Sublinear white noise shrinks supports instantly in SPDEs
For power-law noise, finite mass suffices if gamma<=1/2; weaker noise needs a moment condition that diverges as gamma approaches 1.
math.PR
Probability
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
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Critical loop soup clusters scale in 3
The first scaling limit result for a percolation model on a three-dimensional lattice.
· “Scaling Limit of Critical Loop Soup Clusters in Three, Four, and Five Dimensions”
Particle and mean-field rates are equivalent
Unique minimizer makes uniform log-Sobolev and Polyak-Łojasiewicz inequalities imply each other.
Spectral gap of regular graphs: sharp bound independent of degree
Proof of the Aldous–Fill conjecture gives (1−o(1))·2π²/3n², with cubic graphs extremal for even orders, quartic for odd orders.
· “The maximum relaxation time of a random walk on regular graphs”
Sharp transition for stationary states at quadratic conductance growth
Proof that subquadratic growth forces Gibbsian stationary states, while at quadratic growth non-reversible and periodic behavior appears.
· “On the stationary measures of the critical Ornstein--Uhlenbeck process”
Brownian affine-approx moment formulas involve odd zeta values
First four moments are rational combos of 1 and ζ(2j+1)/π^{2j}; means ~3.538 and ~2.558.
White-noise isoperimetric ratio grows as ln^{3/4} L
Proof uses coarse-graining, geometric linearization, and a (1+1)-dimensional action; fluctuations are superconcentrated.
· “Asymptotics of a planar isoperimetric problem with a white-noise volume term”
Exact constants found for random point matching cost
For every distance exponent q on the flat torus, expected optimal cost is κ_q(log N/2πN)^{q/2}, with κ_1≈0.8523.
· “Exact Asymptotics for the 2D Euclidean Random Matching Problem”
Smartwatch case back EIT hits 91% gesture accuracy
A 12-participant study shows the watch-back contact patch alone supports EIT gesture recognition via multi-depth scanning.
· “Algorithmic threshold for high-dimensional projection pursuit I: general theory”
One projection rank decides radial Marchenko-Pastur laws
No moments or isotropy needed: projected spectra at a single rank fraction force the radial energy condition, and vice versa.
· “Radial Marchenko-Pastur laws: projection characterizations and rigidity”
Random step sizes give MALA a global spectral gap
Uniform step-size randomization recovers the (κ√d)⁻¹ gap scale that no fixed step size can reach.
Rough-path locality breaks exactly at Hurst exponent 1/4
For fractional Brownian motion, no measurable local lift exists at or below H=1/4; above 1/4, automatic integrability and classification…
Compatible path coefficients are forced to be vertical derivatives
One base functional fixes every higher coefficient as a vertical derivative — unique and symmetric.
· “Functional representation and functional calculus for controlled paths”
Stratonovich currents concentrate sub-Gaussianly at equilibrium
Finite-time bounds on current fluctuations yield confidence intervals from just one trajectory.
· “Concentration of additive functionals of Stratonovich-type”
Hilbert space operator with no eigenvalues still has an invariant measure
Construction on weighted shift space settles a 30-year question about invariant measures; companion result gives spectral constraints.
· “Can one feel the existence of a non-trivial invariant measure?”
Tensor averages obey one large-deviation law
If the interaction tensor settles in cut-star norm, rare events and free energies follow a single variational rate function.
One matrix test decides quantum semigroup irreducibility
A drift-plus-diffusion matrix condition separates smoothing from irreducibility, unlike in the classical setting.
· “Irreducibility and regularisation properties of Gaussian quantum Markov semigroups”
Unique solutions for singular damped wave and beam equations
Weak and pathwise uniqueness are proved under explicit parameter conditions, even when the drift is only Hölder continuous
New SPDE scheme doubles spatial rate to 3/2
Fully discrete method for semilinear SPDEs cuts computational cost from δ⁻³ to δ⁻⁵/³.
· “Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise”
Matrix Spencer conjecture proved: O(√n) bound
New technique uses determinantal weights to beat random coloring
Minimum variance equals the closest Laguerre zero-gap
At degree m, the optimal repeated-pole kernel deletes the closest adjacent pair of zeros of L_{m+2}.
· “Least Variability in a Polynomial-Square Class of Rational Kernels”
Passage-time fluctuations shift from Gaussian to Gumbel on cycle networks
As the connectivity range ℓ grows, the first-passage time on an ℓ-spread-out n-cycle changes from Gaussian to extreme-value at ℓ ≍ n/log n.
· “Phase Transition and Fluctuation Results for First-Passage Percolation on Spread-Out Cycle Graphs”
Small-β forest limit detects heavy-cluster phase
Wired β-skewed forests equal FMSF if and only if heavy clusters become unique in Bernoulli percolation.
AMP algorithm beats matrix discrepancy for α<5.74
Free-probabilistic state evolution yields explicit operator norm bound below 2 for all small aspect ratios.
· “Free-Probabilistic State Evolution and Random Matrix Discrepancy”
Unknown rank yields explicit hiring threshold in secretary problem
Only first and last positions matter; the critical q depends solely on employer's rejection rate α.
· “Strategic Interview Positioning under Uncertain Self-Rank”
Small-ball alone fails to guarantee Gaussian-width sample complexity
Explicit counterexample for every n shows that a uniform small-ball condition cannot replace distribution-adapted complexity.
· “Small-Ball Marginals Do Not Control Restricted Eigenvalues by Euclidean Gaussian Width”
Greedy ordering maximizes all circular consecutive-product sums
New proof resolves 2023 conjecture and minimizes speed of cyclic birth-death chains
· “Circular Rearrangement Inequality and Optimal Cyclic Birth and Death Chains”
Boundary geometry sets the Fourier dimension of multiplicative chaos
On tori, GMC Fourier dimension equals the correlation bound; flat and curved boundaries create distinct lower caps.
· “Fourier decay of Gaussian multiplicative chaos and boundary geometry”
This paper develops a cluster representation for one-dimensional CBI processes with jumps…
Total variation convergence rates are obtained for subcritical and critical CBI processes with jumps using a cluster decomposition and…
· “Coupling for one-dimensional subcritical and critical CBI processes with jumps”
Rewarding largest prime factor yields a giant prime and two transitions
Any positive exponent in the reward forces one prime to dominate; the cofactor undergoes a second transition at exponent one.
· “Tilting Billingsley's model toward a giant prime: two phase transitions”
This paper analyzes distributed gradient descent in reproducing kernel Hilbert spaces…
Refined analysis lets more local machines share the load while preserving the minimax optimal learning rate.
· “Generalization Analysis of Distributed Kernel-based Robust Gradient Descent Algorithms”
Explicit scaling law for longest line of sight in random obstacles
Three distinct Poisson obstacle models share a logarithmic scaling for the farthest visible points, with explicit constants from geometry.
· “On the maximum visibility in a ball through the vacant set of Poissonian obstacles”
Optimal moment condition for random determinant CLT found
Uniform integrability of fourth moments weighted by logarithm is exactly the threshold; Berry–Esseen rates proven optimal.
· “The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants”
Cubic trace cancellation hides detectable four-cycle signal in random geometric graphs
A counterexample disproves a spectral conjecture by making signed triangles vanish while signed four-cycles reveal latent geometry.
Lévy SDEs with fading drift converge at explicit polynomial rates
Heavy-tailed noise (α<2) leads to stable limits; above α=2 a diffusive phase transition occurs
Fractional noise memory splits most-probable-path persistence into two regimes
For H ≤ 1/2, deterministic periodic orbits remain likely over many cycles; for H > 1/2, long-range memory destroys their stability.
Eigenvector maxima of Wigner matrices follow Gumbel law
The largest coordinate of any eigenvector matches the GOE/GUE prediction, with sharp universal tail probabilities.
· “Precise Delocalisation and Gumbel Laws for Eigenvectors of Wigner Matrices”
Skew stickiness ratio tends to 2 at short expiry
First rigorous proof under mild conditions settles Bergomi's conjecture using Watanabe expansion.
· “Short-maturity skew stickiness ratio under local volatility”
Strict negative edge correlation in random-cluster model below one
Proved for high-degree expanders, hypercubes, and random regular graphs, including the Q=0 percolation endpoint.
· “Negative correlation for the random-cluster model below one on high-degree regular graphs”
Radial process of p-Brownian motion gets explicit Tanaka-Meyer formula
Local time behavior differs from classical Brownian motion, and finite propagation speed yields sharp exit estimates.
Martingale CLT in p-Wasserstein distance with near-optimal rates
New Gaussian convolution inequality and Lindeberg method yield O(log n/√n) under ellipticity.
· “Martingale central limit theorems in p-Wasserstein distance”
Fourth-order fusion expansion for Sine_β process becomes explicit
Result holds for m≥2, mβ>3 and gives coefficient in terms of power sums of the collision profile.
· “Fourth-Order Fusion Asymptotics for Sine_β Correlation Functions”
Three uniforms exactly sample gamma
New elementary, loop-free algorithms resolve Devroye's open problem and prove all three families are 3-simple.
· “Extended One-Liners for the Gamma, Poisson, and Binomial Distributions”
This paper develops a new method to study how the support of solutions to stochastic…
Finite speed of propagation and near-dichotomy for waiting times are proved for stochastic porous media equations with conservative noise…
· “Parabolic-hyperbolic splitting in support propagation for stochastic porous media equations”
Branching process derived from Brownian tree by erasure
New representation explains the cell mass distribution and dynamics of the Derrida-Retaux model simply.
· “The continuous Derrida-Retaux branching process in the Brownian CRT”
Asymmetric density breaks entropy concavity conjecture
Third-moment product controls endpoint curvature; general log-concave case ruled out without symmetry.
· “Entropy concavity for log-concave random variables: an asymmetric counterexample”
Additive martingale limits converge almost surely with 1‑stable fluctuations
As the parameter approaches criticality, the rescaled limit converges almost surely to twice the derivative martingale, and its…
· “Fluctuations of additive martingale limits of branching Brownian motion”
DSS inequality now covers φ⁴ and sinh-Gordon magnets
Proof uses a parabolic maximum principle and rank-one interpolation to extend the Ising bound to continuous spin systems
Residual pair correlation is capped at n/2 for MTP2 laws
Multivariate total positivity forces conditioned covariances to obey a MaxCut bound and settles the rounding conjecture.
Signed diagonal couplings solve the farthest-from-independence problem
For any number of Gaussian marginals, the law farthest from independence is a single variable repeated with sign flips.
Spectral gap of exchange dynamics is always a 1- or 2-particle mode
On any weighted hypergraph, one or two particles capture the slowest mode; the boundary between regimes is now exact.
· “Aldous' spectral gap phenomena in stochastic exchange models”
This paper proves that a stochastic nonlinear Schrödinger equation with fluctuating…
Local well-posedness in H^1 is established for stochastic nonlinear Schrödinger equations with energy-subcritical deterministic and…
· “The Schr\"odinger equation with fluctuating nonlinearity in the energy space”
Hermite polynomials are the only extremizers of the finite free Stam inequality
Proof uses a matrix defect identity and turns classification into rigidity of root trajectories under convolution.
Periodic exclusion process has an explicit n-step limit shape
For every degree of asymmetry, the density converges to an explicit n-step fan, and a second-class particle picks one of those speeds.
Gaussian cutoff for exclusion on high-dimensional torus
Rigorous proof that symmetric exclusion on D≥2 torus mixes with a Gaussian profile on the spectral-gap time scale.
Full-range moderate deviations for nonlinear Hawkes
Martingale corrector and process-level LDP yield path-space rate identical to Brownian motion.
ANOVA tames conditioning in high-dimensional product Markov chains
New proof shows quasi-stationary measures exist near the product stationary measure with dimension-free error bounds
· “Stability of quasi-stationary measures in high-dimensional products of mixing Markov chains”
Local time splits into fixed space and one random clock
Feller-diffusion local occupation statistics at every position lock onto a single Mittag-Leffler amplitude.
· “Infinite ergodic theory and functional statistics of non-confined Feller process”
Infinite clusters appear at arrow probability 0.99
Origin's infinite-cluster chance passes 11/20 at epsilon=0.99, putting the critical threshold strictly below 1.
Central limit theorem for Dunkl and Bessel processes with drift
Scaled processes converge to standard normal for regular drift, generalizing Dyson Brownian motion asymptotics.
· “A central limit theorem for Bessel and Dunkl processes with drift”
Random hyperbolic polyhedra are exact Laguerre tessellations
Projections of Poisson convex hulls in horoballs yield stationary Laguerre cells that converge to Poisson-Delaunay under critical scaling
Single-ensemble MLMC for McKean-Vlasov: O(ε^{-2-δ}) cost
Geometric coupling decay yields rigorous L^p error bound under global Lipschitz assumptions.
· “Convergence of a single-ensemble multilevel scheme for McKean-Vlasov SDEs”
Tree contact process: new phase between survival and percolation
Proof that for d≥3 the percolation threshold λp exceeds the survival threshold λ1, with full ordering for d≥7.
Dissipation strength sets fluctuation regime in Lévy slow-fast systems
When dissipation is weak, rescaled deviation is a stable process; when strong, it becomes Brownian motion.
Local condition determines global survival for infection processes
A simple inequality on neighbour sets decides which infection model spreads farther, across discrete and continuous time.
· “Comparison inequalities for discrete- and continuous-time infection processes”
This note gives a proof of the Portmanteau theorem
A short exposition attempts a cyclic proof of the seven equivalent forms of the Portmanteau theorem, but the proof is incomplete in two…
Spin-glass pressure limits hinge on 1+1/d moments
Below that threshold, redrawing disorder every volume size makes spikes recur; above it, every coupling converges.
Partial transpose splits into freely independent components for any finite group
A group-theoretic decomposition reveals that the number and distributions of components depend on the group structure, not just its order.
· “Freeness for the G-circulant Decomposition of the Partial Transpose of Random Matrices”
Haar-random unitaries sit at Nielsen distance πD/√3
A random unitary's normalized Nielsen distance is sharply π/√3, with exponential bounds at speed D².
· “Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry”
Fluctuations of polynomial SPDEs converge to a linear equation
Renormalized error field solves an equation driven by singular noise terms.
· “Second-order fields for stochastic partial differential equations”
Optimal $O(n^{-1/2})$ Gaussian rate proven for Markov martingale sums
First $W_p$ bounds, every order $p\ge 2$, for Markov-chain martingale sums, via a two-stage coupling.
· “Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains”
One concentration bound now yields the full anisotropic local law
For sample covariance matrices, no higher-cumulant condition is needed at eigenvalue spacing scale.
· “The anisotropic local law for sample covariance matrices under quadratic-form concentration”
This paper proves sharp new bounds on the size of sample-to-sample fluctuations of the…
For the zero-field Sherrington-Kirkpatrick model at every fixed inverse temperature above one and at zero temperature, the variance of the…
· “Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model”
Noise synchronisation transfers from frozen to law-dependent SDEs
The ensemble Kalman sampler inherits diagonal two-point mixing, the key input for faster coupled sampling.
Generalised KPZ equation solved globally in full subcritical regime
New technique handles initial layer with merely bounded data, ruling out blow-up for any symmetric noise model.
· “Pathwise Global-in-Time Existence for the generalised KPZ Equation in the Full Subcritical Regime”
2n²+1 model runs deliver all four response moments
A vine-copula sigma-point scheme matches million-sample Monte Carlo on skewness and kurtosis.
· “Quadratic Point Estimate Method for Uncertainty Quantification with Dependent Non-Gaussian Inputs”
Zero-count odds derived from fixed k to vanishing density
New formulas give the exponential rate, order-one correction, and Gaussian prefactor for low-density counts in Hardy–Szegő windows.
· “Endpoint and Vanishing-Density Asymptotics for Hardy--SzegH{o} Zero Counts”
Even ferromagnets obey the Ding-Song-Sun inequality
Truncated correlations now peak at zero field for a whole class of single-spin measures, opening the route to log-Sobolev bounds.
· “The Ding-Song-Sun inequality for a class of even ferromagnets”
Bad_2(A) rejects p-spread measures when A is large
When μ_p(A) > (1-p)/(2-p), Bad_2(A) carries no p-spread measure
Recovers the exact causal-dependence graph from a distribution
At zero error, every optimal network provably keeps exactly the links where conditional dependence lives.
· “Tensor Network Moral Graph Recovery of Discrete Probability Distributions”