Probability and statistical mechanics

Using the conformal and metric-measure companion results and the stated Brownian/Liouville quantum gravity inputs, we prove spectral convergence for critical spherical FK–Ising maps to Liouville Brownian motion on the ordinary unit-area \(\sqrt3\)-quantum sphere. The discrete walk has total attempt rate one, uses every map edge including loops and multiplicities, and has the corner measure as its stationary law. Accelerating time by the number of map edges gives joint convergence of the metric-measure space, all ordered eigenvalues with multiplicities and padding, and the heat trace locally uniformly at strictly positive times. The conductivity and clock constant are one in these conventions.

We prove that stationary random walks on critical spherical FK–Ising planar maps converge to Liouville Brownian motion on the ordinary unit-area \(\sqrt3\)-quantum sphere, using the geometric and electrical results of the spectral companion. The walk chooses uniformly among all incident half-edges, retaining loops and multiple edges. With the corner measure as the stationary law, the deterministic time acceleration is exactly the number of map edges. For any deterministic metric scale giving the metric-measure limit, convergence retains that same surface and any fixed finite number of conditionally independent walks with their time parameters.

We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area \(\sqrt2\)-Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges. The result uses the companion contour and metric limits for this same ensemble.

We construct the field and area law of the unit-area critical Liouville quantum sphere as a limit of ordinary subcritical quantum spheres, and equip it with its critical intrinsic metric. We then prove that spherical Fortuin–Kasteleyn planar maps at q = 4, embedded by their equilateral flag uniformizations, converge jointly to this sphere decorated by an independent nested conformal loop ensemble CLE₄. With deterministic distance normalization, the convergence includes the area measure, the full embedded distance function, and every macroscopic interface through all positive integer edge counts.

We resolve the finite spherical cases of Gwynne and Miller's graph-metric conjecture for critical Fortuin–Kasteleyn maps at each fixed \(q\in(0,4)\) and for uniform spanning-tree-decorated maps. After deterministic rescaling of graph distances, these maps, equipped with the vertex probability measure proportional to degree, converge in Gromov–Hausdorff–Prokhorov law to their ordinary unit-area Liouville quantum gravity spheres. Distances use every primal edge, and convergence holds through all positive integer edge counts.

For every fixed \(0\lt q\lt 4\), we prove joint convergence of spherical Fortuin–Kasteleyn planar maps in their flag-triangle uniformization to the corresponding unit-area Liouville quantum gravity sphere decorated by an independent conformal loop ensemble. The convergence includes the area measure, deterministically rescaled graph distances between all vertex pairs, and the full nested interface collection, with interfaces converging uniformly up to reparameterization.

We prove the finite-volume continuum-random-tree prediction for critical Fortuin–Kasteleyn planar maps at every fixed q > 4. After rescaling graph distances by a constant times n^−1/2, an n-edge map with normalized degree measure converges to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.

We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ² with independent exponential edge weights is strictly convex and has a C¹ boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.

We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.

We prove that critical Bernoulli bond percolation has no infinite cluster on any infinite connected locally finite quasi-transitive graph with critical probability less than one. This resolves the bond form of the criticality conjecture of Benjamini and Schramm.

We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ³ have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.

We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing \(p_c\lt p_{2\to2}\le p_u\).

For the nearest-neighbor \(O(4)\) model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic \(\displaystyle m_{\mathrm{lat}}(\beta)\sim 32\exp(\pi/4-1/2)\sqrt\beta\,\exp(-\pi\beta) \qquad (\beta\to\infty).\) Here the mass is the gap of the full Osterwalder–Schrader transfer operator, including rotation-invariant local-observable sectors, in units of one original lattice time step.

We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor \(O(3)\) model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports. Their reconstructed theory has a unique vacuum, a nonzero vacuum complement, and a positive Hamiltonian gap on that entire complement.

We consider the continuum spin field constructed from the nearest-neighbor two-dimensional \(O(3)\) model by the fixed prescription of the companion paper. We prove that its vector two-point spectral measure has a positive atom corresponding to the lowest mass, separated by a positive gap from all remaining mass support.

We prove sharp mass bounds for the two-dimensional nearest-neighbor \(O(4)\) model. For all sufficiently large inverse couplings β, the full transfer gap, including rotation-invariant local-observable sectors, is bounded above and below by positive multiples of \(\sqrt\beta e^{-\pi\beta}\). We also prove that the model has a unique periodic local limit and a positive full gap at every finite β > 0, resolving the all-temperature lattice mass-generation conjecture for this periodic state. Under the stated cutoff scaling, the mass bounds are uniform in independently fixed physical units. A continuum spectral interpretation requires separate convergence hypotheses for the transfer semigroup and the block observables.

We prove exponential decay of two-point correlations for the classical nearest-neighbor \(O(n)\) model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.

We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let \(C_{b_c}(r)\) be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then \(\displaystyle C_{b_c}(r)=B_{\mathrm{XY}}r^{-1/4}(\log r)^{1/8}(1+o(1)), \qquad B_{\mathrm{XY}}\in(0,\infty),\) as \(r\to\infty\).

Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness \(2/\pi\). The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in \(H^{-3}_{\mathrm{loc}}((-1,1)^2)\); all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in \((-1,1)^2\) converge as well.

We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let \(m(b)\) be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches b_c from the massive side, \(\displaystyle \sqrt{b_c-b}\log\frac{1}{m(b)} \longrightarrow A_{\mathrm{XY}},\) where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As \(n\to\infty\), the magnetization is \(A_{\mathrm{XY}}n^{-1/8}(\log n)^{1/16}(1+o(1))\), where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

For the ordinary nearest-neighbor XY model on the square lattice, we prove the Berezinskii–Kosterlitz–Thouless prediction that the critical spin-correlation exponent is 1/4. More precisely, at the mass-defined critical inverse temperature, the infinite-volume correlation is \(n^{-1/4+o(1)}\). The infinite-volume limit through free square boxes is taken before the separation limit.

We prove Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the physical roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbors. After the natural lattice normalization, the critical effective temperature has the universal value \(8\pi\). For the ordinary nearest-neighbor Villain and XY models at sufficiently low fixed temperatures, we also prove that their Green-function-normalized spin fields converge to the imaginary exponential of a Dirichlet Gaussian free field.

For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n^1/3 converges to a finite positive constant.

For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is \(n^{1/6+o(1)}\). The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.

For the planar Ising model with bonds \(J_e=1+\varepsilon\xi_e\), where the ξ_e have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE₃ convergence for sufficiently small ε > 0. The temperature is the spontaneous-magnetization threshold. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

We prove quenched chordal SLE₃ convergence for the planar Ising model with sufficiently weak independent symmetric two-valued ferromagnetic bonds. The temperature is the critical point defined by spontaneous magnetization. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds \(1\pm\varepsilon\) grows as \((\log r)^{1/4+o(1)}\) for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.

We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE₃. The result allows interactions of either sign, arbitrary admissible exterior contours, and uniformly approximated Jordan domains. Convergence holds for the full oriented curve law in uniform distance modulo increasing reparametrization.

We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign. One critical-temperature branch and two field normalizations apply in every bounded simply connected C² Jordan domain with free, plus, or minus boundary conditions, and in the periodic thermodynamic plane state. Energies are centered in their actual states.

We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted. For an interface exploration across a band of fixed positive width, finitely many regular signed barriers approximate every good conditional target law in total variation, uniformly over bounded observables and sufficiently fine meshes.

For every fixed degree d ≥ 3, we prove fixed-energy GOE universality for the adjacency matrix of a uniformly random simple labelled d-regular graph with independent uniform diagonal disorder. The disorder strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process converges to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator. The permitted disorder strength depends on the degree and the distance from the clean spectral edges.

We prove the fixed-degree bulk-universality conjecture for random regular graphs. For every fixed integer d ≥ 3 and every fixed energy in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform simple labelled d-regular graph converges to the GOE bulk process. The convergence holds along all admissible graph sizes, without additional conditioning.

We prove a directional zero–one law for uniformly elliptic nearest-neighbor random walks in stationary, ergodic, finite-range-dependent environments on ℤ^d, d ≥ 3. For every fixed nonzero real direction, the probability of escape in that direction, averaged over the environment, is either zero or one. Finite-range dependence is imposed on the full transition rows: collections of rows at distance greater than a fixed range are independent, including collections indexed by infinite deterministic sets.

We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on ℤ^d, d ≥ 2. This resolves the ballisticity conjecture positively.

We prove the directional zero–one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on ℤ^d, d ≥ 3: the probability of escape in each fixed nonzero real direction is zero or one. Only strict positivity of the transition probabilities is required; no uniform lower bound or moment assumption is imposed.

We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.

We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.

We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.

We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.

We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances \(1-\varepsilon\) and \(1+\varepsilon\), for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy \(\displaystyle G_J(u)=u^{1-\gamma+o(1)},\qquad F_J(s)=s^{1+\theta+o(1)},\) as \(u\downarrow0\) and \(s\downarrow0\), with \(\gamma=(2+\theta)^{-1}\), \(0.4126930\lt \gamma\lt 0.4126934\), and \(0.4231063\lt \theta\lt 0.4231088\). A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.

We prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation. For positive mass it is the unique massive SLE\(_{16/3}\) law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass follows by duality and reversal. The theorem also allows Carathéodory approximations whose diameters diverge.

We prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE_16/3. The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.

We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight \(1\le q\lt 4\). A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.

For every fixed \(1\le q\le4\), critical square-lattice random-cluster Dobrushin interfaces converge to chordal \(\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}\), where \(\kappa(q)=4\pi/\arccos(-\sqrt q/2)\), and the complete nested plane loop collections converge to whole-plane \(\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}\). The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

For every fixed \(1\le q\lt 4\), critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal \(\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}\), where \(\kappa(q)=4\pi/\arccos(-\sqrt q/2)\), confirming the Rohde–Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane \(\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}\) in a spherical matching topology retaining multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

For every fixed \(0\lt q\lt 1\), we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLE_κ, where \(\kappa=4\pi/\arccos(-\sqrt q/2)\in(6,8)\). The approximating domains are simple closed nearest-neighbor lattice polygons with distinct marked vertices, whose marked boundary parametrizations converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence in the uniform metric on oriented curves modulo increasing reparametrization. The proof combines finite connection comparisons below one, localization in irregular tiled disks, and a boundary observable that determines the limiting Loewner driver.

Taking Cardy's crossing formula for critical Poisson–Voronoi percolation as an input, we prove a universal joint limit for the near-critical crossing thresholds of rational polygonal quadrilaterals under the monotone coupling. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice. Each model is normalized by its own expected number of color-pivotal sites for a unit-square crossing.

Taking Cardy's conformal crossing formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral as an input, we prove that the expected number of color-pivotal cells for a unit-square crossing is asymptotic to a positive constant times \(\varepsilon ^{-3/4}\), where the point intensity is \(\varepsilon ^{-2}\). No rate of convergence in Cardy's formula is required.

We prove Cardy's formula for annealed crossing probabilities in critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral. This proves the annealed crossing-probability form of the conformal-invariance conjecture for this model.

We prove that the height function of the square-lattice six-vertex model with weights \(a=b=1\) and \(0\lt c\le2\), in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel \(-(2\pi)^{-1}\log|x-y|\), the squared multiplier is \(1/\arcsin(c/2)\).

We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE₄. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE₄ scaling-limit conjecture.

We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings. Each site has a rate-one clock, mean spin autocorrelations are multiplied by n^1/3, and waiting times and lags are measured in units n^2/3. The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags. The quench limit is selected by these initial states and relaxes to the stationary limiting autocorrelation as the waiting time tends to infinity.

We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings. With rate-one heat-bath clocks at every site, time is scaled by n^2/3 and the mean spin autocorrelation is multiplied by n^1/3. The functions converge in law uniformly on compact positive-time intervals. Their limiting law is not a point mass: it retains sample-to-sample randomness, and its functions decay to zero at large times.

At the critical inverse temperature β = 1, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model. The mixing time is \(n^{2/3+o(1)}\) when every site has rate one, or \(n^{5/3+o(1)}\) attempted uniform-site updates, in probability over the disorder.

At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time \(\exp(n^{1/10000})\) exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.

We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature \(0\leq\beta\lt 1\). With rate-one refresh at each spin, the cutoff location is \(\log n/(2\lambda(\beta))\) for a positive deterministic rate \(\lambda(\beta)\). The location for uniformly chosen single-site update attempts is n times as large. Convergence is in probability over the disorder.

At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence \(t_n=o(n^{2/3})\) in rate-one-per-site time, the Gibbs mass of initial states whose time-t_n total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder. The same statement holds for every deterministic integer sequence \(k_n=o(n^{5/3})\) of uniform-site update attempts.

For every fixed inverse temperature β > 1, we prove subexponential mixing for rate-one-per-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model from a typical equilibrium configuration. If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time \(\exp(n^{1-1/40000000})\), with joint probability tending to one over the disorder and the sampled state.

For every fixed inverse temperature \(0\lt \beta\lt 1\), we prove that the unscaled spectral gap of single-site heat-bath dynamics for the zero-field Gaussian Sherrington–Kirkpatrick model is bounded away from zero with probability tending to one over the disorder. Equivalently, the Gibbs law satisfies a dimension-free Poincaré inequality for all functions.

We construct a bounded, continuous, stable radial pair potential in three dimensions with \(|\phi(r)|\le Cr^{-3-1/32}\) for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.

We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying \(\phi(r)=o(r^{-3})\). Its free-cube canonical free energy is finite at every positive inverse temperature and density.

We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when \(d\lambda^2\gt 1\); we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival. The proof uses reproducible exact-arithmetic verification of polynomial inequalities.

For the ferromagnetic four-state broadcast model with \(0\lt \lambda\lt 1\), we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L³ capacity with edge resistances \(\lambda^{-2|e|}\) is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.

We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when \(d\lambda^2\gt 1\), with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction. Combining the Poisson theorem with known tree-to-graph and algorithmic results gives the exact weak-recovery threshold for the symmetric three-community sparse stochastic block model with independent uniform labels, fixed within- and between-community rates \(a,b\gt 0\), and mean degree \((a+2b)/3\gt 1\): recovery is possible exactly when \((a-b)^2\gt 3(a+2b)\). Above this threshold it is achievable in \(O(n\log n)\) time; at or below it, weak recovery is information-theoretically impossible.

For each fixed \(0\lt \kappa\lt 8\), let \(d=1+\kappa/8\). The gauge \(h(r)=r^d(\log\log(1/r))^{(2-d)/2}\) at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLE_κ. This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.

For each \(0\lt \kappa\lt 8\), we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.

We prove that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.

We prove the Gaussian free field and SLE\(_4\) scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE\(_4\). We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.

We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm's Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.

We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed \(x\in[1/\sqrt2,1]\). The convergence holds as a random distribution on every bounded C² Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations. This includes the uniform height model and the predicted critical endpoint.

For each fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, we prove the conjectured joint scaling limit of the height and both current-cluster collections in every bounded Jordan domain, for every admissible polygonal approximation. The height converges to a Gaussian free field with the predicted coupling constant, and the clusters converge to canonical recursive two-valued local sets of that same field. The joint limit includes all nesting depths and the distinguished wired boundary cluster.

We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.

For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance \(\Theta(n)\). The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson's transition-width theorem.

Let H_n be the index of the first unsatisfiable prefix in random k-SAT on n variables, with independent uniformly signed clauses using k distinct variables and sampled with replacement. For every fixed k ≥ 4, we prove \(\mathop{\mathrm{Var}}\nolimits (H_n)=\Theta_k(n)\). For k = 3, the variance is bounded below by a positive multiple of n and above by a constant multiple of \(n\log n\); the companion paper on random 3-SAT sharpens this to \(\Theta(n)\). The same upper bounds proved here hold after clipping at any fixed positive multiple of n.

The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [4] with priority for resolving the satisfiability conjecture, which establishes the threshold's existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density. A fair search through these certificates halts without requiring a computable rate of finite-size convergence.

For every fixed integer k ≥ 3, random k-SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [5] with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.

For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d-regular tree is a factor of IID exactly when \(\tanh\beta\le(d-1)^{-1/2}\). We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.

We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h^−1/4, first-length mass at most \(Ch^{13/12}\), and mass at least \(ch^{-1/4}\) on bridges with length at least \(ch^{4/3}\).

At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R, with both boundary ports summed and diameter at least \(R/10\), the finite critical partition is \(R^{3/4+o(1)}\) and the normalized mean length is \(R^{4/3+o(1)}\). A separate two-bond estimate on tilted cylinders with controlled site proportions bounds the length-square mass of planar polygons of diameter at most H, modulo translations, by \(H^{2/3+o(1)}\).

For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one. We also determine the near-critical exponential correlation scale and small-force free-energy exponent, and prove spatial local lower bounds that permit conditioning on a prescribed terminal vertex.

We prove critical diameter-tail exponents −2 for unrooted honeycomb polygons and −2/3 for length-weighted polygons, together with a truncated second-length-moment bound of exponent 2/3. Consequently, polygons conditioned to have length at least n have diameter \(n^{3/4+o(1)}\) in probability under either weight.

We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of h layers, with its initial port fixed and its terminal port free, has mean length \(h^{4/3+o(1)}\). We also obtain central visit probability \(R^{-2/3+o(1)}\) and mean length \(R^{4/3+o(1)}\) for macroscopic free-boundary chords in a regular hexagon.

For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4 and simultaneous local-mass and covering exponent 4/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.

At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H, counted modulo translations, is at most \(H^{2/3+o(1)}\). We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.

We prove that critical honeycomb bridges crossing a strip of width R have mean length \(R^{4/3+o(1)}\), with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R^−1/4; the half-plane arch kernel at endpoint separation m is comparable to m^−5/4.

We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is 1/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/12.

We prove a logarithmic window of critical honeycomb bridge lengths: summed through height \(2h(\log h)^{1/64}\), bridges of lengths between \(h^{4/3}(\log h)^{-1/8}\) and \(2h^{4/3}(\log h)^{1/2}\) have mass at least \(h^{3/4}(\log h)^{-C}\).

At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height N is comparable to N^−1/4. The first horizontal-displacement moment of return paths is comparable to N^3/4.

Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length \(R^{4/3+o(1)}\) in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length \(R^{4/3+o(1)}\). A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.

At the critical honeycomb fugacity, self-avoiding port-to-port chords in an equilateral lattice triangle of side R, summed over both boundary endpoints and restricted to diameter at least \(R/100\), have partition sum comparable to R^3/4 and mean length \(R^{4/3+o(1)}\). We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as \(N^{1/6+o(1)}\).

We prove that a fixed number of coordinate sweeps of the Thorp shuffle on \(2^d\) cards makes the squared L² distance of the full permutation density from uniform tend to zero as \(d\to\infty\). In particular, the total-variation mixing time is \(\Theta(d)\) physical shuffles.

We prove that the Thorp shuffle on \(N=2^d\) labeled cards has full-permutation total-variation mixing time \(\Theta(\log N)\) in physical shuffles. An absolute number of coordinate sweeps suffices for the upper bound.

For \(N=2^d\) cards, we prove that an absolute number of coordinate sweeps of the Thorp shuffle brings the full permutation law to total-variation distance tending to zero from uniform. The mixing time therefore has optimal order \(\Theta(\log N)\) in physical shuffles.

We prove that an absolute number of coordinate sweeps of the Thorp shuffle on \(n=2^d\) positions brings the full permutation to total-variation distance at most \(\frac12n^{-5}\) from uniform, uniformly over the initial deck. Thus \(O(d)\) physical shuffles suffice, which is optimal in order.

For the Thorp shuffle on \(2^d\) cards, we prove that the worst-start total-variation distance after \(32800d\) physical shuffles tends to zero as \(d\to\infty\). Combining our fixed-list estimate with the companion Fourier transfer improves this bound to \(512d\) shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.

We prove that the Thorp shuffle on \(2^d\) cards mixes in \(\Theta(d)\) complete shuffles. The full permutation law after \(1600d\) shuffles converges to uniform in total variation as \(d\to\infty\), uniformly over the initial deck, while a support count gives a lower bound of \(2d-O(1)\).

We show how information about partial permutations controls full permutation laws. Let n tend to infinity through multiples of eight. If the images of a uniformly chosen \(7n/8\) labels approach the uniform injection law in average total variation, and the sign mean tends to zero, then the product of two independent permutations with the given law converges to uniform on S_n.

Fix half the labels in a Thorp shuffle on \(2^d\) positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as \(d\to\infty\), uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.

We bound the information remaining after paths of specified cards in the Thorp shuffle on \(n=2^d\) cards have been observed. After a fixed number of deterministic coordinate sweeps, the joint endpoint law of further cards is close to uniform on the available positions, on average over the observed paths, provided a fixed positive fraction of labels lies outside both lists. Combined with a Fourier transfer, these bounds give full-deck mixing after \(2048d\) physical shuffles.

For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on \(2^d\) cards in \(O(d)\) physical steps, matching the order of the support lower bound.

We prove that the Thorp shuffle on \(n=2^d\) cards mixes in \(\Theta(d)\) physical shuffles. After a sufficiently large fixed number of coordinate sweeps, the total-variation distance of the full permutation from uniform tends to zero as \(d\to\infty\).

We prove a permanent inequality with exponent strictly below two for laws on S₄ sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on \(N=2^d\) cards to total-variation distance tending to zero from uniform as \(N\to\infty\). Thus the mixing time has the optimal order \(O(\log N)\) in physical shuffles.

Let A_n be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability \(p\in(0,1)\setminus\{1/2\}\). We prove that \(\mathbb P(\det A_n=0)=(p^2+(1-p)^2+o(1))^n\). The rate is attained by the event that two rows agree.

Let A_n be a symmetric \(n\times n\) matrix whose entries on and above the diagonal are independent uniform signs. We prove \(\displaystyle \Pr(\det A_n=0)=\left(\frac12+o(1)\right)^n.\)