# Probability and statistical mechanics

> 29 AI-made results in probability and statistical mechanics, each explained in plain English with its trust label, Lean status and sources.

Canonical: https://closedform.nishachay.tech/science/mathematics/probability-and-statistical-mechanics/
Last updated: 2026-10-14

29 results in OpenAI Math 2026.

29 AI-made results in probability and statistical mechanics, from 105 papers in the OpenAI math collection, each explained in plain English. 19 have a formal Lean proof listed; 10 are claimed in a paper and not yet machine-checked. The oldest question here dates back to 1943: [The joint critical Ashkin–Teller current limit](https://closedform.nishachay.tech/e/openai-math-2026-233/). Results rated Landmark include [Critical percolation on every quasi-transitive graph](https://closedform.nishachay.tech/e/openai-math-2026-213/) and [The Benjamini–Schramm nonuniqueness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-214/).

## Results (29)

- [The geometric phase diagram, diffusion, and spectra of random planar maps](https://closedform.nishachay.tech/e/openai-math-2026-211/) - Entry 211 · Random FK planar maps are claimed to look like random surfaces for q ≤ 4 and like a random tree above 4. (Formal proof listed)
- [Planar first-passage geometry and the absence of bigeodesics](https://closedform.nishachay.tech/e/openai-math-2026-212/) - Entry 212 · Planar random-metric grids are claimed to have no route that is fastest forever both ways, and a smooth limit shape. (Formal proof listed)
- [Critical percolation on every quasi-transitive graph](https://closedform.nishachay.tech/e/openai-math-2026-213/) - Entry 213 · At the exact tipping point, random networks of a broad, symmetric kind form no endless cluster. (Formal proof listed)
- [The Benjamini–Schramm nonuniqueness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-214/) - Entry 214 · Percolation on every fast-spreading symmetric graph has a range with infinitely many endless clusters. (Formal proof listed)
- [Canonical O(3) continuum limit and exact O(4) mass asymptotics](https://closedform.nishachay.tech/e/openai-math-2026-215/) - Entry 215 · Spin models with three or more components are proved to lose alignment exponentially at every temperature. (Formal proof listed)
- [Critical and near-critical XY scaling and BKT universality](https://closedform.nishachay.tech/e/openai-math-2026-216/) - Entry 216 · Six papers claim exact critical laws for the planar XY model, as the Berezinskii–Kosterlitz–Thouless picture predicts. (Claimed, not checked)
- [The low-temperature Sherrington–Kirkpatrick fluctuation law](https://closedform.nishachay.tech/e/openai-math-2026-217/) - Entry 217 · The typical spread of the low-temperature Sherrington–Kirkpatrick free energy grows like n^(1/6), with a limiting law. (Claimed, not checked)
- [Conformal universality for weakly interacting and random-bond Ising models](https://closedform.nishachay.tech/e/openai-math-2026-218/) - Entry 218 · Weak extra interactions or weak random bonds leave the critical Ising interface's conformal limit, SLE₃, unchanged. (Formal proof listed)
- [GOE bulk universality for regular graphs with weak Anderson disorder](https://closedform.nishachay.tech/e/openai-math-2026-219/) - Entry 219 · Eigenvalue spacings of random regular graphs, even with weak random potentials, match those of Gaussian random matrices. (Claimed, not checked)
- [Directional zero–one laws beyond iid environments and iid ballisticity](https://closedform.nishachay.tech/e/openai-math-2026-220/) - Entry 220 · A walk in a random environment escapes in a given direction with probability 0 or 1, and escape forces positive speed. (Formal proof listed)
- [The Mézard–Parisi formula for diluted spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-221/) - Entry 221 · The Mézard–Parisi cavity formula gives the exact free energy of sparse spin glasses in the Panchenko–Talagrand class. (Formal proof listed)
- [Perceptron free energies and microscopic jamming exponents](https://closedform.nishachay.tech/e/openai-math-2026-222/) - Entry 222 · Exact free energies for Ising and spherical perceptrons at every density, and the gap and force exponents at jamming. (Formal proof listed)
- [Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling](https://closedform.nishachay.tech/e/openai-math-2026-223/) - Entry 223 · Critical random-cluster interfaces on the square grid converge to SLE curves for every cluster weight up to 4. (Claimed, not checked)
- [Critical and quenched near-critical universality for Poisson–Voronoi percolation](https://closedform.nishachay.tech/e/openai-math-2026-224/) - Entry 224 · Cardy's formula holds for Poisson–Voronoi percolation, and near-critical thresholds match the triangular lattice. (Claimed, not checked)
- [Gaussian free field limits throughout the balanced six-vertex regime](https://closedform.nishachay.tech/e/openai-math-2026-225/) - Entry 225 · The balanced six-vertex height surface tends to a Gaussian free field for all c up to 2, with variance 1/arcsin(c/2). (Claimed, not checked)
- [The double-dimer loop ensemble converges to CLE 4](https://closedform.nishachay.tech/e/openai-math-2026-226/) - Entry 226 · Loops from two overlaid random domino tilings of the half-plane converge, as curves, to nested CLE₄. (Claimed, not checked)
- [Critical SK autocorrelation processes and dynamics across the temperature transition](https://closedform.nishachay.tech/e/openai-math-2026-227/) - Entry 227 · Heat-bath mixing in the SK spin glass is fast below β = 1, takes n^(2/3) at β = 1, and slows sharply above it. (Formal proof listed)
- [Continuum phase transitions for radial pair potentials](https://closedform.nishachay.tech/e/openai-math-2026-228/) - Entry 228 · Two built pair potentials give particles in space a first-order phase transition across a range of densities. (Formal proof listed)
- [Exact three- and four-state reconstruction thresholds and four-state tree capacity](https://closedform.nishachay.tech/e/openai-math-2026-229/) - Entry 229 · For three and four colors, guessing a tree's root from far leaves works exactly above the Kesten–Stigum bound. (Formal proof listed)
- [Exact Hausdorff gauges for SLE](https://closedform.nishachay.tech/e/openai-math-2026-230/) - Entry 230 · An explicit size rule gives every piece of an SLE random curve a positive, finite Hausdorff measure. (Formal proof listed)
- [The free uniform spanning forest is a factor of IID](https://closedform.nishachay.tech/e/openai-math-2026-231/) - Entry 231 · One fixed rule builds the free uniform spanning forest from independent random labels on every infinite graph. (Formal proof listed)
- [Gaussian fields and interfaces for triangular-lattice Lipschitz heights](https://closedform.nishachay.tech/e/openai-math-2026-232/) - Entry 232 · Random Lipschitz height surfaces on the triangular lattice converge to the Gaussian free field. (Claimed, not checked)
- [The joint critical Ashkin–Teller current limit](https://closedform.nishachay.tech/e/openai-math-2026-233/) - Entry 233 · Heights and current clusters of the critical Ashkin–Teller model converge together to a free field and its local sets. (Claimed, not checked)
- [All-temperature pressure of orthogonally invariant Ising spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-234/) - Entry 234 · A formula for the pressure of orthogonally invariant Ising spin glasses holds at every temperature. (Formal proof listed)
- [Limiting random SAT thresholds, sharp variance and computability](https://closedform.nishachay.tech/e/openai-math-2026-235/) - Entry 235 · Random k-SAT has a single limiting satisfiability threshold for every k ≥ 3, and the 3-SAT one is computable. (Formal proof listed)
- [The exact factor-of-IID threshold for free Ising spins on trees](https://closedform.nishachay.tech/e/openai-math-2026-236/) - Entry 236 · The free Ising state on a regular tree can be built from independent labels exactly up to tanh β = 1/√(d−1). (Formal proof listed)
- [The three-quarter exponent for honeycomb self-avoiding walk](https://closedform.nishachay.tech/e/openai-math-2026-237/) - Entry 237 · Claims random self-avoiding walks of length n on the honeycomb lattice span about n^(3/4), the long-predicted exponent. (Formal proof listed)
- [Optimal logarithmic mixing of the Thorp shuffle](https://closedform.nishachay.tech/e/openai-math-2026-238/) - Entry 238 · Claims the Thorp shuffle on 2^d cards randomizes the whole deck in order d shuffles, the best possible order. (Formal proof listed)
- [Sharp singularity rates for symmetric random sign matrices](https://closedform.nishachay.tech/e/openai-math-2026-239/) - Entry 239 · Claims a random symmetric ±1 matrix is singular with probability (1/2 + o(1))ⁿ, with a matching biased-sign rate. (Claimed, not checked)

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