpenAI published a Lean proof of part of the claim.

• 212 Planar first-passage geometry and the absence of bigeodesics

  Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C¹ boundary. Differentiability also holds for every Gamma law with positive shape and rate.

  2 papers ▶Episode, 4:53 OpenAI published a Lean proof of part of the claim.

• 213 Critical percolation on every quasi-transitive graph

  Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $p_c\lt 1$: at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ³.

  2 papers ▶Episode, 4:43 OpenAI published a Lean proof of the main theorem.

• 214 The Benjamini–Schramm nonuniqueness conjecture

  Proves $p_c\lt p_u$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.

  ▶Episode, 4:11 OpenAI published a Lean proof of the main theorem.

• 215 Canonical $O(3)$ continuum limit and exact $O(4)$ mass asymptotics

  Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $O(3)$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $O(4)$ model, determines the exact leading asymptotic of the full transfer gap, $m_{\mathrm{lat}}(\beta)\sim32e^{\pi/4-1/2}\sqrt\beta\,e^{-\pi\beta}$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $O(n)$ models with n ≥ 3 at every positive temperature.

  5 papers ▶Episode, 4:48 OpenAI published a Lean proof of part of the claim.

• 216 Critical and near-critical XY scaling and BKT universality

  For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{\beta_c}(r)\sim Ar^{-1/4}(\log r)^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $\sqrt{\beta_c-\beta}\log\xi(\beta)\to B$, with $A,B\gt 0$ after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

  6 papers ▶Episode, 4:36

• 217 The low-temperature Sherrington–Kirkpatrick fluctuation law

  For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to $c_\beta n^{1/3}$, with $c_\beta\gt 0$, confirming the predicted n^1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

  2 papers ▶Episode, 4:13

• 218 Conformal universality for weakly interacting and random-bond Ising models

  Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE₃ interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.

  6 papers ▶Episode, 4:18 OpenAI published a Lean proof of part of the claim.

• 219 GOE bulk universality for regular graphs with weak Anderson disorder

  For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.

  2 papers ▶Episode, 3:49

• 220 Directional zero–one laws beyond iid environments and iid ballisticity

  On ℤ^d, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture.

  3 papers ▶Episode, 4:15 OpenAI published a Lean proof of the main theorem.

• 221 The Mézard–Parisi formula for diluted spin glasses

  Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.

  ▶Episode, 5:08 OpenAI published a Lean proof of the main theorem.

• 222 Perceptron free energies and microscopic jamming exponents

  Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.

  4 papers ▶Episode, 5:45 OpenAI published a Lean proof of part of the claim.

• 223 Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling

  Proves chordal SLE_κ limits for critical square-lattice random-cluster interfaces for $0\lt q\le4$, with $\kappa=4\pi/\arccos(-\sqrt q/2)$: bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For $1\le q\le4$, complete nested plane loops converge to CLE_κ.

  6 papers ▶Episode, 5:15

• 224 Critical and quenched near-critical universality for Poisson–Voronoi percolation

  Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.

  3 papers ▶Episode, 4:54

• 225 Gaussian free field limits throughout the balanced six-vertex regime

  The balanced square-lattice six-vertex height field with $a=b=1$ and $0\lt c\le2$ converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel $-(2\pi)^{-1}\log|x-y|$, the exact variance multiplier is $1/\arcsin(c/2)$.

  ▶Episode, 4:26

• 226 The double-dimer loop ensemble converges to CLE₄

  Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE₄. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.

  ▶Episode, 4:22

• 227 Critical SK autocorrelation processes and dynamics across the temperature transition

  For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $\log n$ scale for fixed $0\le\beta\lt 1$, mixing time $n^{2/3+o(1)}$ at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit.

  8 papers ▶Episode, 5:07 OpenAI published a Lean proof of part of the claim.

• 228 Continuum phase transitions for radial pair potentials

  Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem.

  2 papers ▶Episode, 4:26 OpenAI published a Lean proof of the main theorem.

• 229 Exact three- and four-state reconstruction thresholds and four-state tree capacity

  Proves the exact reconstruction threshold $d\lambda^2\gt 1$, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 and d > 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.

  3 papers ▶Episode, 4:47 OpenAI published a Lean proof of part of the claim.

• 230 Exact Hausdorff gauges for SLE

  Resolves Schramm’s Hausdorff-measure question for chordal SLE_κ, $0\lt \kappa\lt 8$. The explicit gauge $r^d(\log\log(1/r))^{(2-d)/2}$, $d=1+\kappa/8$, gives almost surely positive finite measure to every trace segment $\gamma([s,t])$ with $0\lt s\lt t\lt \infty$, and finite expected measure to the trace in every bounded disk.

  2 papers ▶Episode, 4:35 OpenAI published a Lean proof of part of the claim.

• 231 The free uniform spanning forest is a factor of IID

  On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID.

  ▶Episode, 4:33 OpenAI published a Lean proof of the main theorem.

• 232 Gaussian fields and interfaces for triangular-lattice Lipschitz heights

  Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments $0,\pm2$ and two-arc boundary values $\pm1$, and zero-boundary integer Lipschitz heights weighted by fixed $x\in[1/\sqrt2,1]$. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE₄, establishing Schramm’s real-field/interface predictions.

  3 papers ▶Episode, 4:40

• 233 The joint critical Ashkin–Teller current limit

  Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.

  ▶Episode, 4:32

• 234 All-temperature pressure of orthogonally invariant Ising spin glasses

  Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero.

  ▶Episode, 4:34 OpenAI published a Lean proof of the main theorem.

• 235 Limiting random SAT thresholds, sharp variance and computability

  For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance $\Theta_k(n)$ for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent ECCC TR26-229, made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.

  4 papers ▶Episode, 4:33 OpenAI published a Lean proof of the main theorem.

• 236 The exact factor-of-IID threshold for free Ising spins on trees

  Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when $\tanh\beta\le(d-1)^{-1/2}$, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.

  ▶Episode, 4:45 OpenAI published a Lean proof of the main theorem.

• 237 The three-quarter exponent for honeycomb self-avoiding walk

  Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter $n^{3/4+o(1)}$. Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.

  13 papers ▶Episode, 4:34 OpenAI published a Lean proof of part of the claim.

• 238 Optimal logarithmic mixing of the Thorp shuffle

  Proves that the Thorp shuffle randomizes $N=2^d$ labeled cards in $\Theta(\log N)$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.

  12 papers ▶Episode, 4:33 OpenAI published a Lean proof of the main theorem.

• 239 Sharp singularity rates for symmetric random sign matrices

  Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give $\Pr(\det A_n=0)=(1/2+o(1))^n$; for fixed bias $p\in(0,1)\setminus\{1/2\}$, the rate is $(p^2+(1-p)^2+o(1))^n$. In the biased case, agreeing rows attain this rate.

  2 papers ▶Episode, 4:14

Summary and paper titles from OpenAI’s catalog, used under the Apache License 2.0. Last checked October 19, 2026.