Benjamini–Schramm Conjecture, Mézard–Parisi Formula, Voronoi Percolation and 7 More

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. (Lean)

Proves p_c < 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. (Lean)

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations C_βc(r) ∼ Ar^-1/4(log r)^1/8 and the Berezinskii–Kosterlitz–Thouless essential singularity √(β_c-β) log ξ(β) → B, with A,B > 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.

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_β n^1/3, with c_β > 0, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

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. (Lean)

On ℤd, 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. (Lean)

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. (Lean)

Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for 0 < q ≤ 4, with κ=4π/arccos(-√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 ≤ q ≤ 4, complete nested plane loops converge to CLEκ.

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.

Proves the exact reconstruction threshold dλ² > 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. (Lean)