Probability
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper Directional zero–one laws beyond iid environments and iid ballisticity
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…
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Paper 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…
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Paper Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling
Proves chordal SLEκ 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…
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Paper 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…
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Paper 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…
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Paper 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 \Thetak(n) for every fixed k ≥ 3, and computability of…
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Paper 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)…
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Paper Critical and near-critical XY scaling and BKT universality
For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations C{\betac}(r)\sim Ar^{-1/4}(\log r)^{1/8} and the Berezinskii–Kosterlitz–Thouless essential singularity…
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Paper 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 An=0)=(1/2+o(1))^n; for fixed bias…
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Paper 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.…
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Paper 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…
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Paper 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…
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Paper Exact Hausdorff gauges for SLE
Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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 pc\lt 1: at the critical probability, there is almost surely no…
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Paper The Benjamini–Schramm nonuniqueness conjecture
Proves pc\lt pu 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper The double-dimer loop ensemble converges to \mathrm{CLE}_{4}
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.…
29 posts