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234

All-temperature pressure of orthogonally invariant Ising spin glasses

Probability and statistical mechanics 1 manuscript, 59 pages Lean

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.

221

The Mézard–Parisi formula for diluted spin glasses

Probability and statistical mechanics 1 manuscript, 36 pages Lean Reasoning summary

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.

271

Bloch's law, its lattice correction, and the spherical magnetization law

Mathematical physics 4 manuscripts, 176 pages Lean Reasoning summary

Proves Bloch's T^3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ³. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.

227

Critical SK autocorrelation processes and dynamics across the temperature transition

Probability and statistical mechanics 8 manuscripts, 720 pages Lean

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.

204

Tensor saturation for even spin groups

Algebra 1 manuscript, 46 pages

Proves saturation factor one for $\mathop{\mathrm{Spin}}\nolimits (2n)$ , n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.

216

Critical and near-critical XY scaling and BKT universality

Probability and statistical mechanics 6 manuscripts, 461 pages

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.

Closest manuscript: The Critical Spin Field of the Planar XY Model

218

Conformal universality for weakly interacting and random-bond Ising models

Probability and statistical mechanics 6 manuscripts, 329 pages Lean

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.

222

Perceptron free energies and microscopic jamming exponents

Probability and statistical mechanics 4 manuscripts, 248 pages Lean

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.

215

Canonical

O(3)

continuum limit and exact

O(4)

mass asymptotics

Probability and statistical mechanics 5 manuscripts, 379 pages Lean

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.

238

Optimal logarithmic mixing of the Thorp shuffle

Probability and statistical mechanics 12 manuscripts, 683 pages Lean

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.

268

The spin-one Haldane gap

Mathematical physics 2 manuscripts, 58 pages

Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field $h=3/5$ and gives boundary-selected infinite-volume states with topological index −1.

236

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

Probability and statistical mechanics 1 manuscript, 50 pages Lean

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.

211

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

Probability and statistical mechanics 7 manuscripts, 762 pages Lean

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.

131

Rapid mixing of graph switches for every degree sequence

Theoretical computer science 1 manuscript, 22 pages Lean

Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $O(n^8)$ time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.

217

The low-temperature Sherrington–Kirkpatrick fluctuation law

Probability and statistical mechanics 2 manuscripts, 206 pages

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.

Closest manuscript: The low-temperature Sherrington–Kirkpatrick fluctuation scale

226

The double-dimer loop ensemble converges to CLE₄

Probability and statistical mechanics 1 manuscript, 33 pages

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.

233

The joint critical Ashkin–Teller current limit

Probability and statistical mechanics 1 manuscript, 100 pages

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.

318

Chromatic splitting: filtrations and counterexamples

Topology 5 manuscripts, 372 pages

Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and $p+1$ (p ≥ 7). Nevertheless, for n ≥ 1 and $p\gt n+1$ , the overlap $L_{n-1}L_{K(n)}S_p^\wedge$ admits a $2^n$ -stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of $E(2)$ -local modules over the derived completed sphere.

Closest manuscript: Counterexamples to weak chromatic splitting: sphere kernels and descent exponents

219

GOE bulk universality for regular graphs with weak Anderson disorder

Probability and statistical mechanics 2 manuscripts, 104 pages

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.

223

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

Probability and statistical mechanics 6 manuscripts, 602 pages

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.

Closest manuscript: Square-lattice FK interfaces and nested loops for 1 <= q < 4

281

QAOA attains the SK optimum in the thermodynamic-first limit

Mathematical physics 2 manuscripts, 97 pages Lean

Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.

Closest manuscript: Full support of the zero-temperature Sherrington-Kirkpatrick order parameter

269

Uniform Laughlin gap and stability under bounded scalar disorder

Mathematical physics 2 manuscripts, 91 pages Lean

Proves the fermionic Laughlin spectral-gap conjecture for the full V₁ interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.

229

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

Probability and statistical mechanics 3 manuscripts, 93 pages Lean

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.

213

Critical percolation on every quasi-transitive graph

Probability and statistical mechanics 2 manuscripts, 62 pages Lean

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.

228

Continuum phase transitions for radial pair potentials

Probability and statistical mechanics 2 manuscripts, 64 pages Lean

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.

214

The Benjamini–Schramm nonuniqueness conjecture

Probability and statistical mechanics 1 manuscript, 53 pages Lean

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.

205

Saxl’s conjecture and universal tensor squares

Algebra 2 manuscripts, 54 pages Lean

Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every S_n with $n\notin\{2,4,9\}$ has an irreducible representation whose tensor square contains all irreducibles.

Closest manuscript: A Cyclic Polytabloid Proof of Saxl's Conjecture

265

Area laws and tensor networks for two-dimensional gapped systems

Mathematical physics 2 manuscripts, 146 pages

Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open $L\times L$ squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L⁻¹.

Closest manuscript: Polynomial PEPS approximation of gapped square-grid ground states

178

Deterministic nonbipartite Ramanujan graphs in every fixed degree

Combinatorics 1 manuscript, 85 pages

For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between $-2\sqrt{d-1}$ and $2\sqrt{d-1}$ . A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.

239

Sharp singularity rates for symmetric random sign matrices

Probability and statistical mechanics 2 manuscripts, 101 pages

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.